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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">History of maths</div> <p>This is a <b>timeline of category theory and related mathematics</b>. Its scope ("related mathematics") is taken as: </p> <ul><li><a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Categories</a> of <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebraic</a> structures including <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a> and <a href="/wiki/Universal_algebra" title="Universal algebra">universal algebra</a>;</li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological algebra</a>;</li> <li><a href="/wiki/Homotopical_algebra" title="Homotopical algebra">Homotopical algebra</a>;</li> <li><a href="/wiki/Topology" title="Topology">Topology</a> using categories, including <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>, <a href="/wiki/Categorical_topology" class="mw-redirect" title="Categorical topology">categorical topology</a>, <a href="/wiki/Quantum_topology" title="Quantum topology">quantum topology</a>, <a href="/wiki/Low-dimensional_topology" title="Low-dimensional topology">low-dimensional topology</a>;</li> <li><a href="/wiki/Categorical_logic" title="Categorical logic">Categorical logic</a> and <a href="/wiki/Categorical_set_theory" title="Categorical set theory">set theory</a> in the categorical context such as <a href="/w/index.php?title=Algebraic_set_theory&action=edit&redlink=1" class="new" title="Algebraic set theory (page does not exist)">algebraic set theory</a>;</li> <li><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations of mathematics</a> building on categories, for instance <a href="/wiki/Topos_theory" class="mw-redirect" title="Topos theory">topos theory</a>;</li> <li><a href="/w/index.php?title=Abstract_geometry&action=edit&redlink=1" class="new" title="Abstract geometry (page does not exist)">Abstract geometry</a>, including <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, <a href="/w/index.php?title=Categorical_noncommutative_geometry&action=edit&redlink=1" class="new" title="Categorical noncommutative geometry (page does not exist)">categorical noncommutative geometry</a>, etc.</li> <li>Quantization related to category theory, in particular <a href="/wiki/Categorical_quantization" class="mw-redirect" title="Categorical quantization">categorical quantization</a>;</li> <li><a href="/w/index.php?title=Categorical_physics&action=edit&redlink=1" class="new" title="Categorical physics (page does not exist)">Categorical physics</a> relevant for mathematics.</li></ul> <p>In this article, and in category theory in general, ∞ = <i>ω</i>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Timeline_to_1945:_before_the_definitions">Timeline to 1945: before the definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Timeline_of_category_theory_and_related_mathematics&action=edit&section=1" title="Edit section: Timeline to 1945: before the definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable sortable" width="100%"> <tbody><tr> <th>Year </th> <th style="width:22%">Contributors </th> <th>Event </th></tr> <tr> <td>1890</td> <td><a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a></td> <td><a href="/wiki/Resolution_(algebra)" title="Resolution (algebra)">Resolution</a> of <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a> and <a href="/wiki/Resolution_(algebra)" title="Resolution (algebra)">free resolution</a> of modules. </td></tr> <tr> <td>1890</td> <td><a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a></td> <td><a href="/wiki/Hilbert%27s_syzygy_theorem" title="Hilbert's syzygy theorem">Hilbert's syzygy theorem</a> is a prototype for a concept of dimension in <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a>. </td></tr> <tr> <td>1893</td> <td><a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a></td> <td>A fundamental theorem in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, the <a href="/wiki/Hilbert_Nullstellensatz" class="mw-redirect" title="Hilbert Nullstellensatz">Hilbert Nullstellensatz</a>. It was later reformulated to: the category of <a href="/wiki/Affine_variety" title="Affine variety">affine varieties</a> over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> <i>k</i> is <a href="/wiki/Equivalence_of_categories" title="Equivalence of categories">equivalent</a> to the <a href="/wiki/Opposite_category" title="Opposite category">dual</a> of the category of reduced <a href="/wiki/Finitely_generated_algebra" title="Finitely generated algebra">finitely generated</a> <a href="/wiki/Commutative_algebra_(structure)" class="mw-redirect" title="Commutative algebra (structure)">(commutative) <i>k</i>-algebras</a>. </td></tr> <tr> <td>1894</td> <td><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a></td> <td><a href="/wiki/Fundamental_group" title="Fundamental group">Fundamental group</a> of a <a href="/wiki/Topological_space" title="Topological space">topological space</a>. </td></tr> <tr> <td>1895</td> <td><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a></td> <td><a href="/wiki/Simplicial_homology" title="Simplicial homology">Simplicial homology</a>. </td></tr> <tr> <td>1895</td> <td><a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Henri Poincaré</a></td> <td>Fundamental work <i><a href="/wiki/Analysis_situs_(paper)" class="mw-redirect" title="Analysis situs (paper)">Analysis situs</a></i>, the beginning of <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>. </td></tr> <tr> <td>c.1910</td> <td><a href="/wiki/L._E._J._Brouwer" title="L. E. J. Brouwer">L. E. J. Brouwer</a></td> <td>Brouwer develops <a href="/wiki/Intuitionism" title="Intuitionism">intuitionism</a> as a contribution to <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundational</a> debate in the period roughly 1910 to 1930 on mathematics, with <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a> a by-product of an increasingly sterile discussion on formalism. </td></tr> <tr> <td>1923</td> <td><a href="/wiki/Hermann_K%C3%BCnneth" title="Hermann Künneth">Hermann Künneth</a></td> <td><a href="/wiki/K%C3%BCnneth_formula" class="mw-redirect" title="Künneth formula">Künneth formula</a> for homology of <a href="/wiki/Product_space" class="mw-redirect" title="Product space">product</a> of spaces. </td></tr> <tr> <td>1926</td> <td><a href="/wiki/Heinrich_Brandt" title="Heinrich Brandt">Heinrich Brandt</a></td> <td>defines the notion of <a href="/wiki/Groupoid" title="Groupoid">groupoid</a>. </td></tr> <tr> <td>1928</td> <td><a href="/wiki/Arend_Heyting" title="Arend Heyting">Arend Heyting</a></td> <td>Brouwer's intuitionistic logic made into formal mathematics, as logic in which the <a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a> replaces the <a href="/wiki/Boolean_algebra_(logic)" class="mw-redirect" title="Boolean algebra (logic)">Boolean algebra</a>. </td></tr> <tr> <td>1929</td> <td><a href="/wiki/Walther_Mayer" title="Walther Mayer">Walther Mayer</a></td> <td><a href="/wiki/Chain_complex" title="Chain complex">Chain complexes</a>. </td></tr> <tr> <td>1930</td> <td><a href="/wiki/Ernst_Zermelo" title="Ernst Zermelo">Ernst Zermelo</a>–<a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Abraham Fraenkel</a></td> <td>Statement of the definitive <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">ZF-axioms</a> of <a href="/wiki/Set_theory" title="Set theory">set theory</a>, first stated in 1908 and improved upon since then. </td></tr> <tr> <td>c.1930</td> <td><a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a></td> <td><a href="/wiki/Module_theory" class="mw-redirect" title="Module theory">Module theory</a> is developed by Noether and her students, and algebraic topology starts to be properly founded in <a href="/wiki/Abstract_algebra" title="Abstract algebra">abstract algebra</a> rather than by <i>ad hoc</i> arguments. </td></tr> <tr> <td>1932</td> <td><a href="/wiki/Eduard_%C4%8Cech" title="Eduard Čech">Eduard Čech</a></td> <td><a href="/wiki/%C4%8Cech_cohomology" title="Čech cohomology">Čech cohomology</a>, <a href="/wiki/Homotopy_groups" class="mw-redirect" title="Homotopy groups">homotopy groups</a> of a topological space. </td></tr> <tr> <td>1933</td> <td><a href="/wiki/Solomon_Lefschetz" title="Solomon Lefschetz">Solomon Lefschetz</a></td> <td><a href="/wiki/Singular_homology" title="Singular homology">Singular homology</a> of topological spaces. </td></tr> <tr> <td>1934</td> <td><a href="/wiki/Reinhold_Baer" title="Reinhold Baer">Reinhold Baer</a></td> <td>Ext groups, <a href="/wiki/Ext_functor" title="Ext functor">Ext functor</a> (for <a href="/wiki/Abelian_group" title="Abelian group">abelian groups</a> and with different notation). </td></tr> <tr> <td>1935</td> <td><a href="/wiki/Witold_Hurewicz" title="Witold Hurewicz">Witold Hurewicz</a></td> <td><a href="/wiki/Homotopy_group" title="Homotopy group">Higher homotopy groups</a> of a topological space. </td></tr> <tr> <td>1936</td> <td><a href="/wiki/Marshall_Stone" class="mw-redirect" title="Marshall Stone">Marshall Stone</a></td> <td><a href="/wiki/Stone_representation_theorem" class="mw-redirect" title="Stone representation theorem">Stone representation theorem</a> for Boolean algebras initiates various <a href="/wiki/Stone_duality" title="Stone duality">Stone dualities</a>. </td></tr> <tr> <td>1937</td> <td><a href="/wiki/Richard_Brauer" title="Richard Brauer">Richard Brauer</a>–<a href="/wiki/Cecil_J._Nesbitt" title="Cecil J. Nesbitt">Cecil Nesbitt</a></td> <td><a href="/wiki/Frobenius_algebra" title="Frobenius algebra">Frobenius algebras</a>. </td></tr> <tr> <td>1938</td> <td><a href="/wiki/Hassler_Whitney" title="Hassler Whitney">Hassler Whitney</a></td> <td>"Modern" definition of <a href="/wiki/Cohomology" title="Cohomology">cohomology</a>, summarizing the work since <a href="/wiki/James_Waddell_Alexander_II" title="James Waddell Alexander II">James Alexander</a> and <a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Andrey Kolmogorov</a> first defined <a href="/wiki/Cochain" class="mw-redirect" title="Cochain">cochains</a>. </td></tr> <tr> <td>1940</td> <td><a href="/wiki/Reinhold_Baer" title="Reinhold Baer">Reinhold Baer</a></td> <td><a href="/wiki/Injective_module" title="Injective module">Injective modules</a>. </td></tr> <tr> <td>1940</td> <td><a href="/wiki/Kurt_G%C3%B6del" title="Kurt Gödel">Kurt Gödel</a>–<a href="/wiki/Paul_Bernays" title="Paul Bernays">Paul Bernays</a></td> <td><a href="/wiki/Class_(set_theory)" title="Class (set theory)">Proper classes</a> in set theory. </td></tr> <tr> <td>1940</td> <td><a href="/wiki/Heinz_Hopf" title="Heinz Hopf">Heinz Hopf</a></td> <td><a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebras</a>. </td></tr> <tr> <td>1941</td> <td><a href="/wiki/Witold_Hurewicz" title="Witold Hurewicz">Witold Hurewicz</a></td> <td>First fundamental theorem of homological algebra: Given a <a href="/wiki/Short_exact_sequence" class="mw-redirect" title="Short exact sequence">short exact sequence</a> of spaces there exist a <a href="/wiki/Connecting_homomorphism" class="mw-redirect" title="Connecting homomorphism">connecting homomorphism</a> such that the long sequence of <a href="/wiki/Cohomology" title="Cohomology">cohomology</a> groups of the spaces is exact. </td></tr> <tr> <td>1942</td> <td><a href="/wiki/Samuel_Eilenberg" title="Samuel Eilenberg">Samuel Eilenberg</a>–<a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a></td> <td>Universal coefficient theorem for <a href="/wiki/%C4%8Cech_cohomology" title="Čech cohomology">Čech cohomology</a>; later this became the general <a href="/wiki/Universal_coefficient_theorem" title="Universal coefficient theorem">universal coefficient theorem</a>. The notations Hom and Ext first appear in their paper. </td></tr> <tr> <td>1943</td> <td><a href="/wiki/Norman_Steenrod" title="Norman Steenrod">Norman Steenrod</a></td> <td><a href="/wiki/Homology_with_local_coefficients" class="mw-redirect" title="Homology with local coefficients">Homology with local coefficients</a>. </td></tr> <tr> <td>1943</td> <td><a href="/wiki/Israel_Gelfand" title="Israel Gelfand">Israel Gelfand</a>–<a href="/wiki/Mark_Naimark" title="Mark Naimark">Mark Naimark</a></td> <td><a href="/wiki/Gelfand%E2%80%93Naimark_theorem" title="Gelfand–Naimark theorem">Gelfand–Naimark theorem</a> (sometimes called Gelfand isomorphism theorem): The category <b>Haus</b> of <a href="/wiki/Locally_compact_space" title="Locally compact space">locally compact</a> <a href="/wiki/Hausdorff_space" title="Hausdorff space">Hausdorff spaces</a> with <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> proper maps as <a href="/wiki/Morphism" title="Morphism">morphisms</a> is equivalent to the category C*<b>Alg</b> of commutative <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebras</a> with proper <a href="/wiki/C*-algebra#Abstract_characterization" title="C*-algebra">*-homomorphisms</a> as morphisms. </td></tr> <tr> <td>1944</td> <td><a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Garrett Birkhoff</a>–<a href="/wiki/%C3%98ystein_Ore" title="Øystein Ore">Øystein Ore</a></td> <td><a href="/wiki/Galois_connection" title="Galois connection">Galois connections</a> generalizing the Galois correspondence: a pair of <a href="/wiki/Adjoint_functor" class="mw-redirect" title="Adjoint functor">adjoint functors</a> between two categories that arise from <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered sets</a> (in modern formulation). </td></tr> <tr> <td>1944</td> <td><a href="/wiki/Samuel_Eilenberg" title="Samuel Eilenberg">Samuel Eilenberg</a></td> <td>"Modern" definition of <a href="/wiki/Singular_homology" title="Singular homology">singular homology</a> and singular cohomology. </td></tr> <tr> <td>1945</td> <td><a href="/wiki/Beno_Eckmann" title="Beno Eckmann">Beno Eckmann</a></td> <td>Defines the <a href="/wiki/Cohomology_ring" title="Cohomology ring">cohomology ring</a> building on <a href="/wiki/Heinz_Hopf" title="Heinz Hopf">Heinz Hopf</a>'s work. </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="1945–1970"><span id="1945.E2.80.931970"></span>1945–1970</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Timeline_of_category_theory_and_related_mathematics&action=edit&section=2" title="Edit section: 1945–1970"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable sortable" width="100%"> <tbody><tr> <th>Year </th> <th style="width:22%">Contributors </th> <th>Event </th></tr> <tr> <td>1945</td> <td><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a>–<a href="/wiki/Samuel_Eilenberg" title="Samuel Eilenberg">Samuel Eilenberg</a></td> <td>Start of category theory: axioms for <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">categories</a>, <a href="/wiki/Functor" title="Functor">functors</a> and <a href="/wiki/Natural_transformation" title="Natural transformation">natural transformations</a>. </td></tr> <tr> <td>1945</td> <td><a href="/wiki/Norman_Steenrod" title="Norman Steenrod">Norman Steenrod</a>–<a href="/wiki/Samuel_Eilenberg" title="Samuel Eilenberg">Samuel Eilenberg</a></td> <td><a href="/wiki/Eilenberg%E2%80%93Steenrod_axioms" title="Eilenberg–Steenrod axioms">Eilenberg–Steenrod axioms</a> for homology and cohomology. </td></tr> <tr> <td>1945</td> <td><a href="/wiki/Jean_Leray" title="Jean Leray">Jean Leray</a></td> <td>Starts <a href="/wiki/Sheaf_theory" class="mw-redirect" title="Sheaf theory">sheaf theory</a>: At this time a sheaf was a map that assigned a module or a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> to a closed subspace of a topological space. The first example was the sheaf assigning to a closed subspace its <span class="nowrap">p-th</span> cohomology group. </td></tr> <tr> <td>1945</td> <td><a href="/wiki/Jean_Leray" title="Jean Leray">Jean Leray</a></td> <td>Defines <a href="/wiki/Sheaf_cohomology" title="Sheaf cohomology">Sheaf cohomology</a> using his new concept of sheaf. </td></tr> <tr> <td>1946</td> <td><a href="/wiki/Jean_Leray" title="Jean Leray">Jean Leray</a></td> <td>Invents <a href="/wiki/Spectral_sequences" class="mw-redirect" title="Spectral sequences">spectral sequences</a> as a method for iteratively approximating cohomology groups by previous approximate cohomology groups. In the limiting case it gives the sought cohomology groups. </td></tr> <tr> <td>1948</td> <td>Cartan seminar</td> <td>Writes up <a href="/wiki/Sheaf_theory" class="mw-redirect" title="Sheaf theory">sheaf theory</a> for the first time. </td></tr> <tr> <td>1948</td> <td><a href="/w/index.php?title=A._L._Blakers&action=edit&redlink=1" class="new" title="A. L. Blakers (page does not exist)">A. L. Blakers</a></td> <td><a href="/w/index.php?title=Crossed_complex&action=edit&redlink=1" class="new" title="Crossed complex (page does not exist)">Crossed complexes</a> (called group systems by Blakers), after a suggestion of <a href="/wiki/Samuel_Eilenberg" title="Samuel Eilenberg">Samuel Eilenberg</a>: A nonabelian generalization of <a href="/wiki/Chain_complex" title="Chain complex">chain complexes</a> of abelian groups which are equivalent to strict <span class="nowrap">ω-groupoids</span>. They form a category <b>Crs</b> that has many satisfactory properties such as a <a href="/wiki/Monoidal_category" title="Monoidal category">monoidal structure</a>. </td></tr> <tr> <td>1949</td> <td><a href="/wiki/John_Henry_Whitehead" class="mw-redirect" title="John Henry Whitehead">John Henry Whitehead</a></td> <td><a href="/wiki/Crossed_module" title="Crossed module">Crossed modules</a>. </td></tr> <tr> <td>1949</td> <td><a href="/wiki/Andr%C3%A9_Weil" title="André Weil">André Weil</a></td> <td>Formulates the <a href="/wiki/Weil_conjectures" title="Weil conjectures">Weil conjectures</a> on remarkable relations between the cohomological structure of <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic varieties</a> over <b>C</b> and the diophantine structure of algebraic varieties over <a href="/wiki/Finite_field" title="Finite field">finite fields</a>. </td></tr> <tr> <td>1950</td> <td><a href="/wiki/Henri_Cartan" title="Henri Cartan">Henri Cartan</a></td> <td>In the book Sheaf theory from the Cartan seminar he defines: <a href="/wiki/Sheaf_(mathematics)#The_étale_space_of_a_sheaf" title="Sheaf (mathematics)">Sheaf space</a> (étale space), <a href="/wiki/Support_(mathematics)#Family_of_supports" title="Support (mathematics)">support</a> of sheaves axiomatically, <a href="/wiki/Sheaf_cohomology" title="Sheaf cohomology">sheaf cohomology</a> with support in an axiomatic form and more. </td></tr> <tr> <td>1950</td> <td><a href="/wiki/John_Henry_Whitehead" class="mw-redirect" title="John Henry Whitehead">John Henry Whitehead</a></td> <td>Outlines <a href="/wiki/Algebraic_homotopy" title="Algebraic homotopy">algebraic homotopy</a> program for describing, understanding and calculating <a href="/wiki/Homotopy_type" class="mw-redirect" title="Homotopy type">homotopy types</a> of spaces and homotopy classes of mappings </td></tr> <tr> <td>1950</td> <td><a href="/wiki/Samuel_Eilenberg" title="Samuel Eilenberg">Samuel Eilenberg</a>–Joe Zilber</td> <td><a href="/wiki/Simplicial_set" title="Simplicial set">Simplicial sets</a> as a purely algebraic model of well behaved topological spaces. A simplicial set can also be seen as a presheaf on the <a href="/wiki/Simplicial_set" title="Simplicial set">simplex category</a>. A category is a simplicial set such that the <a href="/wiki/Segal_map" class="mw-redirect" title="Segal map">Segal maps</a> are isomorphisms. </td></tr> <tr> <td>1951</td> <td><a href="/wiki/Henri_Cartan" title="Henri Cartan">Henri Cartan</a></td> <td>Modern definition of <a href="/wiki/Sheaf_theory" class="mw-redirect" title="Sheaf theory">sheaf theory</a> in which a <a href="/wiki/Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> is defined using <a href="/wiki/Open_subset" class="mw-redirect" title="Open subset">open subsets</a> instead of <a href="/wiki/Closed_subset" class="mw-redirect" title="Closed subset">closed subsets</a> of a topological space and all the open subsets are treated at once. A sheaf on a topological space <i>X</i> becomes a functor resembling a function defined locally on <i>X</i>, and taking values in sets, abelian groups, <a href="/wiki/Commutative_ring" title="Commutative ring">commutative rings</a>, modules or generally in any category <span class="texhtml">C</span>. In fact <a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a> later made a <a href="/w/index.php?title=Dictionary_between_sheaves_and_functions&action=edit&redlink=1" class="new" title="Dictionary between sheaves and functions (page does not exist)">dictionary between sheaves and functions</a>. Another interpretation of sheaves is as continuously <a href="/w/index.php?title=Varying_set&action=edit&redlink=1" class="new" title="Varying set (page does not exist)">varying sets</a> (a generalization of <a href="/w/index.php?title=Abstract_set&action=edit&redlink=1" class="new" title="Abstract set (page does not exist)">abstract sets</a>). Its purpose is to provide a unified approach to connect local and global properties of topological spaces and to classify the obstructions for passing from local objects to global objects on a topological space by pasting together the local pieces. The <span class="texhtml">C</span>-valued sheaves on a topological space and their homomorphisms form a category. </td></tr> <tr> <td>1952</td> <td><a href="/wiki/William_S._Massey" title="William S. Massey">William Massey</a></td> <td>Invents <a href="/wiki/Spectral_sequence#Exact_couples" title="Spectral sequence">exact couples</a> for calculating spectral sequences. </td></tr> <tr> <td>1953</td> <td><a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a></td> <td><a href="/wiki/Serre_C-theory" class="mw-redirect" title="Serre C-theory">Serre <i>C</i>-theory</a> and <a href="/wiki/Subcategory#Types_of_subcategories" title="Subcategory">Serre subcategories</a>. </td></tr> <tr> <td>1952</td> <td><a href="/wiki/Nobuo_Yoneda" title="Nobuo Yoneda">Nobuo Yoneda</a></td> <td>Yoneda publishes his <a href="/wiki/Yoneda_lemma" title="Yoneda lemma">famous lemma</a>. Yoneda's Lemma allows one to consider objects in a (small) category as a <a href="/wiki/Presheaf_(category_theory)" title="Presheaf (category theory)">presheaves</a>. Yoneda lemma plays a critical role in the study of representable functors in algebraic geometry. For example, even though it is never mentioned explicitly, it is central to the ideas of Grothendieck's "Fondements de la Géométrie Algébrique". </td></tr> <tr> <td>1955</td> <td><a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a></td> <td>Shows there is a 1−1 correspondence between <a href="/wiki/Algebraic_vector_bundle" class="mw-redirect" title="Algebraic vector bundle">algebraic vector bundles</a> over an affine variety and <a href="/wiki/Finitely_generated_projective_module" class="mw-redirect" title="Finitely generated projective module">finitely generated projective modules</a> over its <a href="/wiki/Affine_variety#Introduction" title="Affine variety">coordinate ring</a> (<a href="/wiki/Serre%E2%80%93Swan_theorem" title="Serre–Swan theorem">Serre–Swan theorem</a>). </td></tr> <tr> <td>1955</td> <td><a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a></td> <td><a href="/wiki/Coherent_sheaf_cohomology" title="Coherent sheaf cohomology">Coherent sheaf cohomology</a> in algebraic geometry. </td></tr> <tr> <td>1956</td> <td><a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a></td> <td><a href="/wiki/Algebraic_geometry_and_analytic_geometry" title="Algebraic geometry and analytic geometry">GAGA correspondence</a>. </td></tr> <tr> <td>1956</td> <td><a href="/wiki/Henri_Cartan" title="Henri Cartan">Henri Cartan</a>–<a href="/wiki/Samuel_Eilenberg" title="Samuel Eilenberg">Samuel Eilenberg</a></td> <td>Influential book: <i>Homological Algebra</i>, summarizing the state of the art in its topic at that time. The notation <a href="/wiki/Tor_functor" title="Tor functor">Tor</a><sub><i>n</i></sub> and <a href="/wiki/Ext_functors" class="mw-redirect" title="Ext functors">Ext</a><sup><i>n</i></sup>, as well as the concepts of <a href="/wiki/Projective_module" title="Projective module">projective module</a>, <a href="/wiki/Projective_resolution#Projective_resolutions" class="mw-redirect" title="Projective resolution">projective</a> and <a href="/wiki/Injective_resolution#Injective_resolutions" class="mw-redirect" title="Injective resolution">injective resolution</a> of a module, <a href="/wiki/Derived_functor" title="Derived functor">derived functor</a> and <a href="/wiki/Hyperhomology" title="Hyperhomology">hyperhomology</a> appear in this book for the first time. </td></tr> <tr> <td>1956</td> <td><a href="/wiki/Daniel_Kan" title="Daniel Kan">Daniel Kan</a></td> <td><a href="/wiki/Simplicial_homotopy_theory" class="mw-redirect" title="Simplicial homotopy theory">Simplicial homotopy theory</a> also called categorical homotopy theory: A homotopy theory completely internal to the <a href="/wiki/Simplicial_object" class="mw-redirect" title="Simplicial object">category of simplicial sets</a>. </td></tr> <tr> <td>1957</td> <td><a href="/wiki/Charles_Ehresmann" title="Charles Ehresmann">Charles Ehresmann</a>–<a href="/wiki/Jean_B%C3%A9nabou" title="Jean Bénabou">Jean Bénabou</a></td> <td><a href="/wiki/Pointless_topology" title="Pointless topology">Pointless topology</a> building on <a href="/wiki/Marshall_Stone" class="mw-redirect" title="Marshall Stone">Marshall Stone</a>'s work. </td></tr> <tr> <td>1957</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Abelian_categories" class="mw-redirect" title="Abelian categories">Abelian categories</a> in homological algebra that combine exactness and linearity. </td></tr> <tr> <td>1957</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td>Influential <a href="/wiki/Tohoku_paper" class="mw-redirect" title="Tohoku paper"><i>Tohoku</i> paper</a> rewrites <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a>; proving <a href="/wiki/Coherent_duality" title="Coherent duality">Grothendieck duality</a> (Serre duality for possibly singular algebraic varieties). He also showed that the conceptual basis for homological algebra over a ring also holds for linear objects varying as sheaves over a space. </td></tr> <tr> <td>1957</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Grothendieck%27s_relative_point_of_view" title="Grothendieck's relative point of view">Grothendieck's relative point of view</a>, <a href="/wiki/S-scheme" class="mw-redirect" title="S-scheme">S-schemes</a>. </td></tr> <tr> <td>1957</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Grothendieck%E2%80%93Hirzebruch%E2%80%93Riemann%E2%80%93Roch_theorem" class="mw-redirect" title="Grothendieck–Hirzebruch–Riemann–Roch theorem">Grothendieck–Hirzebruch–Riemann–Roch theorem</a> for smooth; the proof introduces <a href="/wiki/K-theory" title="K-theory">K-theory</a>. </td></tr> <tr> <td>1957</td> <td><a href="/wiki/Daniel_Kan" title="Daniel Kan">Daniel Kan</a></td> <td><a href="/wiki/Kan_complex" class="mw-redirect" title="Kan complex">Kan complexes</a>: <a href="/wiki/Simplicial_set" title="Simplicial set">Simplicial sets</a> (in which every horn has a filler) that are geometric models of simplicial <a href="/wiki/%E2%88%9E-groupoid" title="∞-groupoid">∞-groupoids</a>. Kan complexes are also the fibrant (and cofibrant) objects of <a href="/wiki/Model_category" title="Model category">model categories</a> of simplicial sets for which the fibrations are <a href="/wiki/Kan_fibration" title="Kan fibration">Kan fibrations</a>. </td></tr> <tr> <td>1958</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td>Starts new foundation of <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> by generalizing varieties and other spaces in algebraic geometry to <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> which have the structure of a category with open subsets as objects and restrictions as morphisms. form a category that is a <a href="/wiki/Toposes#Grothendieck_topoi_.28topoi_in_geometry.29" class="mw-redirect" title="Toposes">Grothendieck topos</a>, and to a scheme and even a stack one may associate a Zariski topos, an étale topos, a fppf topos, a fpqc topos, a Nisnevich topos, a flat topos, ... depending on the topology imposed on the scheme. The whole of algebraic geometry was categorized with time. </td></tr> <tr> <td>1958</td> <td><a href="/wiki/Roger_Godement" title="Roger Godement">Roger Godement</a></td> <td><a href="/wiki/Monad_(category_theory)" title="Monad (category theory)">Monads</a> in category theory (then called standard constructions and triples). Monads generalize classical notions from <a href="/wiki/Universal_algebra" title="Universal algebra">universal algebra</a> and can in this sense be thought of as an <a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">algebraic theory</a> over a category: the theory of the category of T-algebras. An algebra for a monad subsumes and generalizes the notion of a model for an algebraic theory. </td></tr> <tr> <td>1958</td> <td><a href="/wiki/Daniel_Kan" title="Daniel Kan">Daniel Kan</a></td> <td>Daniel Kan introduces <a href="/wiki/Adjoint_functors" title="Adjoint functors">Adjoint functors</a>. They are critical, for example, in the theory of sheaves. </td></tr> <tr> <td>1958</td> <td><a href="/wiki/Daniel_Kan" title="Daniel Kan">Daniel Kan</a></td> <td><a href="/wiki/Limit_(category_theory)" title="Limit (category theory)">Limits</a> in category theory. </td></tr> <tr> <td>1958</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Fibred_category" title="Fibred category">Fibred categories</a>. </td></tr> <tr> <td>1959</td> <td><a href="/wiki/Bernard_Dwork" title="Bernard Dwork">Bernard Dwork</a></td> <td>Proves the rationality part of the <a href="/wiki/Weil_conjectures" title="Weil conjectures">Weil conjectures</a> (the first conjecture). </td></tr> <tr> <td>1959</td> <td><a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a></td> <td><a href="/wiki/Algebraic_K-theory" title="Algebraic K-theory">Algebraic K-theory</a> launched by explicit analogy of <a href="/wiki/Ring_theory" title="Ring theory">ring theory</a> with geometric cases. </td></tr> <tr> <td>1960</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Fiber_functor" title="Fiber functor">Fiber functors</a> </td></tr> <tr> <td>1960</td> <td><a href="/wiki/Daniel_Kan" title="Daniel Kan">Daniel Kan</a></td> <td><a href="/wiki/Kan_extension" title="Kan extension">Kan extensions</a> </td></tr> <tr> <td>1960</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Formal_algebraic_geometry" class="mw-redirect" title="Formal algebraic geometry">Formal algebraic geometry</a> and <a href="/wiki/Formal_scheme" title="Formal scheme">formal schemes</a> </td></tr> <tr> <td>1960</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Representable_functor" title="Representable functor">Representable functors</a> </td></tr> <tr> <td>1960</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td>Categorizes <a href="/wiki/Galois_theory" title="Galois theory">Galois theory</a> (<a href="/wiki/Grothendieck%27s_Galois_theory" title="Grothendieck's Galois theory">Grothendieck's Galois theory</a>) </td></tr> <tr> <td>1960</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Descent_(category_theory)" class="mw-redirect" title="Descent (category theory)">Descent theory</a>: An idea extending the notion of <a href="/wiki/Quotient_space_(topology)" title="Quotient space (topology)">gluing</a> in topology to <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> to get around the brute equivalence relations. It also generalizes <a href="/wiki/Localization_of_a_topological_space" title="Localization of a topological space">localization</a> in topology </td></tr> <tr> <td>1961</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Local_cohomology" title="Local cohomology">Local cohomology</a>. Introduced at a seminar in 1961 but the notes are published in 1967 </td></tr> <tr> <td>1961</td> <td><a href="/wiki/Jim_Stasheff" title="Jim Stasheff">Jim Stasheff</a></td> <td><a href="/wiki/Associahedron" title="Associahedron">Associahedra</a> later used in the definition of <a href="/wiki/Weak_n-category" title="Weak n-category">weak <i>n</i>-categories</a> </td></tr> <tr> <td>1961</td> <td><a href="/wiki/Richard_Swan" title="Richard Swan">Richard Swan</a></td> <td>Shows there is a 1−1 correspondence between topological vector bundles over a <a href="/wiki/Compact_space" title="Compact space">compact</a> Hausdorff space <i>X</i> and finitely generated projective modules over the ring <i>C</i>(<i>X</i>) of continuous functions on <i>X</i> (<a href="/wiki/Serre%E2%80%93Swan_theorem" title="Serre–Swan theorem">Serre–Swan theorem</a>) </td></tr> <tr> <td>1963</td> <td>Frank Adams–<a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a></td> <td><a href="/wiki/PROP_(category_theory)" title="PROP (category theory)">PROP categories</a> and PACT categories for higher homotopies. PROPs are categories for describing families of operations with any number of inputs and outputs. <a href="/wiki/Operad_(category_theory)" class="mw-redirect" title="Operad (category theory)">Operads</a> are special PROPs with operations with only one output </td></tr> <tr> <td>1963</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/%C3%89tale_topology" title="Étale topology">Étale topology</a>, a special Grothendieck topology on </td></tr> <tr> <td>1963</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/%C3%89tale_cohomology" title="Étale cohomology">Étale cohomology</a> </td></tr> <tr> <td>1963</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td>Grothendieck <a href="/wiki/Topos" title="Topos">toposes</a>, which are categories which are like universes (generalized spaces) of sets in which one can do mathematics </td></tr> <tr> <td>1963</td> <td><a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a></td> <td><a href="/wiki/Algebraic_theory" title="Algebraic theory">Algebraic theories</a> and <a href="/wiki/Algebraic_category" class="mw-redirect" title="Algebraic category">algebraic categories</a> </td></tr> <tr> <td>1963</td> <td><a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a></td> <td>Founds <a href="/wiki/Categorical_logic" title="Categorical logic">categorical logic</a>, discovers <a href="/w/index.php?title=Internal_logic_(category_theory)&action=edit&redlink=1" class="new" title="Internal logic (category theory) (page does not exist)">internal logics</a> of categories and recognizes its importance and introduces <a href="/wiki/Lawvere_theory" title="Lawvere theory">Lawvere theories</a>. Essentially categorical logic is a lift of different logics to being internal logics of categories. Each kind of category with extra structure corresponds to a system of logic with its own inference rules. A Lawvere theory is an <a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">algebraic theory</a> as a category with finite <a href="/wiki/Product_(category_theory)" title="Product (category theory)">products</a> and possessing a "generic algebra" (a generic group). The structures described by a Lawvere theory are models of the Lawvere theory </td></tr> <tr> <td>1963</td> <td><a href="/wiki/Jean-Louis_Verdier" title="Jean-Louis Verdier">Jean-Louis Verdier</a></td> <td><a href="/wiki/Triangulated_category" title="Triangulated category">Triangulated categories</a> and <a href="/wiki/Triangulated_functor" class="mw-redirect" title="Triangulated functor">triangulated functors</a>. <a href="/wiki/Derived_category" title="Derived category">Derived categories</a> and <a href="/wiki/Derived_functor" title="Derived functor">derived functors</a> are special cases of these </td></tr> <tr> <td>1963</td> <td><a href="/wiki/Jim_Stasheff" title="Jim Stasheff">Jim Stasheff</a></td> <td><a href="/wiki/A%E2%88%9E-algebra" class="mw-redirect" title="A∞-algebra"><i>A</i><sub>∞</sub>-algebras</a>: <a href="/wiki/Dg-algebra" class="mw-redirect" title="Dg-algebra">dg-algebra</a> analogs of <a href="/wiki/Topological_monoid" title="Topological monoid">topological monoids</a> associative up to homotopy appearing in topology (i.e. <a href="/wiki/H-space" title="H-space">H-spaces</a>) </td></tr> <tr> <td>1963</td> <td><a href="/wiki/Jean_Giraud_(mathematician)" title="Jean Giraud (mathematician)">Jean Giraud</a></td> <td><a href="/wiki/Giraud_characterization_theorem" class="mw-redirect" title="Giraud characterization theorem">Giraud characterization theorem</a> characterizing Grothendieck toposes as categories of sheaves over a small site </td></tr> <tr> <td>1963</td> <td><a href="/wiki/Charles_Ehresmann" title="Charles Ehresmann">Charles Ehresmann</a></td> <td><a href="/w/index.php?title=Internal_category_theory&action=edit&redlink=1" class="new" title="Internal category theory (page does not exist)">Internal category theory</a>: Internalization of categories in a category <b>V</b> with <a href="/wiki/Pullback_(category_theory)" title="Pullback (category theory)">pullbacks</a> is replacing the category <b>Set</b> (same for classes instead of sets) by <b>V</b> in the definition of a category. Internalization is a way to rise the <a href="/w/index.php?title=Categorical_dimension&action=edit&redlink=1" class="new" title="Categorical dimension (page does not exist)">categorical dimension</a> </td></tr> <tr> <td>1963</td> <td><a href="/wiki/Charles_Ehresmann" title="Charles Ehresmann">Charles Ehresmann</a></td> <td><a href="/w/index.php?title=Multiple_category&action=edit&redlink=1" class="new" title="Multiple category (page does not exist)">Multiple categories</a> and <a href="/w/index.php?title=Multiple_functor&action=edit&redlink=1" class="new" title="Multiple functor (page does not exist)">multiple functors</a> </td></tr> <tr> <td>1963</td> <td><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a></td> <td><a href="/wiki/Monoidal_category" title="Monoidal category">Monoidal categories</a>, also called tensor categories: Strict 2-categories with one object made by a <a href="/w/index.php?title=Relabelling_trick&action=edit&redlink=1" class="new" title="Relabelling trick (page does not exist)">relabelling trick</a> to categories with a <a href="/wiki/Tensor_product" title="Tensor product">tensor product</a> of objects that is secretly the composition of morphisms in the 2-category. There are several object in a monoidal category since the relabelling trick makes 2-morphisms of the 2-category to morphisms, morphisms of the 2-category to objects and forgets about the single object. In general a higher relabelling trick works for <a href="/wiki/N-category" class="mw-redirect" title="N-category"><b>n</b>-categories</a> with one object to make general monoidal categories. The most common examples include: <a href="/wiki/Ribbon_category" title="Ribbon category">ribbon categories</a>, <a href="/wiki/Braided_tensor_category" class="mw-redirect" title="Braided tensor category">braided tensor categories</a>, <a href="/wiki/Spherical_category" title="Spherical category">spherical categories</a>, <a href="/wiki/Compact_closed_category" title="Compact closed category">compact closed categories</a>, <a href="/wiki/Symmetric_monoidal_category" title="Symmetric monoidal category">symmetric tensor categories</a>, <a href="/w/index.php?title=Modular_category&action=edit&redlink=1" class="new" title="Modular category (page does not exist)">modular categories</a>, <a href="/wiki/Autonomous_category" title="Autonomous category">autonomous categories</a>, <a href="/w/index.php?title=Category_with_duality&action=edit&redlink=1" class="new" title="Category with duality (page does not exist)">categories with duality</a> </td></tr> <tr> <td>1963</td> <td><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a></td> <td><a href="/wiki/Monoidal_category" title="Monoidal category">Mac Lane coherence theorem</a> for determining <a href="/wiki/Commutative_diagram" title="Commutative diagram">commutativity of diagrams</a> in <a href="/wiki/Monoidal_categories" class="mw-redirect" title="Monoidal categories">monoidal categories</a> </td></tr> <tr> <td>1964</td> <td><a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a></td> <td>ETCS <a href="/wiki/Elementary_Theory_of_the_Category_of_Sets" title="Elementary Theory of the Category of Sets">Elementary Theory of the Category of Sets</a>: An axiomatization of the <a href="/wiki/Category_of_sets" title="Category of sets">category of sets</a> which is also the constant case of an <a href="/wiki/Toposes#Elementary_toposes_.28toposes_in_logic.29" class="mw-redirect" title="Toposes">elementary topos</a> </td></tr> <tr> <td>1964</td> <td>Barry Mitchell–<a href="/wiki/Peter_Freyd" class="mw-redirect" title="Peter Freyd">Peter Freyd</a></td> <td><a href="/wiki/Mitchell%27s_embedding_theorem" title="Mitchell's embedding theorem">Mitchell–Freyd embedding theorem</a>: Every <a href="/wiki/Small_category" class="mw-redirect" title="Small category">small</a> <a href="/wiki/Abelian_category" title="Abelian category">abelian category</a> admits an exact and full embedding into the <a href="/wiki/Category_of_modules" title="Category of modules">category of (left) modules</a> <b>Mod</b><sub><i>R</i></sub> over some ring <i>R</i> </td></tr> <tr> <td>1964</td> <td><a href="/wiki/Rudolf_Haag" title="Rudolf Haag">Rudolf Haag</a>–<a href="/wiki/Daniel_Kastler" title="Daniel Kastler">Daniel Kastler</a></td> <td><a href="/wiki/Algebraic_quantum_field_theory" title="Algebraic quantum field theory">Algebraic quantum field theory</a> after ideas of <a href="/wiki/Irving_Segal" title="Irving Segal">Irving Segal</a> </td></tr> <tr> <td>1964</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td>Topologizes categories axiomatically by imposing a <a href="/wiki/Grothendieck_topology" title="Grothendieck topology">Grothendieck topology</a> on categories which are then called <a href="/wiki/Grothendieck_topology" title="Grothendieck topology">sites</a>. The purpose of sites is to define coverings on them so sheaves over sites can be defined. The other "spaces" one can define sheaves for except topological spaces are locales </td></tr> <tr> <td>1964</td> <td><a href="/wiki/Michael_Artin" title="Michael Artin">Michael Artin</a>–<a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/%C3%89tale_cohomology#ℓ-adic_cohomology_groups" title="Étale cohomology">ℓ-adic cohomology</a>, technical development in SGA4 of the long-anticipated <a href="/wiki/Weil_cohomology" class="mw-redirect" title="Weil cohomology">Weil cohomology</a>. </td></tr> <tr> <td>1964</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td>Proves the <a href="/wiki/Weil_conjectures" title="Weil conjectures">Weil conjectures</a> except the analogue of the <a href="/wiki/Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a> </td></tr> <tr> <td>1964</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Six_operations" title="Six operations">Six operations</a> formalism in <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a>; R<i>f</i><sub>*</sub>, <i>f</i><sup>−1</sup>, R<i>f</i><sub>!</sub>, <i>f</i><sup>!</sup>, ⊗<sup>L</sup>, RHom, and proof of its closedness </td></tr> <tr> <td>1964</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td>Introduced in a letter to <a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Jean-Pierre Serre</a> conjectural <a href="/wiki/Motive_(algebraic_geometry)" title="Motive (algebraic geometry)">motives</a> to express the idea that there is a single universal cohomology theory underlying the various cohomology theories for algebraic varieties. According to Grothendieck's philosophy there should be a universal cohomology functor attaching a <a href="/wiki/Motive_(algebraic_geometry)#pure_motives" title="Motive (algebraic geometry)">pure motive</a> h(<i>X</i>) to each smooth <a href="/wiki/Projective_variety" title="Projective variety">projective variety</a> <i>X</i>. When <i>X</i> is not smooth or projective h(<i>X</i>) must be replaced by a more general <a href="/wiki/Motive_(algebraic_geometry)#Mixed_motives" title="Motive (algebraic geometry)">mixed motive</a> which has a weight filtration whose quotients are pure motives. The <a href="/wiki/Motive_(algebraic_geometry)" title="Motive (algebraic geometry)">category of motives</a> (the categorical framework for the universal cohomology theory) may be used as an abstract substitute for singular cohomology (and rational cohomology) to compare, relate and unite "motivated" properties and parallel phenomena of the various cohomology theories and to detect topological structure of algebraic varieties. The categories of pure motives and of mixed motives are abelian tensor categories and the category of pure motives is also a <a href="/wiki/Tannakian_category" class="mw-redirect" title="Tannakian category">Tannakian category</a>. Categories of motives are made by replacing the category of varieties by a category with the same objects but whose morphisms are <a href="/wiki/Correspondence_(algebraic_geometry)" title="Correspondence (algebraic geometry)">correspondences</a>, modulo a suitable <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a>; different equivalences give different theories. <a href="/wiki/Rational_equivalence" class="mw-redirect" title="Rational equivalence">Rational equivalence</a> gives the category of <a href="/wiki/Chow_motive" class="mw-redirect" title="Chow motive">Chow motives</a> with <a href="/wiki/Chow_group" title="Chow group">Chow groups</a> as morphisms which are in some sense universal. Every geometric cohomology theory is a functor on the category of motives. Each induced functor ρ:motives modulo numerical equivalence→graded <b>Q</b>-vector spaces is called a <a href="/wiki/Realization_(probability)" title="Realization (probability)">realization</a> of the category of motives, the inverse functors are called <a href="/w/index.php?title=Improvement_(mathematics)&action=edit&redlink=1" class="new" title="Improvement (mathematics) (page does not exist)">improvements</a>. Mixed motives explain phenomena in as diverse areas as: Hodge theory, algebraic K-theory, polylogarithms, regulator maps, automorphic forms, L-functions, ℓ-adic representations, trigonometric sums, homotopy of algebraic varieties, algebraic cycles, moduli spaces and thus has the potential of enriching each area and of unifying them all. </td></tr> <tr> <td>1965</td> <td>Edgar Brown</td> <td>Abstract <a href="/wiki/Homotopy_category" title="Homotopy category">homotopy categories</a>: A proper framework for the study of homotopy theory of <a href="/wiki/CW_complex" title="CW complex">CW complexes</a> </td></tr> <tr> <td>1965</td> <td><a href="/wiki/Max_Kelly" title="Max Kelly">Max Kelly</a></td> <td><a href="/wiki/Differential_graded_category" title="Differential graded category">dg-categories</a> </td></tr> <tr> <td>1965</td> <td><a href="/wiki/Max_Kelly" title="Max Kelly">Max Kelly</a>–<a href="/wiki/Samuel_Eilenberg" title="Samuel Eilenberg">Samuel Eilenberg</a></td> <td><a href="/wiki/Enriched_category_theory" class="mw-redirect" title="Enriched category theory">Enriched category theory</a>: Categories <span class="texhtml">C</span> enriched over a category <b>V</b> are categories with <a href="/wiki/Hom-set" class="mw-redirect" title="Hom-set">Hom-sets</a> Hom<sub><span class="texhtml">C</span></sub> not just a set or class but with the structure of objects in the category <b>V</b>. Enrichment over <b>V</b> is a way to rise the <a href="/w/index.php?title=Categorical_dimension&action=edit&redlink=1" class="new" title="Categorical dimension (page does not exist)">categorical dimension</a> </td></tr> <tr> <td>1965</td> <td><a href="/wiki/Charles_Ehresmann" title="Charles Ehresmann">Charles Ehresmann</a></td> <td>Defines both <a href="/wiki/Strict_2-category" title="Strict 2-category">strict 2-categories</a> and <a href="/wiki/Strict_n-category" class="mw-redirect" title="Strict n-category">strict <i>n</i>-categories</a> </td></tr> <tr> <td>1966</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Crystalline_cohomology#Crystals" title="Crystalline cohomology">Crystals</a> (a kind of sheaf used in <a href="/wiki/Crystalline_cohomology" title="Crystalline cohomology">crystalline cohomology</a>) </td></tr> <tr> <td>1966</td> <td><a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a></td> <td>ETAC <a href="/w/index.php?title=Elementary_theory_of_abstract_categories&action=edit&redlink=1" class="new" title="Elementary theory of abstract categories (page does not exist)">Elementary theory of abstract categories</a>, first proposed axioms for <b>Cat</b> or category theory using <a href="/wiki/First-order_logic" title="First-order logic">first-order logic</a> </td></tr> <tr> <td>1967</td> <td><a href="/wiki/Jean_B%C3%A9nabou" title="Jean Bénabou">Jean Bénabou</a></td> <td><a href="/wiki/Bicategory" title="Bicategory">Bicategories</a> (weak 2-categories) and weak 2-functors </td></tr> <tr> <td>1967</td> <td><a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a></td> <td>Founds <a href="/wiki/Synthetic_differential_geometry" title="Synthetic differential geometry">synthetic differential geometry</a> </td></tr> <tr> <td>1967</td> <td>Simon Kochen–Ernst Specker</td> <td><a href="/wiki/Kochen%E2%80%93Specker_theorem" title="Kochen–Specker theorem">Kochen–Specker theorem</a> in quantum mechanics </td></tr> <tr> <td>1967</td> <td><a href="/wiki/Jean-Louis_Verdier" title="Jean-Louis Verdier">Jean-Louis Verdier</a></td> <td>Defines <a href="/wiki/Derived_categories" class="mw-redirect" title="Derived categories">derived categories</a> and redefines <a href="/wiki/Derived_functor" title="Derived functor">derived functors</a> in terms of derived categories </td></tr> <tr> <td>1967</td> <td>Peter Gabriel–Michel Zisman</td> <td>Axiomatizes <a href="/wiki/Simplicial_homotopy_theory" class="mw-redirect" title="Simplicial homotopy theory">simplicial homotopy theory</a> </td></tr> <tr> <td>1967</td> <td><a href="/wiki/Daniel_Quillen" title="Daniel Quillen">Daniel Quillen</a></td> <td><a href="/wiki/Model_category" title="Model category">Quillen model categories</a> and <a href="/wiki/Model_category" title="Model category">Quillen model functors</a>: A framework for doing homotopy theory in an axiomatic way in categories and an abstraction of <a href="/wiki/Homotopy_category" title="Homotopy category">homotopy categories</a> in such a way that <i>hC</i> = <i>C</i>[<i>W</i><sup>−1</sup>] where <i>W</i><sup>−1</sup> are the inverted <a href="/wiki/Model_category" title="Model category">weak equivalences</a> of the Quillen model category C. Quillen model categories are homotopically complete and cocomplete, and come with a built-in <a href="/wiki/Eckmann%E2%80%93Hilton_duality" title="Eckmann–Hilton duality">Eckmann–Hilton duality</a> </td></tr> <tr> <td>1967</td> <td><a href="/wiki/Daniel_Quillen" title="Daniel Quillen">Daniel Quillen</a></td> <td><a href="/wiki/Homotopical_algebra" title="Homotopical algebra">Homotopical algebra</a> (published as a book and also sometimes called noncommutative homological algebra): The study of various <a href="/wiki/Model_category" title="Model category">model categories</a> and the interplay between fibrations, cofibrations and weak equivalences in arbitrary closed model categories </td></tr> <tr> <td>1967</td> <td><a href="/wiki/Daniel_Quillen" title="Daniel Quillen">Daniel Quillen</a></td> <td><a href="/wiki/Quillen_axioms" class="mw-redirect" title="Quillen axioms">Quillen axioms</a> for homotopy theory in <a href="/wiki/Model_category" title="Model category">model categories</a> </td></tr> <tr> <td>1967</td> <td><a href="/wiki/Daniel_Quillen" title="Daniel Quillen">Daniel Quillen</a></td> <td>First <a href="/w/index.php?title=Fundamental_theorem_of_simplicial_homotopy_theory&action=edit&redlink=1" class="new" title="Fundamental theorem of simplicial homotopy theory (page does not exist)">fundamental theorem of simplicial homotopy theory</a>: The <a href="/wiki/Category_of_simplicial_sets" class="mw-redirect" title="Category of simplicial sets">category of simplicial sets</a> is a (proper) closed (simplicial) <a href="/wiki/Model_category" title="Model category">model category</a> </td></tr> <tr> <td>1967</td> <td><a href="/wiki/Daniel_Quillen" title="Daniel Quillen">Daniel Quillen</a></td> <td>Second <a href="/w/index.php?title=Fundamental_theorem_of_simplicial_homotopy_theory&action=edit&redlink=1" class="new" title="Fundamental theorem of simplicial homotopy theory (page does not exist)">fundamental theorem of simplicial homotopy theory</a>: The <a href="/w/index.php?title=Realization_functor&action=edit&redlink=1" class="new" title="Realization functor (page does not exist)">realization functor</a> and the <a href="/w/index.php?title=Singular_functor&action=edit&redlink=1" class="new" title="Singular functor (page does not exist)">singular functor</a> is an equivalence of categories <b>hΔ</b> and <b>hTop</b> (<b>Δ</b> the <a href="/wiki/Category_of_simplicial_sets" class="mw-redirect" title="Category of simplicial sets">category of simplicial sets</a>) </td></tr> <tr> <td>1967</td> <td><a href="/wiki/Jean_B%C3%A9nabou" title="Jean Bénabou">Jean Bénabou</a></td> <td><a href="/wiki/V-category" class="mw-redirect" title="V-category"><b>V</b>-categories</a>: A category <span class="texhtml">C</span> with an action ⊗ :<b>V</b> × <span class="texhtml">C</span> → <span class="texhtml">C</span> which is associative and unital up to coherent isomorphism, for <b>V</b> a <a href="/wiki/Symmetric_monoidal_category" title="Symmetric monoidal category">symmetric monoidal category</a>. V-categories can be seen as the categorification of R-modules over a commutative ring <i>R</i> </td></tr> <tr> <td>1968</td> <td><a href="/wiki/Chen_Ning_Yang" class="mw-redirect" title="Chen Ning Yang">Chen-Ning Yang</a>-<a href="/wiki/Rodney_Baxter" title="Rodney Baxter">Rodney Baxter</a></td> <td><a href="/wiki/Yang%E2%80%93Baxter_equation" title="Yang–Baxter equation">Yang–Baxter equation</a>, later used as a relation in <a href="/wiki/Braided_monoidal_category" title="Braided monoidal category">braided monoidal categories</a> for crossings of braids </td></tr> <tr> <td>1968</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Crystalline_cohomology" title="Crystalline cohomology">Crystalline cohomology</a>: A <a href="/wiki/P-adic_cohomology" title="P-adic cohomology"><i>p</i>-adic cohomology</a> theory in <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> <i>p</i> invented to fill the gap left by <a href="/wiki/%C3%89tale_cohomology" title="Étale cohomology">étale cohomology</a> which is deficient in using mod <i>p</i> coefficients for this case. It is sometimes referred to by Grothendieck as the yoga of de Rham coefficients and Hodge coefficients since crystalline cohomology of a variety <i>X</i> in characteristic <i>p</i> is like <a href="/wiki/De_Rham_cohomology" title="De Rham cohomology">de Rham cohomology</a> mod <i>p</i> of <i>X</i> and there is an isomorphism between de Rham cohomology groups and Hodge cohomology groups of harmonic forms </td></tr> <tr> <td>1968</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Grothendieck_connection" title="Grothendieck connection">Grothendieck connection</a> </td></tr> <tr> <td>1968</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td>Formulates the <a href="/wiki/Standard_conjectures_on_algebraic_cycles" title="Standard conjectures on algebraic cycles">standard conjectures on algebraic cycles</a> </td></tr> <tr> <td>1968</td> <td><a href="/wiki/Michael_Artin" title="Michael Artin">Michael Artin</a></td> <td><a href="/wiki/Algebraic_space" title="Algebraic space">Algebraic spaces</a> in algebraic geometry as a generalization of <a href="/wiki/Scheme_(algebraic_geometry)" class="mw-redirect" title="Scheme (algebraic geometry)">scheme</a> </td></tr> <tr> <td>1968</td> <td><a href="/wiki/Charles_Ehresmann" title="Charles Ehresmann">Charles Ehresmann</a></td> <td><a href="/wiki/Sketch_(category_theory)" class="mw-redirect" title="Sketch (category theory)">Sketches</a>: An alternative way of presenting a theory (which is categorical in character as opposed to linguistic) whose models are to study in appropriate categories. A sketch is a small category with a set of distinguished cones and a set of distinguished cocones satisfying some axioms. A model of a sketch is a set-valued functor transforming the distinguished cones into limit cones and the distinguished cocones into colimit cones. The categories of models of sketches are exactly the <a href="/wiki/Accessible_category" title="Accessible category">accessible categories</a> </td></tr> <tr> <td>1968</td> <td><a href="/wiki/Joachim_Lambek" title="Joachim Lambek">Joachim Lambek</a></td> <td><a href="/wiki/Multicategory" title="Multicategory">Multicategories</a> </td></tr> <tr> <td>1968-1972</td> <td><a href="/wiki/Michael_Boardman" title="Michael Boardman">Michael Boardman</a> and Rainer Vogt (1968), <a href="/wiki/J._Peter_May" title="J. Peter May">Peter May</a> (1972)</td> <td><a href="/wiki/Operad" title="Operad">Operads</a>: An abstraction of the family of composable functions of several variables together with an action of permutation of variables. Operads can be seen as algebraic theories and algebras over operads are then models of the theories. Each operad gives a <a href="/wiki/Monad_(category_theory)" title="Monad (category theory)">monad</a> on <b>Top</b>. <a href="/wiki/Multicategory_(category_theory)" class="mw-redirect" title="Multicategory (category theory)">Multicategories</a> with one object are operads. <a href="/wiki/PROP_(category_theory)" title="PROP (category theory)">PROPs</a> generalize operads to admit operations with several inputs and several outputs. Operads are used in defining <a href="/wiki/Opetope" title="Opetope">opetopes</a>, higher category theory, homotopy theory, homological algebra, algebraic geometry, string theory and many other areas. </td></tr> <tr> <td>1969</td> <td><a href="/wiki/Max_Kelly" title="Max Kelly">Max Kelly</a>-<a href="/wiki/Nobuo_Yoneda" title="Nobuo Yoneda">Nobuo Yoneda</a></td> <td><a href="/wiki/End_(category_theory)" title="End (category theory)">Ends and coends</a> </td></tr> <tr> <td>1969</td> <td><a href="/wiki/Pierre_Deligne" title="Pierre Deligne">Pierre Deligne</a>-<a href="/wiki/David_Mumford" title="David Mumford">David Mumford</a></td> <td><a href="/wiki/Deligne%E2%80%93Mumford_stacks" class="mw-redirect" title="Deligne–Mumford stacks">Deligne–Mumford stacks</a> as a generalization of <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> </td></tr> <tr> <td>1969</td> <td><a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a></td> <td><a href="/wiki/Doctrine_(mathematics)" class="mw-redirect" title="Doctrine (mathematics)">Doctrines (category theory)</a>, a doctrine is a monad on a 2-category </td></tr> <tr> <td>1970</td> <td><a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a>-<a href="/wiki/Myles_Tierney" title="Myles Tierney">Myles Tierney</a></td> <td><a href="/wiki/Elementary_topoi" class="mw-redirect" title="Elementary topoi">Elementary topoi</a>: Categories modeled after the category of sets which are like <a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">universes</a> (generalized spaces) of sets in which one can do mathematics. One of many ways to define a topos is: a properly <a href="/wiki/Cartesian_closed_category" title="Cartesian closed category">cartesian closed category</a> with a <a href="/wiki/Subobject_classifier" title="Subobject classifier">subobject classifier</a>. Every <a href="/wiki/Grothendieck_topos" class="mw-redirect" title="Grothendieck topos">Grothendieck topos</a> is an elementary topos </td></tr> <tr> <td>1970</td> <td><a href="/wiki/John_Horton_Conway" title="John Horton Conway">John Conway</a></td> <td><a href="/wiki/Skein_theory" class="mw-redirect" title="Skein theory">Skein theory</a> of <a href="/wiki/Knot_(mathematics)" title="Knot (mathematics)">knots</a>: The computation of knot invariants by <a href="/wiki/Skein_module" class="mw-redirect" title="Skein module">skein modules</a>. Skein modules can be based on <a href="/wiki/Quantum_invariant" title="Quantum invariant">quantum invariants</a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="1971–1980"><span id="1971.E2.80.931980"></span>1971–1980</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Timeline_of_category_theory_and_related_mathematics&action=edit&section=3" title="Edit section: 1971–1980"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable sortable" width="100%"> <tbody><tr> <th>Year </th> <th style="width:22%">Contributors </th> <th>Event </th></tr> <tr> <td>1971</td> <td><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a></td> <td>Influential book: <i>Categories for the Working Mathematician</i>, which became the standard reference in category theory </td></tr> <tr> <td>1971</td> <td><a href="/wiki/Horst_Herrlich" title="Horst Herrlich">Horst Herrlich</a>–<a href="/w/index.php?title=Oswald_Wyler&action=edit&redlink=1" class="new" title="Oswald Wyler (page does not exist)">Oswald Wyler</a></td> <td><a href="/wiki/Categorical_topology" class="mw-redirect" title="Categorical topology">Categorical topology</a>: The study of <a href="/wiki/Topological_category" title="Topological category">topological categories</a> of <a href="/w/index.php?title=Structured_set&action=edit&redlink=1" class="new" title="Structured set (page does not exist)">structured sets</a> (generalizations of topological spaces, <a href="/wiki/Uniform_space" title="Uniform space">uniform spaces</a> and the various other spaces in topology) and relations between them, culminating in <a href="/w/index.php?title=Universal_topology&action=edit&redlink=1" class="new" title="Universal topology (page does not exist)">universal topology</a>. General categorical topology study and uses structured sets in a topological category as general topology study and uses topological spaces. Algebraic categorical topology tries to apply the machinery of algebraic topology for topological spaces to structured sets in a topological category. </td></tr> <tr> <td>1971</td> <td><a href="/wiki/Harold_Neville_Vazeille_Temperley" class="mw-redirect" title="Harold Neville Vazeille Temperley">Harold Temperley</a>–<a href="/wiki/Elliott_Lieb" class="mw-redirect" title="Elliott Lieb">Elliott Lieb</a></td> <td><a href="/wiki/Temperley%E2%80%93Lieb_algebra" title="Temperley–Lieb algebra">Temperley–Lieb algebras</a>: Algebras of <a href="/wiki/Tangle_(mathematics)" title="Tangle (mathematics)">tangles</a> defined by generators of tangles and relations among them </td></tr> <tr> <td>1971</td> <td><a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a>–<a href="/wiki/Myles_Tierney" title="Myles Tierney">Myles Tierney</a></td> <td><a href="/wiki/Lawvere%E2%80%93Tierney_topology" title="Lawvere–Tierney topology">Lawvere–Tierney topology</a> on a topos </td></tr> <tr> <td>1971</td> <td><a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a>–<a href="/wiki/Myles_Tierney" title="Myles Tierney">Myles Tierney</a></td> <td><a href="/w/index.php?title=Topos_theoretic_forcing&action=edit&redlink=1" class="new" title="Topos theoretic forcing (page does not exist)">Topos theoretic forcing</a> (forcing in toposes): Categorization of the <a href="/wiki/Forcing_(mathematics)" title="Forcing (mathematics)">set theoretic forcing</a> method to toposes for attempts to prove or disprove the <a href="/wiki/Continuum_hypothesis" title="Continuum hypothesis">continuum hypothesis</a>, independence of the <a href="/wiki/Axiom_of_choice" title="Axiom of choice">axiom of choice</a>, etc. in toposes </td></tr> <tr> <td>1971</td> <td>Bob Walters–<a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td><a href="/w/index.php?title=Yoneda_structure&action=edit&redlink=1" class="new" title="Yoneda structure (page does not exist)">Yoneda structures</a> on 2-categories </td></tr> <tr> <td>1971</td> <td><a href="/wiki/Roger_Penrose" title="Roger Penrose">Roger Penrose</a></td> <td><a href="/wiki/String_diagram" title="String diagram">String diagrams</a> to manipulate morphisms in a monoidal category </td></tr> <tr> <td>1971</td> <td><a href="/wiki/Jean_Giraud_(mathematician)" title="Jean Giraud (mathematician)">Jean Giraud</a></td> <td><a href="/wiki/Gerbe" title="Gerbe">Gerbes</a>: Categorified principal bundles that are also special cases of stacks </td></tr> <tr> <td>1971</td> <td><a href="/wiki/Joachim_Lambek" title="Joachim Lambek">Joachim Lambek</a></td> <td>Generalizes the <a href="/wiki/Haskell%E2%80%93Curry%E2%80%93William%E2%80%93Howard_correspondence" class="mw-redirect" title="Haskell–Curry–William–Howard correspondence">Haskell–Curry–William–Howard correspondence</a> to a three way isomorphism between types, propositions and objects of a cartesian closed category </td></tr> <tr> <td>1972</td> <td><a href="/wiki/Max_Kelly" title="Max Kelly">Max Kelly</a></td> <td><a href="/w/index.php?title=Clubs_(category_theory)&action=edit&redlink=1" class="new" title="Clubs (category theory) (page does not exist)">Clubs (category theory)</a> and <a href="/wiki/Coherence_(category_theory)" class="mw-redirect" title="Coherence (category theory)">coherence (category theory)</a>. A club is a special kind of 2-dimensional theory or a monoid in <b>Cat</b>/(category of finite sets and permutations <i>P</i>), each club giving a 2-monad on <b>Cat</b> </td></tr> <tr> <td>1972</td> <td>John Isbell</td> <td><a href="/wiki/Complete_Heyting_algebra#Frames_and_locales" title="Complete Heyting algebra">Locales</a>: A "generalized topological space" or "pointless spaces" defined by a <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a> (a complete <a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a> also called a Brouwer lattice) just as for a topological space the open subsets form a lattice. If the lattice possess enough points it is a topological space. Locales are the main objects of <a href="/wiki/Pointless_topology" title="Pointless topology">pointless topology</a>, the dual objects being <a href="/wiki/Complete_Heyting_algebra#Frames_and_locales" title="Complete Heyting algebra">frames</a>. Both locales and frames form categories that are each other's <a href="/wiki/Opposite_category" title="Opposite category">opposite</a>. Sheaves can be defined over locales. The other "spaces" one can define sheaves over are sites. Although locales were known earlier John Isbell first named them </td></tr> <tr> <td>1972</td> <td><a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td><a href="/w/index.php?title=Formal_theory_of_monads&action=edit&redlink=1" class="new" title="Formal theory of monads (page does not exist)">Formal theory of monads</a>: The theory of <a href="/wiki/Monad_(category_theory)" title="Monad (category theory)">monads</a> in 2-categories </td></tr> <tr> <td>1972</td> <td><a href="/wiki/Peter_Freyd" class="mw-redirect" title="Peter Freyd">Peter Freyd</a></td> <td><a href="/wiki/Fundamental_theorem_of_topos_theory" title="Fundamental theorem of topos theory">Fundamental theorem of topos theory</a>: Every <a href="/wiki/Slice_category" class="mw-redirect" title="Slice category">slice category</a> (<i>E</i>,<i>Y</i>) of a topos <i>E</i> is a topos and the functor <i>f</i>*: (<i>E</i>,<i>X</i>) → (<i>E</i>,<i>Y</i>) preserves <a href="/wiki/Exponential_object" title="Exponential object">exponentials</a> and the subobject classifier object Ω and has a right and left adjoint functor </td></tr> <tr> <td>1972</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Grothendieck_universe" title="Grothendieck universe">Grothendieck universes</a> for sets as part of <a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">foundations</a> for categories </td></tr> <tr> <td>1972</td> <td><a href="/wiki/Jean_B%C3%A9nabou" title="Jean Bénabou">Jean Bénabou</a>–<a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td><a href="/wiki/Cosmos_(category_theory)" title="Cosmos (category theory)">Cosmoses</a> which categorize <a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">universes</a>: A cosmos is a generalized universe of 1-categories in which you can do category theory. When set theory is generalized to the study of a <a href="/wiki/Topos" title="Topos">Grothendieck topos</a>, the analogous generalization of category theory is the study of a cosmos. <ol><li>Ross Street definition: A <a href="/wiki/Bicategory" title="Bicategory">bicategory</a> such that</li> <li>small bicoproducts exist;</li> <li>each <a href="/wiki/Monad_(category_theory)" title="Monad (category theory)">monad</a> admits a <a href="/wiki/Kleisli_category" title="Kleisli category">Kleisli construction</a> (analogous to the quotient of an equivalence relation in a topos);</li> <li>it is locally small-cocomplete; and</li> <li>there exists a small <a href="/wiki/Generator_(category_theory)" title="Generator (category theory)">Cauchy generator</a>.</li></ol> <p>Cosmoses are closed under dualization, parametrization and localization. Ross Street also introduces <a href="/wiki/Cosmos_(category_theory)" title="Cosmos (category theory)">elementary cosmoses</a>. </p><p>Jean Bénabou definition: A bicomplete <a href="/wiki/Symmetric_monoidal_closed_category" class="mw-redirect" title="Symmetric monoidal closed category">symmetric monoidal closed category</a> </p> </td></tr> <tr> <td>1972</td> <td>William Mitchell–<a href="/wiki/Jean_B%C3%A9nabou" title="Jean Bénabou">Jean Bénabou</a></td> <td><a href="/wiki/Mitchell%E2%80%93B%C3%A9nabou_internal_language" class="mw-redirect" title="Mitchell–Bénabou internal language">Mitchell–Bénabou internal language</a> of a <a href="/wiki/Topos" title="Topos">toposes</a>: For a topos <i>E</i> with <a href="/wiki/Subobject_classifier" title="Subobject classifier">subobject classifier</a> object Ω a language (or <a href="/wiki/Type_theory" title="Type theory">type theory</a>) L(<i>E</i>) where: <ol><li>the types are the objects of <i>E</i></li> <li>terms of type <i>X</i> in the variables <i>x</i><sub><i>i</i></sub> of type <i>X</i><sub><i>i</i></sub> are polynomial expressions φ(<i>x</i><sub>1</sub>,...,<i>x</i><sub><i>m</i></sub>): 1→<i>X</i> in the arrows <i>x</i><sub><i>i</i></sub>: 1→<i>X</i><sub><i>i</i></sub> in <i>E</i></li> <li>formulas are terms of type Ω (arrows from types to Ω)</li> <li>connectives are induced from the internal <a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a> structure of Ω</li> <li>quantifiers bounded by types and applied to formulas are also treated</li> <li>for each type <i>X</i> there are also two binary relations =<sub><i>X</i></sub> (defined applying the diagonal map to the product term of the arguments) and ∈<sub><i>X</i></sub> (defined applying the evaluation map to the product of the term and the power term of the arguments).</li></ol> <p>A formula is true if the arrow which interprets it factor through the arrow true:1→Ω. The Mitchell-Bénabou internal language is a powerful way to describe various objects in a topos as if they were sets and hence is a way of making the topos into a generalized set theory, to write and prove statements in a topos using first order intuitionistic predicate logic, to consider toposes as type theories and to express properties of a topos. Any language L also generates a <a href="/w/index.php?title=Linguistic_topos&action=edit&redlink=1" class="new" title="Linguistic topos (page does not exist)">linguistic topos</a> <i>E</i>(L) </p> </td></tr> <tr> <td>1973</td> <td>Chris Reedy</td> <td><a href="/w/index.php?title=Reedy_category&action=edit&redlink=1" class="new" title="Reedy category (page does not exist)">Reedy categories</a>: Categories of "shapes" that can be used to do homotopy theory. A Reedy category is a category <b>R</b> equipped with a structure enabling the inductive construction of diagrams and natural transformations of shape <b>R</b>. The most important consequence of a Reedy structure on <b>R</b> is the existence of a model structure on the <a href="/wiki/Functor_category" title="Functor category">functor category</a> <b>M</b><sup><b>R</b></sup> whenever <b>M</b> is a <a href="/wiki/Model_category" title="Model category">model category</a>. Another advantage of the Reedy structure is that its cofibrations, fibrations and factorizations are explicit. In a Reedy category there is a notion of an injective and a surjective morphism such that any morphism can be factored uniquely as a surjection followed by an injection. Examples are the <a href="/wiki/Ordinal_number" title="Ordinal number">ordinal</a> α considered as a <a href="/wiki/Partially_ordered_set" title="Partially ordered set">poset</a> and hence a category. The opposite <b>R</b>° of a Reedy category <b>R</b> is also a Reedy category. The <a href="/wiki/Simplex_category" title="Simplex category">simplex category</a> <b>Δ</b> and more generally for any <a href="/wiki/Simplicial_set" title="Simplicial set">simplicial set</a> <i>X</i> its category of simplices <b>Δ</b>/<i>X</i> is a Reedy category. The model structure on <b>M</b><sup><b>Δ</b></sup> for a model category <b>M</b> is described in an unpublished manuscript by Chris Reedy </td></tr> <tr> <td>1973</td> <td><a href="/wiki/Kenneth_Brown_(mathematician)" title="Kenneth Brown (mathematician)">Kenneth Brown</a>–Stephen Gersten</td> <td>Shows the existence of a global closed <a href="/wiki/Model_category" title="Model category">model structure</a> on the category of <a href="/wiki/Simplicial_sheaf" class="mw-redirect" title="Simplicial sheaf">simplicial sheaves</a> on a topological space, with weak assumptions on the topological space </td></tr> <tr> <td>1973</td> <td><a href="/wiki/Kenneth_Brown_(mathematician)" title="Kenneth Brown (mathematician)">Kenneth Brown</a></td> <td><a href="/w/index.php?title=Generalized_sheaf_cohomology&action=edit&redlink=1" class="new" title="Generalized sheaf cohomology (page does not exist)">Generalized sheaf cohomology</a> of a topological space <i>X</i> with coefficients a sheaf on <i>X</i> with values in Kans <a href="/wiki/Spectrum_(homotopy_theory)" class="mw-redirect" title="Spectrum (homotopy theory)">category of spectra</a> with some finiteness conditions. It generalizes <a href="/wiki/Generalized_cohomology_theory" class="mw-redirect" title="Generalized cohomology theory">generalized cohomology theory</a> and <a href="/wiki/Sheaf_cohomology" title="Sheaf cohomology">sheaf cohomology</a> with coefficients in a complex of abelian sheaves </td></tr> <tr> <td>1973</td> <td><a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a></td> <td>Finds that <a href="/wiki/Cauchy_completeness" class="mw-redirect" title="Cauchy completeness">Cauchy completeness</a> can be expressed for general <a href="/wiki/Enriched_category" title="Enriched category">enriched categories</a> with the <a href="/w/index.php?title=Category_of_generalized_metric_spaces&action=edit&redlink=1" class="new" title="Category of generalized metric spaces (page does not exist)">category of generalized metric spaces</a> as a special case. Cauchy sequences become left adjoint modules and convergence become representability </td></tr> <tr> <td>1973</td> <td><a href="/wiki/Jean_B%C3%A9nabou" title="Jean Bénabou">Jean Bénabou</a></td> <td><a href="/wiki/Profunctor" title="Profunctor">Distributors</a> (also called modules, profunctors, <a href="/w/index.php?title=Categorical_bridge&action=edit&redlink=1" class="new" title="Categorical bridge (page does not exist)">directed bridges</a>) </td></tr> <tr> <td>1973</td> <td><a href="/wiki/Pierre_Deligne" title="Pierre Deligne">Pierre Deligne</a></td> <td>Proves the last of the <a href="/wiki/Weil_conjectures" title="Weil conjectures">Weil conjectures</a>, the analogue of the Riemann hypothesis </td></tr> <tr> <td>1973</td> <td><a href="/wiki/Michael_Boardman" title="Michael Boardman">Michael Boardman</a>–Rainer Vogt</td> <td><a href="/wiki/Segal_categories" class="mw-redirect" title="Segal categories">Segal categories</a>: Simplicial analogues of <a href="/wiki/Fukaya_category" title="Fukaya category">A</a><sub><a href="/wiki/Fukaya_category" title="Fukaya category">∞</a></sub><a href="/wiki/Fukaya_category" title="Fukaya category">-categories</a>. They naturally generalize <a href="/wiki/Simplicial_object" class="mw-redirect" title="Simplicial object">simplicial categories</a>, in that they can be regarded as simplicial categories with composition only given up to homotopy.<br /> <p>Def: A <a href="/wiki/Simplicial_space" title="Simplicial space">simplicial space</a> <i>X</i> such that <i>X</i><sub>0</sub> (the set of points) is a discrete <a href="/wiki/Simplicial_set" title="Simplicial set">simplicial set</a> and the <a href="/wiki/Segal_map" class="mw-redirect" title="Segal map">Segal map</a> </p> <dl><dd>φ<sub><i>k</i></sub> : <i>X</i><sub><i>k</i></sub> → <i>X</i><sub>1</sub> × <sub> <i>X</i><sub>0</sub></sub> ... × <sub> <i>X</i><sub>0</sub></sub> <i>X</i><sub>1</sub> (induced by <i>X</i>(α<sub><i>i</i></sub>): <i>X</i><sub><i>k</i></sub> → <i>X</i><sub>1</sub>) assigned to <i>X</i></dd></dl> <p>is a weak equivalence of simplicial sets for <i>k</i> ≥ 2. </p><p>Segal categories are a weak form of <a href="/w/index.php?title=S-category&action=edit&redlink=1" class="new" title="S-category (page does not exist)">S-categories</a>, in which composition is only defined up to a coherent system of equivalences.<br /> Segal categories were defined one year later implicitly by <a href="/wiki/Graeme_Segal" title="Graeme Segal">Graeme Segal</a>. They were named Segal categories first by William Dwyer–<a href="/wiki/Daniel_Kan" title="Daniel Kan">Daniel Kan</a>–Jeffrey Smith 1989. In their famous book Homotopy invariant algebraic structures on topological spaces, J. Michael Boardman and Rainer Vogt called them <a href="/wiki/Quasi-category" title="Quasi-category">quasi-categories</a>. A quasi-category is a simplicial set satisfying the weak Kan condition, so quasi-categories are also called <a href="/wiki/Weak_Kan_complex" class="mw-redirect" title="Weak Kan complex">weak Kan complexes</a> </p> </td></tr> <tr> <td>1973</td> <td><a href="/wiki/Daniel_Quillen" title="Daniel Quillen">Daniel Quillen</a></td> <td><a href="/wiki/Frobenius_categories" class="mw-redirect" title="Frobenius categories">Frobenius categories</a>: An <a href="/wiki/Exact_category" title="Exact category">exact category</a> in which the classes of <a href="/wiki/Injective_object" title="Injective object">injective</a> and <a href="/wiki/Projective_object" title="Projective object">projective objects</a> coincide and for all objects <i>x</i> in the category there is a deflation P(<i>x</i>)→<i>x</i> (the projective cover of x) and an inflation <i>x</i>→I(<i>x</i>) (the injective hull of <i>x</i>) such that both P(x) and I(<i>x</i>) are in the category of pro/injective objects. A Frobenius category <b>E</b> is an example of a <a href="/wiki/Model_category" title="Model category">model category</a> and the quotient <b>E</b>/P (P is the class of projective/injective objects) is its <a href="/wiki/Homotopy_category" title="Homotopy category">homotopy category</a> <b>hE</b> </td></tr> <tr> <td>1974</td> <td><a href="/wiki/Michael_Artin" title="Michael Artin">Michael Artin</a></td> <td>Generalizes <a href="/wiki/Deligne%E2%80%93Mumford_stacks" class="mw-redirect" title="Deligne–Mumford stacks">Deligne–Mumford stacks</a> to <a href="/wiki/Artin_stacks" class="mw-redirect" title="Artin stacks">Artin stacks</a> </td></tr> <tr> <td>1974</td> <td>Robert Paré</td> <td><a href="/w/index.php?title=Par%C3%A9_monadicity_theorem&action=edit&redlink=1" class="new" title="Paré monadicity theorem (page does not exist)">Paré monadicity theorem</a>: <b>E</b> is a topos → <b>E</b>° is monadic over <b>E</b> </td></tr> <tr> <td>1974</td> <td><a href="/wiki/Andy_Magid" title="Andy Magid">Andy Magid</a></td> <td>Generalizes <a href="/wiki/Grothendieck%27s_Galois_theory" title="Grothendieck's Galois theory">Grothendieck's Galois theory</a> from <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">groups</a> to the case of rings using Galois groupoids </td></tr> <tr> <td>1974</td> <td><a href="/wiki/Jean_B%C3%A9nabou" title="Jean Bénabou">Jean Bénabou</a></td> <td>Logic of <a href="/wiki/Fibred_category" title="Fibred category">fibred categories</a> </td></tr> <tr> <td>1974</td> <td><a href="/wiki/John_Gray_(mathematician)" title="John Gray (mathematician)">John Gray</a></td> <td><a href="/w/index.php?title=Gray_category&action=edit&redlink=1" class="new" title="Gray category (page does not exist)">Gray categories</a> with <a href="/w/index.php?title=Gray_tensor_product&action=edit&redlink=1" class="new" title="Gray tensor product (page does not exist)">Gray tensor product</a> </td></tr> <tr> <td>1974</td> <td><a href="/wiki/Kenneth_Brown_(mathematician)" title="Kenneth Brown (mathematician)">Kenneth Brown</a></td> <td>Writes a very influential paper that defines <a href="/w/index.php?title=Brown_category&action=edit&redlink=1" class="new" title="Brown category (page does not exist)">Browns categories</a> of fibrant objects and dually Brown categories of cofibrant objects </td></tr> <tr> <td>1974</td> <td><a href="/wiki/Shiing-Shen_Chern" title="Shiing-Shen Chern">Shiing-Shen Chern</a>–<a href="/wiki/James_Simons" class="mw-redirect" title="James Simons">James Simons</a></td> <td><a href="/wiki/Chern%E2%80%93Simons_theory" title="Chern–Simons theory">Chern–Simons theory</a>: A particular TQFT which describe knot and <a href="/wiki/Manifold" title="Manifold">manifold</a> invariants, at that time only in 3D </td></tr> <tr> <td>1975</td> <td><a href="/wiki/Saul_Kripke" title="Saul Kripke">Saul Kripke</a>–<a href="/wiki/Andr%C3%A9_Joyal" title="André Joyal">André Joyal</a></td> <td><a href="/wiki/Kripke%E2%80%93Joyal_semantics" class="mw-redirect" title="Kripke–Joyal semantics">Kripke–Joyal semantics</a> of the <a href="/w/index.php?title=Mitchell%E2%80%93Benabou_internal_language&action=edit&redlink=1" class="new" title="Mitchell–Benabou internal language (page does not exist)">Mitchell–Bénabou internal language</a> for toposes: The logic in categories of sheaves is first-order intuitionistic predicate logic </td></tr> <tr> <td>1975</td> <td>Radu Diaconescu</td> <td><a href="/wiki/Diaconescu_theorem" class="mw-redirect" title="Diaconescu theorem">Diaconescu theorem</a>: The internal axiom of choice holds in a <a href="/wiki/Topos" title="Topos">topos</a> → the topos is a boolean topos. So in IZF the axiom of choice implies the <a href="/wiki/Law_of_excluded_middle" title="Law of excluded middle">law of excluded middle</a> </td></tr> <tr> <td>1975</td> <td>Manfred Szabo</td> <td><a href="/w/index.php?title=Polycategory&action=edit&redlink=1" class="new" title="Polycategory (page does not exist)">Polycategories</a> </td></tr> <tr> <td>1975</td> <td><a href="/wiki/William_Lawvere" title="William Lawvere">William Lawvere</a></td> <td>Observes that <a href="/w/index.php?title=Deligne%27s_theorem_on_topos&action=edit&redlink=1" class="new" title="Deligne's theorem on topos (page does not exist)">Deligne's theorem</a> about enough points in a <a href="/wiki/Coherent_topos" title="Coherent topos">coherent topos</a> implies the <a href="/wiki/G%C3%B6del%27s_completeness_theorem" title="Gödel's completeness theorem">Gödel completeness theorem</a> for first-order logic in that topos </td></tr> <tr> <td>1976</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/w/index.php?title=Schematic_homotopy_type&action=edit&redlink=1" class="new" title="Schematic homotopy type (page does not exist)">Schematic homotopy types</a> </td></tr> <tr> <td>1976</td> <td>Marcel Crabbe</td> <td><a href="/w/index.php?title=Heyting_category&action=edit&redlink=1" class="new" title="Heyting category (page does not exist)">Heyting categories</a> also called <a href="/w/index.php?title=Logos_(mathematics)&action=edit&redlink=1" class="new" title="Logos (mathematics) (page does not exist)">logoses</a>: <a href="/wiki/Regular_category" title="Regular category">Regular categories</a> in which the subobjects of an object form a lattice, and in which each inverse image map has a right adjoint. More precisely a <a href="/w/index.php?title=Coherent_category&action=edit&redlink=1" class="new" title="Coherent category (page does not exist)">coherent category</a> <span class="texhtml">C</span> such that for all morphisms <i>f</i>:<i>A</i>→<i>B</i> in <span class="texhtml">C</span> the functor <i>f</i>*:Sub<sub><span class="texhtml">C</span></sub>(<i>B</i>)→Sub<sub><span class="texhtml">C</span></sub>(<i>A</i>) has a left adjoint and a right adjoint. Sub<sub><span class="texhtml">C</span></sub>(<i>A</i>) is the <a href="/wiki/Preorder" title="Preorder">preorder</a> of subobjects of <i>A</i> (the full subcategory of <span class="texhtml">C</span>/<i>A</i> whose objects are subobjects of <i>A</i>) in <span class="texhtml">C</span>. Every <a href="/wiki/Topos" title="Topos">topos</a> is a logos. Heyting categories generalize <a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebras</a>. </td></tr> <tr> <td>1976</td> <td><a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td><a href="/wiki/Computad" class="mw-redirect" title="Computad">Computads</a> </td></tr> <tr> <td>1977</td> <td><a href="/wiki/Michael_Makkai" title="Michael Makkai">Michael Makkai</a>–Gonzalo Reyes</td> <td>Develops the <a href="/wiki/Mitchell%E2%80%93B%C3%A9nabou_internal_language" class="mw-redirect" title="Mitchell–Bénabou internal language">Mitchell–Bénabou internal language</a> of a topos thoroughly in a more general setting </td></tr> <tr> <td>1977</td> <td>Andre Boileau–<a href="/wiki/Andr%C3%A9_Joyal" title="André Joyal">André Joyal</a>–John Zangwill</td> <td>LST, <a href="/w/index.php?title=Local_set_theory&action=edit&redlink=1" class="new" title="Local set theory (page does not exist)">local set theory</a>: Local set theory is a <a href="/wiki/Typed_set_theory" class="mw-redirect" title="Typed set theory">typed set theory</a> whose underlying logic is higher-order <a href="/wiki/Intuitionistic_logic" title="Intuitionistic logic">intuitionistic logic</a>. It is a generalization of classical set theory, in which sets are replaced by terms of certain types. The category C(S) built out of a local theory S whose objects are the local sets (or S-sets) and whose arrows are the local maps (or S-maps) is a <a href="/w/index.php?title=Linguistic_topos&action=edit&redlink=1" class="new" title="Linguistic topos (page does not exist)">linguistic topos</a>. Every topos <b>E</b> is equivalent to a linguistic topos C(S(<b>E</b>)) </td></tr> <tr> <td>1977</td> <td><a href="/w/index.php?title=John_Roberts_(mathematician)&action=edit&redlink=1" class="new" title="John Roberts (mathematician) (page does not exist)">John Roberts</a></td> <td>Introduces most general <a href="/wiki/Nonabelian_cohomology" title="Nonabelian cohomology">nonabelian cohomology</a> of ω-categories with ω-categories as coefficients when he realized that general cohomology is about coloring simplices in <a href="/w/index.php?title=%CE%A9-category&action=edit&redlink=1" class="new" title="Ω-category (page does not exist)">ω-categories</a>. There are two methods of constructing general nonabelian cohomology, as <a href="/wiki/Nonabelian_sheaf_cohomology" class="mw-redirect" title="Nonabelian sheaf cohomology">nonabelian sheaf cohomology</a> in terms of <a href="/wiki/Descent_(category_theory)" class="mw-redirect" title="Descent (category theory)">descent</a> for ω-category valued sheaves, and in terms of <a href="/w/index.php?title=Homotopical_cohomology_theory&action=edit&redlink=1" class="new" title="Homotopical cohomology theory (page does not exist)">homotopical cohomology theory</a> which realizes the cocycles. The two approaches are related by <a href="/wiki/Descent_(category_theory)" class="mw-redirect" title="Descent (category theory)">codescent</a> </td></tr> <tr> <td>1978</td> <td><a href="/w/index.php?title=John_Roberts_(mathematician)&action=edit&redlink=1" class="new" title="John Roberts (mathematician) (page does not exist)">John Roberts</a></td> <td><a href="/w/index.php?title=Complicial_set&action=edit&redlink=1" class="new" title="Complicial set (page does not exist)">Complicial sets</a> (simplicial sets with structure or enchantment) </td></tr> <tr> <td>1978</td> <td>Francois Bayen–Moshe Flato–Chris Fronsdal–<a href="/wiki/Andr%C3%A9_Lichnerowicz" title="André Lichnerowicz">André Lichnerowicz</a>–Daniel Sternheimer</td> <td><a href="/wiki/Weyl_quantization#Deformation_quantization" class="mw-redirect" title="Weyl quantization">Deformation quantization</a>, later to be a part of categorical quantization </td></tr> <tr> <td>1978</td> <td><a href="/wiki/Andr%C3%A9_Joyal" title="André Joyal">André Joyal</a></td> <td><a href="/wiki/Combinatorial_species" title="Combinatorial species">Combinatorial species</a> in <a href="/wiki/Enumerative_combinatorics" title="Enumerative combinatorics">enumerative combinatorics</a> </td></tr> <tr> <td>1978</td> <td>Don Anderson</td> <td>Building on work of <a href="/wiki/Kenneth_Brown_(mathematician)" title="Kenneth Brown (mathematician)">Kenneth Brown</a> defines <a href="/w/index.php?title=ABC_categories&action=edit&redlink=1" class="new" title="ABC categories (page does not exist)">ABC (co)fibration categories</a> for doing homotopy theory and more general <a href="/w/index.php?title=ABC_model_category&action=edit&redlink=1" class="new" title="ABC model category (page does not exist)">ABC model categories</a>, but the theory lies dormant until 2003. Every <a href="/wiki/Model_category" title="Model category">Quillen model category</a> is an ABC model category. A difference to Quillen model categories is that in ABC model categories fibrations and cofibrations are independent and that for an ABC model category M<sup>D</sup> is an ABC model category. To an ABC (co)fibration category is canonically associated a (left) right <a href="/wiki/Derivator" title="Derivator">Heller derivator</a>. Topological spaces with homotopy equivalences as weak equivalences, Hurewicz cofibrations as cofibrations and Hurewicz fibrations as fibrations form an ABC model category, the <a href="/w/index.php?title=Hurewicz_model_structure&action=edit&redlink=1" class="new" title="Hurewicz model structure (page does not exist)">Hurewicz model structure</a> on <b>Top</b>. Complexes of objects in an abelian category with quasi-isomorphisms as weak equivalences and monomorphisms as cofibrations form an ABC precofibration category </td></tr> <tr> <td>1979</td> <td>Don Anderson</td> <td><a href="/w/index.php?title=Anderson_axioms&action=edit&redlink=1" class="new" title="Anderson axioms (page does not exist)">Anderson axioms</a> for homotopy theory in categories with a <a href="/w/index.php?title=Fraction_functor&action=edit&redlink=1" class="new" title="Fraction functor (page does not exist)">fraction functor</a> </td></tr> <tr> <td>1980</td> <td><a href="/wiki/Alexander_Zamolodchikov" title="Alexander Zamolodchikov">Alexander Zamolodchikov</a></td> <td><a href="/wiki/Zamolodchikov_equation" class="mw-redirect" title="Zamolodchikov equation">Zamolodchikov equation</a> also called <a href="/w/index.php?title=Tetrahedron_equation&action=edit&redlink=1" class="new" title="Tetrahedron equation (page does not exist)">tetrahedron equation</a> </td></tr> <tr> <td>1980</td> <td><a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td>Bicategorical <a href="/wiki/Yoneda_lemma" title="Yoneda lemma">Yoneda lemma</a> </td></tr> <tr> <td>1980</td> <td><a href="/wiki/Masaki_Kashiwara" title="Masaki Kashiwara">Masaki Kashiwara</a>–Zoghman Mebkhout</td> <td>Proves the <a href="/wiki/Riemann%E2%80%93Hilbert_correspondence" title="Riemann–Hilbert correspondence">Riemann–Hilbert correspondence</a> for <a href="/wiki/Complex_manifold" title="Complex manifold">complex manifolds</a> </td></tr> <tr> <td>1980</td> <td><a href="/wiki/Peter_Freyd" class="mw-redirect" title="Peter Freyd">Peter Freyd</a></td> <td><a href="/w/index.php?title=Numerals_(topos_theory)&action=edit&redlink=1" class="new" title="Numerals (topos theory) (page does not exist)">Numerals</a> in a topos </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="1981–1990"><span id="1981.E2.80.931990"></span>1981–1990</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Timeline_of_category_theory_and_related_mathematics&action=edit&section=4" title="Edit section: 1981–1990"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable sortable" width="100%"> <tbody><tr> <th>Year </th> <th style="width:22%">Contributors </th> <th>Event </th></tr> <tr> <td>1981</td> <td><a href="/wiki/Shigeru_Mukai" title="Shigeru Mukai">Shigeru Mukai</a></td> <td><a href="/wiki/Mukai%E2%80%93Fourier_transform" class="mw-redirect" title="Mukai–Fourier transform">Mukai–Fourier transform</a> </td></tr> <tr> <td>1982</td> <td>Bob Walters</td> <td><a href="/wiki/Enriched_category" title="Enriched category">Enriched categories</a> with bicategories as a base </td></tr> <tr> <td>1983</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Pursuing_stacks" class="mw-redirect" title="Pursuing stacks">Pursuing stacks</a>: Manuscript circulated from Bangor, written in English in response to a correspondence in English with <a href="/wiki/Ronald_Brown_(mathematician)" title="Ronald Brown (mathematician)">Ronald Brown</a> and <a href="/wiki/Tim_Porter" title="Tim Porter">Tim Porter</a>, starting with a letter addressed to <a href="/wiki/Daniel_Quillen" title="Daniel Quillen">Daniel Quillen</a>, developing mathematical visions in a 629 pages manuscript, a kind of diary, and to be published by the Société Mathématique de France, edited by G. Maltsiniotis. </td></tr> <tr> <td>1983</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td>First appearance of <a href="/wiki/Strict_%E2%88%9E-categories" class="mw-redirect" title="Strict ∞-categories">strict ∞-categories</a> in pursuing stacks, following a 1981 published definition by <a href="/wiki/Ronald_Brown_(mathematician)" title="Ronald Brown (mathematician)">Ronald Brown</a> and <a href="/w/index.php?title=Philip_J._Higgins&action=edit&redlink=1" class="new" title="Philip J. Higgins (page does not exist)">Philip J. Higgins</a>. </td></tr> <tr> <td>1983</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Fundamental_infinity_groupoid" class="mw-redirect" title="Fundamental infinity groupoid">Fundamental infinity groupoid</a>: A complete homotopy invariant Π<sub>∞</sub>(<i>X</i>) for CW-complexes <i>X</i>. The inverse functor is the <a href="/wiki/Abstract_simplicial_complex" title="Abstract simplicial complex">geometric realization functor</a> | . | and together they form an "equivalence" between the <a href="/wiki/Category_of_CW-complexes" class="mw-redirect" title="Category of CW-complexes">category of CW-complexes</a> and the category of ω-groupoids </td></tr> <tr> <td>1983</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Homotopy_hypothesis" title="Homotopy hypothesis">Homotopy hypothesis</a>: The <a href="/wiki/Homotopy_category" title="Homotopy category">homotopy category</a> of CW-complexes is <a href="/wiki/Quillen_adjunction" title="Quillen adjunction">Quillen equivalent</a> to a homotopy category of reasonable weak <a href="/wiki/%E2%88%9E-groupoid" title="∞-groupoid">∞-groupoids</a> </td></tr> <tr> <td>1983</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Derivator" title="Derivator">Grothendieck derivators</a>: A model for homotopy theory similar to <a href="/wiki/Model_category" title="Model category">Quilen model categories</a> but more satisfactory. Grothendieck derivators are dual to <a href="/wiki/Derivator" title="Derivator">Heller derivators</a> </td></tr> <tr> <td>1983</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/w/index.php?title=Elementary_modelizer&action=edit&redlink=1" class="new" title="Elementary modelizer (page does not exist)">Elementary modelizers</a>: Categories of presheaves that modelize <a href="/wiki/Homotopy_type" class="mw-redirect" title="Homotopy type">homotopy types</a> (thus generalizing the theory of <a href="/wiki/Simplicial_set" title="Simplicial set">simplicial sets</a>). <a href="/w/index.php?title=Canonical_modelizer&action=edit&redlink=1" class="new" title="Canonical modelizer (page does not exist)">Canonical modelizers</a> are also used in pursuing stacks </td></tr> <tr> <td>1983</td> <td><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td> <td><a href="/wiki/Smooth_functor" title="Smooth functor">Smooth functors</a> and <a href="/w/index.php?title=Proper_functor&action=edit&redlink=1" class="new" title="Proper functor (page does not exist)">proper functors</a> </td></tr> <tr> <td>1984</td> <td>Vladimir Bazhanov–Razumov Stroganov</td> <td><a href="/w/index.php?title=Bazhanov%E2%80%93Stroganov_d-simplex_equation&action=edit&redlink=1" class="new" title="Bazhanov–Stroganov d-simplex equation (page does not exist)">Bazhanov–Stroganov d-simplex equation</a> generalizing the Yang–Baxter equation and the Zamolodchikov equation </td></tr> <tr> <td>1984</td> <td><a href="/wiki/Horst_Herrlich" title="Horst Herrlich">Horst Herrlich</a></td> <td><a href="/w/index.php?title=Universal_topology&action=edit&redlink=1" class="new" title="Universal topology (page does not exist)">Universal topology</a> in <a href="/wiki/Categorical_topology" class="mw-redirect" title="Categorical topology">categorical topology</a>: A unifying categorical approach to the different structured sets (topological structures such as topological spaces and uniform spaces) whose class form a topological category similar as universal algebra is for algebraic structures </td></tr> <tr> <td>1984</td> <td><a href="/wiki/Andr%C3%A9_Joyal" title="André Joyal">André Joyal</a></td> <td><a href="/wiki/Simplicial_sheaf" class="mw-redirect" title="Simplicial sheaf">Simplicial sheaves</a> (sheaves with values in simplicial sets). Simplicial sheaves on a topological space <i>X</i> is a model for the <a href="/w/index.php?title=Hypercomplete_topos&action=edit&redlink=1" class="new" title="Hypercomplete topos (page does not exist)">hypercomplete</a> <a href="/wiki/%E2%88%9E-topos" title="∞-topos">∞-topos</a> Sh(<i>X</i>)<sup>^</sup> </td></tr> <tr> <td>1984</td> <td><a href="/wiki/Andr%C3%A9_Joyal" title="André Joyal">André Joyal</a></td> <td>Shows that the category of <a href="/wiki/Simplicial_object" class="mw-redirect" title="Simplicial object">simplicial objects</a> in a <a href="/wiki/Topos_(mathematics)#Grothendieck_topoi_.28topoi_in_geometry.29" class="mw-redirect" title="Topos (mathematics)">Grothendieck topos</a> has a closed <a href="/wiki/Model_category" title="Model category">model structure</a> </td></tr> <tr> <td>1984</td> <td><a href="/wiki/Andr%C3%A9_Joyal" title="André Joyal">André Joyal</a>–<a href="/wiki/Myles_Tierney" title="Myles Tierney">Myles Tierney</a></td> <td><a href="/w/index.php?title=Main_Galois_theorem_for_toposes&action=edit&redlink=1" class="new" title="Main Galois theorem for toposes (page does not exist)">Main Galois theorem for toposes</a>: Every topos is equivalent to a category of étale presheaves on an open étale groupoid </td></tr> <tr> <td>1985</td> <td>Michael Schlessinger–<a href="/wiki/Jim_Stasheff" title="Jim Stasheff">Jim Stasheff</a></td> <td>L<sub>∞</sub>-algebras </td></tr> <tr> <td>1985</td> <td><a href="/wiki/Andr%C3%A9_Joyal" title="André Joyal">André Joyal</a>–<a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td><a href="/wiki/Braided_monoidal_category" title="Braided monoidal category">Braided monoidal categories</a> </td></tr> <tr> <td>1985</td> <td><a href="/wiki/Andr%C3%A9_Joyal" title="André Joyal">André Joyal</a>–<a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td><a href="/w/index.php?title=Joyal%E2%80%93Street_coherence_theorem&action=edit&redlink=1" class="new" title="Joyal–Street coherence theorem (page does not exist)">Joyal–Street coherence theorem</a> for braided monoidal categories </td></tr> <tr> <td>1985</td> <td>Paul Ghez–Ricardo Lima–<a href="/w/index.php?title=John_Roberts_(mathematician)&action=edit&redlink=1" class="new" title="John Roberts (mathematician) (page does not exist)">John Roberts</a></td> <td><a href="/w/index.php?title=C*-category&action=edit&redlink=1" class="new" title="C*-category (page does not exist)">C*-categories</a> </td></tr> <tr> <td>1986</td> <td><a href="/wiki/Joachim_Lambek" title="Joachim Lambek">Joachim Lambek</a>–Phil Scott</td> <td>Influential book: Introduction to higher-order categorical logic </td></tr> <tr> <td>1986</td> <td><a href="/wiki/Joachim_Lambek" title="Joachim Lambek">Joachim Lambek</a>–Phil Scott</td> <td><a href="/w/index.php?title=Fundamental_theorem_of_topology&action=edit&redlink=1" class="new" title="Fundamental theorem of topology (page does not exist)">Fundamental theorem of topology</a>: The section-functor Γ and the germ-functor Λ establish a dual adjunction between the category of presheaves and the category of bundles (over the same topological space) which restricts to a dual equivalence of categories (or duality) between corresponding full subcategories of sheaves and of étale bundles </td></tr> <tr> <td>1986</td> <td><a href="/wiki/Peter_Freyd" class="mw-redirect" title="Peter Freyd">Peter Freyd</a>–<a href="/w/index.php?title=David_Yetter&action=edit&redlink=1" class="new" title="David Yetter (page does not exist)">David Yetter</a></td> <td>Constructs the (compact braided) monoidal <a href="/wiki/Category_of_tangles" class="mw-redirect" title="Category of tangles">category of tangles</a> </td></tr> <tr> <td>1986</td> <td><a href="/wiki/Vladimir_Drinfeld" title="Vladimir Drinfeld">Vladimir Drinfeld</a>–<a href="/wiki/Michio_Jimbo" title="Michio Jimbo">Michio Jimbo</a></td> <td><a href="/wiki/Quantum_groups" class="mw-redirect" title="Quantum groups">Quantum groups</a>: In other words, quasitriangular <a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebras</a>. The point is that the categories of representations of quantum groups are <a href="/wiki/Monoidal_category" title="Monoidal category">tensor categories</a> with extra structure. They are used in construction of <a href="/wiki/Quantum_invariant" title="Quantum invariant">quantum invariants</a> of knots and links and low-dimensional manifolds, representation theory, <a href="/w/index.php?title=Q-deformation_theory&action=edit&redlink=1" class="new" title="Q-deformation theory (page does not exist)">q-deformation theory</a>, <a href="/wiki/Conformal_field_theory" title="Conformal field theory">CFT</a>, <a href="/wiki/Integrable_system" title="Integrable system">integrable systems</a>. The invariants are constructed from <a href="/wiki/Braided_monoidal_category" title="Braided monoidal category">braided monoidal categories</a> that are categories of representations of quantum groups. The underlying structure of a <a href="/wiki/TQFT" class="mw-redirect" title="TQFT">TQFT</a> is a <a href="/w/index.php?title=Modular_category&action=edit&redlink=1" class="new" title="Modular category (page does not exist)">modular category</a> of representations of a quantum group </td></tr> <tr> <td>1986</td> <td><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a></td> <td><a href="/wiki/Mathematics,_Form_and_Function" title="Mathematics, Form and Function">Mathematics, form and function</a> (a foundation of mathematics) </td></tr> <tr> <td>1987</td> <td><a href="/wiki/Jean-Yves_Girard" title="Jean-Yves Girard">Jean-Yves Girard</a></td> <td><a href="/wiki/Linear_logic" title="Linear logic">Linear logic</a>: The internal logic of a <a href="/wiki/Preadditive_category#R-linear_categories" title="Preadditive category">linear category</a> (an <a href="/wiki/Enriched_category" title="Enriched category">enriched category</a> with its <a href="/wiki/Hom-set" class="mw-redirect" title="Hom-set">Hom-sets</a> being <a href="/wiki/Vector_space" title="Vector space">linear spaces</a>) </td></tr> <tr> <td>1987</td> <td><a href="/wiki/Peter_Freyd" class="mw-redirect" title="Peter Freyd">Peter Freyd</a></td> <td><a href="/w/index.php?title=Freyd_representation_theorem&action=edit&redlink=1" class="new" title="Freyd representation theorem (page does not exist)">Freyd representation theorem</a> for <a href="/wiki/Topos#Grothendieck_topoi_(topoi_in_geometry)" title="Topos">Grothendieck toposes</a> </td></tr> <tr> <td>1987</td> <td><a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td>Definition of the <a href="/wiki/Nerve_of_a_category" class="mw-redirect" title="Nerve of a category">nerve of a weak <i>n</i>-category</a> and thus obtaining the first definition of <a href="/wiki/Weak_n-category" title="Weak n-category">weak <i>n</i>-category</a> using simplices </td></tr> <tr> <td>1987</td> <td><a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a>–<a href="/w/index.php?title=John_Roberts_(mathematician)&action=edit&redlink=1" class="new" title="John Roberts (mathematician) (page does not exist)">John Roberts</a></td> <td>Formulates <a href="/w/index.php?title=Street%E2%80%93Roberts_conjecture&action=edit&redlink=1" class="new" title="Street–Roberts conjecture (page does not exist)">Street–Roberts conjecture</a>: Strict <a href="/w/index.php?title=%CE%A9-category&action=edit&redlink=1" class="new" title="Ω-category (page does not exist)">ω-categories</a> are equivalent to <a href="/w/index.php?title=Complicial_set&action=edit&redlink=1" class="new" title="Complicial set (page does not exist)">complicial sets</a> </td></tr> <tr> <td>1987</td> <td><a href="/wiki/Andr%C3%A9_Joyal" title="André Joyal">André Joyal</a>–<a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a>–Mei Chee Shum</td> <td><a href="/wiki/Ribbon_category" title="Ribbon category">Ribbon categories</a>: A balanced rigid braided <a href="/wiki/Monoidal_category" title="Monoidal category">monoidal category</a> </td></tr> <tr> <td>1987</td> <td><a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td><a href="/w/index.php?title=N-computad&action=edit&redlink=1" class="new" title="N-computad (page does not exist)"><i>n</i>-computads</a> </td></tr> <tr> <td>1987</td> <td>Iain Aitchison</td> <td>Bottom up <a href="/w/index.php?title=Pascal_triangle_algorithm&action=edit&redlink=1" class="new" title="Pascal triangle algorithm (page does not exist)">Pascal triangle algorithm</a> for computing nonabelian <i>n</i>-cocycle conditions for <a href="/wiki/Nonabelian_cohomology" title="Nonabelian cohomology">nonabelian cohomology</a> </td></tr> <tr> <td>1987</td> <td><a href="/wiki/Vladimir_Drinfeld" title="Vladimir Drinfeld">Vladimir Drinfeld</a>-<a href="/wiki/G%C3%A9rard_Laumon" title="Gérard Laumon">Gérard Laumon</a></td> <td>Formulates <a href="/wiki/Langlands_program" title="Langlands program">geometric Langlands program</a> </td></tr> <tr> <td>1987</td> <td><a href="/wiki/Vladimir_Turaev" title="Vladimir Turaev">Vladimir Turaev</a></td> <td>Starts <a href="/wiki/Quantum_topology" title="Quantum topology">quantum topology</a> by using <a href="/wiki/Quantum_groups" class="mw-redirect" title="Quantum groups">quantum groups</a> and <a href="/wiki/R-matrix" title="R-matrix">R-matrices</a> to giving an algebraic unification of most of the known <a href="/wiki/Knot_polynomial" title="Knot polynomial">knot polynomials</a>. Especially important was <a href="/wiki/Vaughan_Jones" title="Vaughan Jones">Vaughan Jones</a> and <a href="/wiki/Edward_Witten" title="Edward Witten">Edward Wittens</a> work on the <a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a> </td></tr> <tr> <td>1988</td> <td><a href="/w/index.php?title=Alex_Heller&action=edit&redlink=1" class="new" title="Alex Heller (page does not exist)">Alex Heller</a></td> <td><a href="/w/index.php?title=Heller_axioms&action=edit&redlink=1" class="new" title="Heller axioms (page does not exist)">Heller axioms</a> for homotopy theory as a special abstract <a href="/w/index.php?title=Hyperfunctor&action=edit&redlink=1" class="new" title="Hyperfunctor (page does not exist)">hyperfunctor</a>. A feature of this approach is a very general <a href="/wiki/Localization_of_a_category" title="Localization of a category">localization</a> </td></tr> <tr> <td>1988</td> <td><a href="/w/index.php?title=Alex_Heller&action=edit&redlink=1" class="new" title="Alex Heller (page does not exist)">Alex Heller</a></td> <td><a href="/wiki/Derivator" title="Derivator">Heller derivators</a>, the dual of <a href="/wiki/Derivator" title="Derivator">Grothendieck derivators</a> </td></tr> <tr> <td>1988</td> <td><a href="/w/index.php?title=Alex_Heller&action=edit&redlink=1" class="new" title="Alex Heller (page does not exist)">Alex Heller</a></td> <td>Gives a global closed <a href="/wiki/Model_category" title="Model category">model structure</a> on the category of <a href="/wiki/Simplicial_sheaf" class="mw-redirect" title="Simplicial sheaf">simplicial presheaves</a>. John Jardine has also given a model structure in the category of simplicial presheaves </td></tr> <tr> <td>1988 </td> <td><a href="/wiki/Greg_Moore_(physicist)" title="Greg Moore (physicist)">Gregory Moore</a>-<a href="/wiki/Nathan_Seiberg" title="Nathan Seiberg">Nathan Seiberg</a> </td> <td><a href="/wiki/Two-dimensional_conformal_field_theory" title="Two-dimensional conformal field theory">Rational Conformal Field Theories</a> lead to modular tensor categories </td></tr> <tr> <td>1988</td> <td><a href="/wiki/Graeme_Segal" title="Graeme Segal">Graeme Segal</a></td> <td><a href="/w/index.php?title=Elliptic_object&action=edit&redlink=1" class="new" title="Elliptic object (page does not exist)">Elliptic objects</a>: A functor that is a categorified version of a vector bundle equipped with a connection, it is a 2D parallel transport for strings </td></tr> <tr> <td>1988</td> <td><a href="/wiki/Graeme_Segal" title="Graeme Segal">Graeme Segal</a></td> <td>Conformal field theory <a href="/wiki/Conformal_field_theory" title="Conformal field theory">CFT</a>: A symmetric monoidal functor Z: <b>nCob</b><sub><b>C</b></sub>→<b>Hilb</b> satisfying some axioms </td></tr> <tr> <td>1988</td> <td><a href="/wiki/Edward_Witten" title="Edward Witten">Edward Witten</a></td> <td>Topological quantum field theory <a href="/wiki/TQFT" class="mw-redirect" title="TQFT">TQFT</a>: A monoidal functor Z: <b>nCob</b>→<b>Hilb</b> satisfying some axioms </td></tr> <tr> <td>1988</td> <td><a href="/wiki/Edward_Witten" title="Edward Witten">Edward Witten</a></td> <td><a href="/wiki/Topological_string_theory" title="Topological string theory">Topological string theory</a> </td></tr> <tr> <td>1989</td> <td>Hans Baues</td> <td>Influential book: <a href="/wiki/Algebraic_homotopy" title="Algebraic homotopy">Algebraic homotopy</a> </td></tr> <tr> <td>1989</td> <td><a href="/wiki/Michael_Makkai" title="Michael Makkai">Michael Makkai</a>-Robert Paré</td> <td><a href="/wiki/Accessible_category" title="Accessible category">Accessible categories</a>: Categories with a "good" set of <a href="/wiki/Generator_(category_theory)" title="Generator (category theory)">generators</a> allowing to manipulate <a href="/wiki/Large_category" class="mw-redirect" title="Large category">large categories</a> as if they were <a href="/wiki/Small_category" class="mw-redirect" title="Small category">small categories</a>, without the fear of encountering any set-theoretic paradoxes. <a href="/wiki/Locally_presentable_category" class="mw-redirect" title="Locally presentable category">Locally presentable categories</a> are complete accessible categories. Accessible categories are the categories of models of <a href="/wiki/Sketch_(category_theory)" class="mw-redirect" title="Sketch (category theory)">sketches</a>. The name comes from that these categories are accessible as models of sketches. </td></tr> <tr> <td>1989</td> <td><a href="/wiki/Edward_Witten" title="Edward Witten">Edward Witten</a></td> <td><a href="/w/index.php?title=Witten_functional_integral&action=edit&redlink=1" class="new" title="Witten functional integral (page does not exist)">Witten functional integral</a> formalism and <a href="/wiki/Quantum_invariant" title="Quantum invariant">Witten invariants</a> for manifolds. </td></tr> <tr> <td>1990</td> <td><a href="/wiki/Peter_Freyd" class="mw-redirect" title="Peter Freyd">Peter Freyd</a></td> <td><a href="/wiki/Allegory_(category_theory)" class="mw-redirect" title="Allegory (category theory)">Allegories</a>: An abstraction of the <a href="/wiki/Category_of_sets_and_relations" class="mw-redirect" title="Category of sets and relations">category of sets with relations as morphisms</a>, it bears the same resemblance to binary relations as categories do to functions and sets. It is a category in which one has in addition to composition a unary operation reciprocation <i>R</i>° and a partial binary operation intersection <i>R</i> ∩ <i>S</i>, like in the category of sets with relations as morphisms (instead of functions) for which a number of axioms are required. It generalizes the <a href="/wiki/Relation_algebra" title="Relation algebra">relation algebra</a> to relations between different sorts. </td></tr> <tr> <td>1990</td> <td><a href="/wiki/Nicolai_Reshetikhin" title="Nicolai Reshetikhin">Nicolai Reshetikhin</a>–<a href="/wiki/Vladimir_Turaev" title="Vladimir Turaev">Vladimir Turaev</a>–<a href="/wiki/Edward_Witten" title="Edward Witten">Edward Witten</a></td> <td><a href="/wiki/Reshetikhin%E2%80%93Turaev%E2%80%93Witten_invariant" class="mw-redirect" title="Reshetikhin–Turaev–Witten invariant">Reshetikhin–Turaev–Witten invariants</a> of knots from <a href="/w/index.php?title=Modular_tensor_category&action=edit&redlink=1" class="new" title="Modular tensor category (page does not exist)">modular tensor categories</a> of representations of <a href="/wiki/Quantum_group" title="Quantum group">quantum groups</a>. </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="1991–2000"><span id="1991.E2.80.932000"></span>1991–2000</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Timeline_of_category_theory_and_related_mathematics&action=edit&section=5" title="Edit section: 1991–2000"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable sortable" width="100%"> <tbody><tr> <th>Year </th> <th style="width:22%">Contributors </th> <th>Event </th></tr> <tr> <td>1991</td> <td><a href="/wiki/Jean-Yves_Girard" title="Jean-Yves Girard">Jean-Yves Girard</a></td> <td><a href="/w/index.php?title=Polarization_(logic)&action=edit&redlink=1" class="new" title="Polarization (logic) (page does not exist)">Polarization</a> of <a href="/wiki/Linear_logic" title="Linear logic">linear logic</a>. </td></tr> <tr> <td>1991</td> <td><a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td><a href="/w/index.php?title=Parity_complex&action=edit&redlink=1" class="new" title="Parity complex (page does not exist)">Parity complexes</a>. A parity complex generates a free <a href="/w/index.php?title=%CE%A9-category&action=edit&redlink=1" class="new" title="Ω-category (page does not exist)">ω-category</a>. </td></tr> <tr> <td>1991</td> <td><a href="/wiki/Andr%C3%A9_Joyal" title="André Joyal">André Joyal</a>-<a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td>Formalization of Penrose <a href="/wiki/String_diagram" title="String diagram">string diagrams</a> to calculate with <a href="/w/index.php?title=Abstract_tensor&action=edit&redlink=1" class="new" title="Abstract tensor (page does not exist)">abstract tensors</a> in various <a href="/wiki/Monoidal_category" title="Monoidal category">monoidal categories</a> with extra structure. The calculus now depends on the connection with <a href="/wiki/Low-dimensional_topology" title="Low-dimensional topology">low-dimensional topology</a>. </td></tr> <tr> <td>1991</td> <td><a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td>Definition of the descent strict ω-category of a cosimplicial strict ω-category. </td></tr> <tr> <td>1991</td> <td><a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td>Top down <a href="/w/index.php?title=Excision_of_extremals_algorithm&action=edit&redlink=1" class="new" title="Excision of extremals algorithm (page does not exist)">excision of extremals algorithm</a> for computing nonabelian <i>n</i>-cocycle conditions for <a href="/wiki/Nonabelian_cohomology" title="Nonabelian cohomology">nonabelian cohomology</a>. </td></tr> <tr> <td>1992</td> <td>Yves Diers</td> <td><a href="/w/index.php?title=Axiomatic_categorical_geometry&action=edit&redlink=1" class="new" title="Axiomatic categorical geometry (page does not exist)">Axiomatic categorical geometry</a> using <a href="/w/index.php?title=Algebraic-geometric_category&action=edit&redlink=1" class="new" title="Algebraic-geometric category (page does not exist)">algebraic-geometric categories</a> and <a href="/w/index.php?title=Algebraic-geometric_functor&action=edit&redlink=1" class="new" title="Algebraic-geometric functor (page does not exist)">algebraic-geometric functors</a>. </td></tr> <tr> <td>1992</td> <td><a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Saunders Mac Lane</a>-<a href="/wiki/Ieke_Moerdijk" title="Ieke Moerdijk">Ieke Moerdijk</a></td> <td>Influential book: <i>Sheaves in geometry and logic</i>. </td></tr> <tr> <td>1992</td> <td>John Greenlees-<a href="/wiki/J._Peter_May" title="J. Peter May">Peter May</a></td> <td><a href="/w/index.php?title=Greenlees-May_duality&action=edit&redlink=1" class="new" title="Greenlees-May duality (page does not exist)">Greenlees-May duality</a> </td></tr> <tr> <td>1992</td> <td><a href="/wiki/Vladimir_Turaev" title="Vladimir Turaev">Vladimir Turaev</a></td> <td><a href="/w/index.php?title=Modular_tensor_category&action=edit&redlink=1" class="new" title="Modular tensor category (page does not exist)">Modular tensor categories</a>. Special <a href="/wiki/Monoidal_category" title="Monoidal category">tensor categories</a> that arise in constructing <a href="/wiki/Knot_invariant" title="Knot invariant">knot invariants</a>, in constructing <a href="/wiki/Topological_quantum_field_theory" title="Topological quantum field theory">TQFTs</a> and <a href="/wiki/Conformal_field_theory" title="Conformal field theory">CFTs</a>, as truncation (semisimple quotient) of the category of representations of a <a href="/wiki/Quantum_group" title="Quantum group">quantum group</a> (at roots of unity), as categories of representations of weak <a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebras</a>, as category of representations of a <a href="/wiki/Conformal_field_theory" title="Conformal field theory">RCFT</a>. </td></tr> <tr> <td>1992</td> <td><a href="/wiki/Vladimir_Turaev" title="Vladimir Turaev">Vladimir Turaev</a>-<a href="/wiki/Oleg_Viro" title="Oleg Viro">Oleg Viro</a></td> <td><a href="/w/index.php?title=Turaev-Viro_state_sum_model&action=edit&redlink=1" class="new" title="Turaev-Viro state sum model (page does not exist)">Turaev-Viro state sum models</a> based on <a href="/wiki/Spherical_category" title="Spherical category">spherical categories</a> (the first state sum models) and <a href="/w/index.php?title=Turaev-Viro_invariant&action=edit&redlink=1" class="new" title="Turaev-Viro invariant (page does not exist)">Turaev-Viro state sum invariants</a> for 3-manifolds. </td></tr> <tr> <td>1992</td> <td><a href="/wiki/Vladimir_Turaev" title="Vladimir Turaev">Vladimir Turaev</a></td> <td>Shadow world of links: <a href="/w/index.php?title=Shadow_(links)&action=edit&redlink=1" class="new" title="Shadow (links) (page does not exist)">Shadows of links</a> give shadow invariants of links by shadow <a href="/w/index.php?title=State_sum&action=edit&redlink=1" class="new" title="State sum (page does not exist)">state sums</a>. </td></tr> <tr> <td>1993</td> <td><a href="/wiki/Ruth_Lawrence" title="Ruth Lawrence">Ruth Lawrence</a></td> <td><a href="/w/index.php?title=Extended_topological_quantum_field_theory&action=edit&redlink=1" class="new" title="Extended topological quantum field theory (page does not exist)">Extended TQFTs</a> </td></tr> <tr> <td>1993</td> <td><a href="/w/index.php?title=David_Yetter&action=edit&redlink=1" class="new" title="David Yetter (page does not exist)">David Yetter</a>-<a href="/wiki/Louis_Crane" class="mw-redirect" title="Louis Crane">Louis Crane</a></td> <td><a href="/w/index.php?title=Crane-Yetter_state_sum_model&action=edit&redlink=1" class="new" title="Crane-Yetter state sum model (page does not exist)">Crane-Yetter state sum models</a> based on <a href="/wiki/Ribbon_category" title="Ribbon category">ribbon categories</a> and <a href="/w/index.php?title=Crane-Yetter_invariant&action=edit&redlink=1" class="new" title="Crane-Yetter invariant (page does not exist)">Crane-Yetter state sum invariants</a> for 4-manifolds. </td></tr> <tr> <td>1993</td> <td><a href="/wiki/Kenji_Fukaya" title="Kenji Fukaya">Kenji Fukaya</a></td> <td><a href="/wiki/Fukaya_category" title="Fukaya category"><i>A</i></a><sub><a href="/wiki/Fukaya_category" title="Fukaya category">∞</a></sub><a href="/wiki/Fukaya_category" title="Fukaya category">-categories</a> and <a href="/wiki/Fukaya_category" title="Fukaya category"><i>A</i></a><sub><a href="/wiki/Fukaya_category" title="Fukaya category">∞</a></sub><a href="/wiki/Fukaya_category" title="Fukaya category">-functors</a>: Most commonly in <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a>, a category with several compositions such that the first composition is associative up to homotopy which satisfies an equation that holds up to another homotopy, etc. (associative up to higher homotopy). A stands for associative. <p>Def: A category <span class="texhtml">C</span> such that </p> <ol><li>for all <i>X</i>, <i>Y</i> in Ob(<span class="texhtml">C</span>) the <a href="/wiki/Hom-set" class="mw-redirect" title="Hom-set">Hom-sets</a> Hom<sub><span class="texhtml">C</span></sub>(<i>X</i>,<i>Y</i>) are finite-dimensional <a href="/wiki/Chain_complex" title="Chain complex">chain complexes</a> of <b>Z</b>-graded modules</li> <li>for all objects <i>X</i><sub>1</sub>, ..., <i>X</i><sub><i>n</i></sub> in Ob(<span class="texhtml">C</span>) there is a family of linear composition maps (the higher compositions)</li></ol> <dl><dd><dl><dd><i>m</i><sub><i>n</i></sub> : Hom<sub><span class="texhtml">C</span></sub>(<i>X</i><sub>0</sub>,<i>X</i><sub>1</sub>) ⊗ Hom<sub><span class="texhtml">C</span></sub>(<i>X</i><sub>1</sub>,<i>X</i><sub>2</sub>) ⊗ ... ⊗ Hom<sub><span class="texhtml">C</span></sub>(<i>X</i><sub><i>n</i>−1</sub>,<i>X</i><sub><i>n</i></sub>) → Hom<sub><span class="texhtml">C</span></sub>(<i>X</i><sub>0</sub>,<i>X</i><sub><i>n</i></sub>)</dd></dl></dd></dl> <dl><dd><dl><dd>of degree <i>n</i> − 2 (homological grading convention is used) for <i>n</i> ≥ 1</dd></dl></dd></dl> <ol><li><i>m</i><sub>1</sub> is the differential on the chain complex Hom<sub><span class="texhtml">C</span></sub>(<i>X</i>,<i>Y</i>)</li> <li><i>m</i><sub><i>n</i></sub> satisfy the quadratic <i>A</i><sub>∞</sub>-associativity equation for all <i>n</i> ≥ 0.</li></ol> <p><i>m</i><sub>1</sub> and <i>m</i><sub>2</sub> will be <a href="/wiki/Chain_map" class="mw-redirect" title="Chain map">chain maps</a> but the compositions <i>m</i><sub><i>i</i></sub> of higher order are not chain maps; nevertheless they are <a href="/wiki/Massey_product" title="Massey product">Massey products</a>. In particular it is a <a href="/wiki/Preadditive_category#R-linear_categories" title="Preadditive category">linear category</a>. </p><p>Examples are the <a href="/wiki/Fukaya_category" title="Fukaya category">Fukaya category</a> Fuk(<i>X</i>) and <a href="/wiki/Loop_space" title="Loop space">loop space</a> Ω<i>X</i> where <i>X</i> is a topological space and <a href="/wiki/A%E2%88%9E-algebra" class="mw-redirect" title="A∞-algebra"><i>A</i></a><sub><a href="/wiki/A%E2%88%9E-algebra" class="mw-redirect" title="A∞-algebra">∞</a></sub><a href="/wiki/A%E2%88%9E-algebra" class="mw-redirect" title="A∞-algebra">-algebras</a> as <i>A</i><sub>∞</sub>-categories with one object. </p><p>When there are no higher maps (trivial homotopies) <i>C</i> is a <a href="/wiki/Dg-category" class="mw-redirect" title="Dg-category">dg-category</a>. Every <i>A</i><sub>∞</sub>-category is quasiisomorphic in a functorial way to a dg-category. A quasiisomorphism is a chain map that is an isomorphism in homology. </p><p>The framework of dg-categories and dg-functors is too narrow for many problems, and it is preferable to consider the wider class of <i>A</i><sub>∞</sub>-categories and <i>A</i><sub>∞</sub>-functors. Many features of <i>A</i><sub>∞</sub>-categories and <i>A</i><sub>∞</sub>-functors come from the fact that they form a symmetric closed <a href="/wiki/Multicategory" title="Multicategory">multicategory</a>, which is revealed in the language of <a href="/wiki/Monad_(category_theory)#Comonads" title="Monad (category theory)">comonads</a>. From a higher-dimensional perspective <i>A</i><sub>∞</sub>-categories are weak <span class="nowrap"><i>ω</i>-categories</span> with all morphisms invertible. <i>A</i><sub>∞</sub>-categories can also be viewed as <a href="/w/index.php?title=Noncommutative_formal_dg-manifold&action=edit&redlink=1" class="new" title="Noncommutative formal dg-manifold (page does not exist)">noncommutative formal dg-manifolds</a> with a closed marked subscheme of objects. </p> </td></tr> <tr> <td>1993</td> <td><a href="/w/index.php?title=John_Barret_(mathematician)&action=edit&redlink=1" class="new" title="John Barret (mathematician) (page does not exist)">John Barret</a>-Bruce Westbury</td> <td><a href="/wiki/Spherical_categories" class="mw-redirect" title="Spherical categories">Spherical categories</a>: <a href="/wiki/Monoidal_category" title="Monoidal category">Monoidal categories</a> with duals for diagrams on spheres instead for in the plane. </td></tr> <tr> <td>1993</td> <td><a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Maxim Kontsevich</a></td> <td><a href="/wiki/Kontsevich_invariant" title="Kontsevich invariant">Kontsevich invariants</a> for knots (are perturbation expansion Feynman integrals for the <a href="/w/index.php?title=Witten_functional_integral&action=edit&redlink=1" class="new" title="Witten functional integral (page does not exist)">Witten functional integral</a>) defined by the Kontsevich integral. They are the universal <a href="/wiki/Vassiliev_invariant" class="mw-redirect" title="Vassiliev invariant">Vassiliev invariants</a> for knots. </td></tr> <tr> <td>1993</td> <td>Daniel Freed</td> <td>A new view on <a href="/wiki/TQFT" class="mw-redirect" title="TQFT">TQFT</a> using <a href="/w/index.php?title=Modular_tensor_category&action=edit&redlink=1" class="new" title="Modular tensor category (page does not exist)">modular tensor categories</a> that unifies three approaches to TQFT (modular tensor categories from path integrals). </td></tr> <tr> <td>1994</td> <td>Francis Borceux</td> <td><i>Handbook of <a href="/wiki/Categorical_Algebra" class="mw-redirect" title="Categorical Algebra">Categorical Algebra</a></i> (3 volumes). </td></tr> <tr> <td>1994</td> <td><a href="/wiki/Jean_B%C3%A9nabou" title="Jean Bénabou">Jean Bénabou</a>–Bruno Loiseau</td> <td><a href="/w/index.php?title=Orbital_(topos_theory)&action=edit&redlink=1" class="new" title="Orbital (topos theory) (page does not exist)">Orbitals</a> in a topos. </td></tr> <tr> <td>1994</td> <td><a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Maxim Kontsevich</a></td> <td>Formulates the <a href="/wiki/Homological_mirror_symmetry" title="Homological mirror symmetry">homological mirror symmetry</a> conjecture: <i>X</i> a compact <a href="/wiki/Symplectic_manifold" title="Symplectic manifold">symplectic manifold</a> with first <a href="/wiki/Chern_class" title="Chern class">Chern class</a> <i>c</i><sub>1</sub>(<i>X</i>) = 0 and <i>Y</i> a compact Calabi–Yau manifold are mirror pairs if and only if <i>D</i>(Fuk<sub><i>X</i></sub>) (the derived category of the <a href="/wiki/Fukaya_category" title="Fukaya category">Fukaya triangulated category</a> of <i>X</i> concocted out of <a href="/w/index.php?title=Lagrangian_cycle&action=edit&redlink=1" class="new" title="Lagrangian cycle (page does not exist)">Lagrangian cycles</a> with local systems) is equivalent to a subcategory of <i>D</i><sup><i>b</i></sup>(Coh<sub><i>Y</i></sub>) (the bounded derived category of coherent sheaves on <i>Y</i>). </td></tr> <tr> <td>1994</td> <td><a href="/wiki/Louis_Crane" class="mw-redirect" title="Louis Crane">Louis Crane</a>-<a href="/wiki/Igor_Frenkel" title="Igor Frenkel">Igor Frenkel</a></td> <td><a href="/w/index.php?title=Hopf_category&action=edit&redlink=1" class="new" title="Hopf category (page does not exist)">Hopf categories</a> and construction of 4D <a href="/wiki/TQFT" class="mw-redirect" title="TQFT">TQFTs</a> by them. </td></tr> <tr> <td>1994</td> <td>John Fischer</td> <td>Defines the <a href="/wiki/2-category" class="mw-redirect" title="2-category">2-category</a> of <a href="/w/index.php?title=2-knot&action=edit&redlink=1" class="new" title="2-knot (page does not exist)">2-knots</a> (knotted surfaces). </td></tr> <tr> <td>1995</td> <td>Bob Gordon-John Power-<a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td><a href="/wiki/Tricategory" title="Tricategory">Tricategories</a> and a corresponding <a href="/wiki/Coherence_theorem" class="mw-redirect" title="Coherence theorem">coherence theorem</a>: Every weak 3-category is equivalent to a <a href="/w/index.php?title=Gray_category&action=edit&redlink=1" class="new" title="Gray category (page does not exist)">Gray 3-category</a>. </td></tr> <tr> <td>1995</td> <td><a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a>–<a href="/w/index.php?title=Dominic_Verity&action=edit&redlink=1" class="new" title="Dominic Verity (page does not exist)">Dominic Verity</a></td> <td><a href="/w/index.php?title=Surface_diagram&action=edit&redlink=1" class="new" title="Surface diagram (page does not exist)">Surface diagrams</a> for tricategories. </td></tr> <tr> <td>1995</td> <td><a href="/wiki/Louis_Crane" class="mw-redirect" title="Louis Crane">Louis Crane</a></td> <td>Coins <a href="/wiki/Categorification" title="Categorification">categorification</a> leading to the <a href="/w/index.php?title=Categorical_ladder&action=edit&redlink=1" class="new" title="Categorical ladder (page does not exist)">categorical ladder</a>. </td></tr> <tr> <td>1995</td> <td>Sjoerd Crans</td> <td>A general procedure of transferring closed <a href="/wiki/Model_category" title="Model category">model structures</a> on a category along <a href="/wiki/Adjoint_functor" class="mw-redirect" title="Adjoint functor">adjoint functor</a> pairs to another category. </td></tr> <tr> <td>1995</td> <td><a href="/wiki/Andr%C3%A9_Joyal" title="André Joyal">André Joyal</a>-<a href="/wiki/Ieke_Moerdijk" title="Ieke Moerdijk">Ieke Moerdijk</a></td> <td>AST, <a href="/w/index.php?title=Algebraic_set_theory&action=edit&redlink=1" class="new" title="Algebraic set theory (page does not exist)">Algebraic set theory</a>: Also sometimes called categorical set theory. It was developed from 1988 by André Joyal and Ieke Moerdijk, and was first presented in detail as a book in 1995 by them. AST is a framework based on category theory to study and organize <a href="/wiki/Set_theory#Axiomatic_set_theory" title="Set theory">set theories</a> and to construct <a href="/w/index.php?title=Model_of_a_set_theory&action=edit&redlink=1" class="new" title="Model of a set theory (page does not exist)">models of set theories</a>. The aim of AST is to provide a uniform <a href="/wiki/Categorical_semantics" class="mw-redirect" title="Categorical semantics">categorical semantics</a> or description of set theories of different kinds (classical or constructive, bounded, predicative or impredicative, well-founded or non-well-founded, ...), the various constructions of the <a href="/w/index.php?title=Cumulative_hierarchy_of_sets&action=edit&redlink=1" class="new" title="Cumulative hierarchy of sets (page does not exist)">cumulative hierarchy of sets</a>, forcing models, sheaf models and realisability models. Instead of focusing on categories of sets AST focuses on categories of classes. The basic tool of AST is the notion of a <a href="/w/index.php?title=Category_with_class_structure&action=edit&redlink=1" class="new" title="Category with class structure (page does not exist)">category with class structure</a> (a category of classes equipped with a class of small maps (the intuition being that their fibres are small in some sense), powerclasses and a universal object (a <a href="/wiki/Universe_(mathematics)" title="Universe (mathematics)">universe</a>)) which provides an axiomatic framework in which models of set theory can be constructed. The notion of a class category permits both the definition of ZF-algebras (<a href="/w/index.php?title=Zermelo-Fraenkel_algebra&action=edit&redlink=1" class="new" title="Zermelo-Fraenkel algebra (page does not exist)">Zermelo-Fraenkel algebras</a>) and related structures expressing the idea that the hierarchy of sets is an algebraic structure on the one hand and the interpretation of the first-order logic of elementary set theory on the other. The subcategory of sets in a class category is an <a href="/wiki/Topos_(mathematics)#Elementary_toposes_.28toposes_in_logic.29" class="mw-redirect" title="Topos (mathematics)">elementary topos</a> and every elementary topos occurs as sets in a class category. The class category itself always embeds into the <a href="/w/index.php?title=Ideal_completion&action=edit&redlink=1" class="new" title="Ideal completion (page does not exist)">ideal completion</a> of a topos. The interpretation of the logic is that in every class category the universe is a model of basic intuitionistic set theory (BIST) that is logically complete with respect to class category models. Therefore, class categories generalize both topos theory and intuitionistic set theory. AST founds and formalizes set theory on the ZF-algebra with operations union and successor (singleton) instead of on the membership relation. The <a href="/wiki/Zermelo%E2%80%93Fraenkel_set_theory" title="Zermelo–Fraenkel set theory">ZF-axioms</a> are nothing but a description of the free ZF-algebra just as the <a href="/wiki/Peano_axioms" title="Peano axioms">Peano axioms</a> are a description of the <a href="/wiki/Free_monoid" title="Free monoid">free monoid</a> on one generator. In this perspective the models of set theory are algebras for a suitably presented <a href="/wiki/Theory_(mathematical_logic)" title="Theory (mathematical logic)">algebraic theory</a> and many familiar set theoretic conditions (such as well-foundedness) are related to familiar algebraic conditions (such as freeness). Using an auxiliary notion of small map it is possible to extend the axioms of a topos and provide a general theory for uniformly constructing models of set theory out of toposes. </td></tr> <tr> <td>1995</td> <td><a href="/wiki/Michael_Makkai" title="Michael Makkai">Michael Makkai</a></td> <td>SFAM, <a href="/w/index.php?title=Structuralist_foundation_of_abstract_mathematics&action=edit&redlink=1" class="new" title="Structuralist foundation of abstract mathematics (page does not exist)">Structuralist foundation of abstract mathematics</a>. In SFAM the universe consists of higher-dimensional categories, functors are replaced by saturated <a href="/wiki/Anafunctor" title="Anafunctor">anafunctors</a>, sets are <a href="/w/index.php?title=Abstract_set&action=edit&redlink=1" class="new" title="Abstract set (page does not exist)">abstract sets</a>, the formal logic for entities is <a href="/w/index.php?title=FOLDS&action=edit&redlink=1" class="new" title="FOLDS (page does not exist)">FOLDS</a> (first-order logic with dependent sorts) in which the identity relation is not given a priori by first-order axioms but derived from within a context. </td></tr> <tr> <td>1995</td> <td><a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a>-<a href="/w/index.php?title=James_Dolan_(mathematician)&action=edit&redlink=1" class="new" title="James Dolan (mathematician) (page does not exist)">James Dolan</a></td> <td><a href="/wiki/Opetopic_set" class="mw-redirect" title="Opetopic set">Opetopic sets</a> (<a href="/wiki/Opetope" title="Opetope">opetopes</a>) based on <a href="/wiki/Operad" title="Operad">operads</a>. <a href="/wiki/Weak_n-category" title="Weak n-category">Weak <i>n</i>-categories</a> are <i>n</i>-opetopic sets. </td></tr> <tr> <td>1995</td> <td><a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a>-<a href="/w/index.php?title=James_Dolan_(mathematician)&action=edit&redlink=1" class="new" title="James Dolan (mathematician) (page does not exist)">James Dolan</a></td> <td>Introduced the <a href="/w/index.php?title=Periodic_table_of_mathematics&action=edit&redlink=1" class="new" title="Periodic table of mathematics (page does not exist)">periodic table of mathematics</a> which identifies <a href="/w/index.php?title=K-tuply_monoidal_n-category&action=edit&redlink=1" class="new" title="K-tuply monoidal n-category (page does not exist)"><i>k</i>-tuply monoidal <i>n</i>-categories</a>. It mirrors the table of <a href="/wiki/Homotopy_groups_of_the_spheres" class="mw-redirect" title="Homotopy groups of the spheres">homotopy groups of the spheres</a>. </td></tr> <tr> <td>1995</td> <td><a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a>–<a href="/w/index.php?title=James_Dolan_(mathematician)&action=edit&redlink=1" class="new" title="James Dolan (mathematician) (page does not exist)">James Dolan</a></td> <td>Outlined a program in which <i>n</i>-dimensional <a href="/wiki/TQFT" class="mw-redirect" title="TQFT">TQFTs</a> are described as <a href="/w/index.php?title=N-category_representation&action=edit&redlink=1" class="new" title="N-category representation (page does not exist)"><i>n</i>-category representations</a>. </td></tr> <tr> <td>1995</td> <td><a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a>–<a href="/w/index.php?title=James_Dolan_(mathematician)&action=edit&redlink=1" class="new" title="James Dolan (mathematician) (page does not exist)">James Dolan</a></td> <td>Proposed <i>n</i>-dimensional <a href="/wiki/Deformation_quantization" title="Deformation quantization">deformation quantization</a>. </td></tr> <tr> <td>1995</td> <td><a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a>–<a href="/w/index.php?title=James_Dolan_(mathematician)&action=edit&redlink=1" class="new" title="James Dolan (mathematician) (page does not exist)">James Dolan</a></td> <td><a href="/wiki/Tangle_hypothesis" class="mw-redirect" title="Tangle hypothesis">Tangle hypothesis</a>: The <i>n</i>-category of framed <i>n</i>-tangles in <i>n</i> + <i>k</i> dimensions is (<i>n</i> + <i>k</i>)-equivalent to the free weak <i>k</i>-tuply monoidal <i>n</i>-category with duals on one object. </td></tr> <tr> <td>1995</td> <td><a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a>-<a href="/w/index.php?title=James_Dolan_(mathematician)&action=edit&redlink=1" class="new" title="James Dolan (mathematician) (page does not exist)">James Dolan</a></td> <td><a href="/wiki/Cobordism_hypothesis" title="Cobordism hypothesis">Cobordism hypothesis</a> (Extended TQFT hypothesis I): The <i>n</i>-category of which <i>n</i>-dimensional extended TQFTs are representations, <b>nCob</b>, is the free stable weak <i>n</i>-category with duals on one object. </td></tr> <tr> <td>1995</td> <td><a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a>-<a href="/w/index.php?title=James_Dolan_(mathematician)&action=edit&redlink=1" class="new" title="James Dolan (mathematician) (page does not exist)">James Dolan</a></td> <td><a href="/wiki/Stabilization_hypothesis" title="Stabilization hypothesis">Stabilization hypothesis</a>: After suspending a weak <i>n</i>-category <i>n</i> + 2 times, further suspensions have no essential effect. The suspension functor <i>S</i>: <b>nCat</b><sub><i>k</i></sub>→<b>nCat</b><sub><i>k</i>+1</sub> is an equivalence of categories for <i>k</i> = <i>n</i> + 2. </td></tr> <tr> <td>1995</td> <td><a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a>-<a href="/w/index.php?title=James_Dolan_(mathematician)&action=edit&redlink=1" class="new" title="James Dolan (mathematician) (page does not exist)">James Dolan</a></td> <td><a href="/wiki/Extended_TQFT_hypothesis" class="mw-redirect" title="Extended TQFT hypothesis">Extended TQFT hypothesis</a> II: An <i>n</i>-dimensional unitary extended TQFT is a weak <i>n</i>-functor, preserving all levels of duality, from the free stable weak <i>n</i>-category with duals on one object to <b>nHilb</b>. </td></tr> <tr> <td>1995</td> <td>Valentin Lychagin</td> <td><a href="/wiki/Categorical_quantization" class="mw-redirect" title="Categorical quantization">Categorical quantization</a> </td></tr> <tr> <td>1995</td> <td><a href="/wiki/Pierre_Deligne" title="Pierre Deligne">Pierre Deligne</a>-<a href="/wiki/Vladimir_Drinfeld" title="Vladimir Drinfeld">Vladimir Drinfeld</a>-<a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Maxim Kontsevich</a></td> <td><a href="/wiki/Derived_algebraic_geometry" title="Derived algebraic geometry">Derived algebraic geometry</a> with <a href="/wiki/Derived_scheme" title="Derived scheme">derived schemes</a> and <a href="/wiki/Derived_moduli_stacks" class="mw-redirect" title="Derived moduli stacks">derived moduli stacks</a>. A program of doing algebraic geometry and especially <a href="/wiki/Moduli_problem" class="mw-redirect" title="Moduli problem">moduli problems</a> in the <a href="/wiki/Derived_category" title="Derived category">derived category</a> of schemes or algebraic varieties instead of in their normal categories. </td></tr> <tr> <td>1997</td> <td><a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Maxim Kontsevich</a></td> <td>Formal <a href="/wiki/Deformation_quantization" title="Deformation quantization">deformation quantization</a> theorem: Every <a href="/wiki/Poisson_manifold" title="Poisson manifold">Poisson manifold</a> admits a differentiable <a href="/wiki/Star_product_(quantization)" class="mw-redirect" title="Star product (quantization)">star product</a> and they are classified up to equivalence by formal deformations of the Poisson structure. </td></tr> <tr> <td>1998</td> <td>Claudio Hermida-<a href="/w/index.php?title=Michael-Makkai&action=edit&redlink=1" class="new" title="Michael-Makkai (page does not exist)">Michael-Makkai</a>-John Power</td> <td><a href="/w/index.php?title=Multitope&action=edit&redlink=1" class="new" title="Multitope (page does not exist)">Multitopes</a>, Multitopic sets. </td></tr> <tr> <td>1998</td> <td><a href="/wiki/Carlos_Simpson" title="Carlos Simpson">Carlos Simpson</a></td> <td><a href="/w/index.php?title=Simpson_conjecture&action=edit&redlink=1" class="new" title="Simpson conjecture (page does not exist)">Simpson conjecture</a>: Every weak ∞-category is equivalent to a ∞-category in which composition and exchange laws are strict and only the unit laws are allowed to hold weakly. It is proven for <span class="nowrap">1,2,3</span>-categories with a single object. </td></tr> <tr> <td>1998</td> <td>André Hirschowitz-Carlos Simpson</td> <td>Give a <a href="/wiki/Model_category" title="Model category">model category</a> structure on the category of Segal categories. <a href="/wiki/Segal_category" title="Segal category">Segal categories</a> are the fibrant-cofibrant objects and <a href="/wiki/Segal_map" class="mw-redirect" title="Segal map">Segal maps</a> are the <a href="/wiki/Model_category" title="Model category">weak equivalences</a>. In fact they generalize the definition to that of a <a href="/wiki/Segal_n-category" class="mw-redirect" title="Segal n-category">Segal <i>n</i>-category</a> and give a model structure for Segal <i>n</i>-categories for any <i>n</i> ≥ 1. </td></tr> <tr> <td>1998</td> <td><a href="/wiki/Christopher_Isham" title="Christopher Isham">Chris Isham</a>–Jeremy Butterfield</td> <td><a href="/wiki/Kochen%E2%80%93Specker_theorem" title="Kochen–Specker theorem">Kochen–Specker theorem</a> in topos theory of presheaves: The <a href="/wiki/Spectral_presheaf" class="mw-redirect" title="Spectral presheaf">spectral presheaf</a> (the presheaf that assigns to each operator its spectrum) has no <a href="/wiki/Global_element" title="Global element">global elements</a> (<a href="/wiki/Global_section" class="mw-redirect" title="Global section">global sections</a>) but may have partial elements or <a href="/w/index.php?title=Local_element&action=edit&redlink=1" class="new" title="Local element (page does not exist)">local elements</a>. A global element is the analogue for presheaves of the ordinary idea of an element of a set. This is equivalent in quantum theory to the spectrum of the <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a> of observables in a topos having no points. </td></tr> <tr> <td>1998</td> <td><a href="/wiki/Richard_Thomas_(mathematician)" title="Richard Thomas (mathematician)">Richard Thomas</a></td> <td>Richard Thomas, a student of <a href="/wiki/Simon_Donaldson" title="Simon Donaldson">Simon Donaldson</a>, introduces <a href="/wiki/Donaldson%E2%80%93Thomas_invariant" class="mw-redirect" title="Donaldson–Thomas invariant">Donaldson–Thomas invariants</a> which are systems of numerical invariants of complex oriented 3-manifolds <i>X</i>, analogous to <a href="/wiki/Donaldson_invariant" class="mw-redirect" title="Donaldson invariant">Donaldson invariants</a> in the theory of 4-manifolds. They are certain <a href="/wiki/Weighted_Euler_characteristic" class="mw-redirect" title="Weighted Euler characteristic">weighted Euler characteristics</a> of the <a href="/w/index.php?title=Moduli_space_of_sheaves&action=edit&redlink=1" class="new" title="Moduli space of sheaves (page does not exist)">moduli space of sheaves</a> on <i>X</i> and "count" Gieseker semistable <a href="/wiki/Coherent_sheaf" title="Coherent sheaf">coherent sheaves</a> with fixed <a href="/wiki/Chern_character" class="mw-redirect" title="Chern character">Chern character</a> on <i>X</i>. Ideally the moduli spaces should be a critical sets of <a href="/w/index.php?title=Holomorphic_Chern%E2%80%93Simons_functions&action=edit&redlink=1" class="new" title="Holomorphic Chern–Simons functions (page does not exist)">holomorphic Chern–Simons functions</a> and the Donaldson–Thomas invariants should be the number of critical points of this function, counted correctly. Currently such holomorphic Chern–Simons functions exist at best locally. </td></tr> <tr> <td>1998</td> <td><a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a></td> <td><a href="/wiki/Spin_foam" title="Spin foam">Spin foam models</a>: A 2-dimensional <a href="/wiki/Cell_complex" class="mw-redirect" title="Cell complex">cell complex</a> with faces labeled by representations and edges labeled by <a href="/wiki/Intertwining_operator" class="mw-redirect" title="Intertwining operator">intertwining operators</a>. Spin foams are functors between <a href="/w/index.php?title=Spin_network_category&action=edit&redlink=1" class="new" title="Spin network category (page does not exist)">spin network categories</a>. Any slice of a spin foam gives a spin network. </td></tr> <tr> <td>1998</td> <td><a href="/wiki/John_Baez" class="mw-redirect" title="John Baez">John Baez</a>–<a href="/w/index.php?title=James_Dolan_(mathematician)&action=edit&redlink=1" class="new" title="James Dolan (mathematician) (page does not exist)">James Dolan</a></td> <td><a href="/w/index.php?title=Microcosm_principle&action=edit&redlink=1" class="new" title="Microcosm principle (page does not exist)">Microcosm principle</a>: Certain algebraic structures can be defined in any category equipped with a categorified version of the same structure. </td></tr> <tr> <td>1998</td> <td><a href="/wiki/Alexander_Rosenberg_(mathematician)" class="mw-redirect" title="Alexander Rosenberg (mathematician)">Alexander Rosenberg</a></td> <td><a href="/wiki/Noncommutative_scheme" class="mw-redirect" title="Noncommutative scheme">Noncommutative schemes</a>: The pair (Spec(<b>A</b>),O<sub><b>A</b></sub>) where <b>A</b> is an <a href="/wiki/Abelian_category" title="Abelian category">abelian category</a> and to it is associated a topological space Spec(<b>A</b>) together with a sheaf of rings O<sub><b>A</b></sub> on it. In the case when <b>A</b> = <b>QCoh</b>(<i>X</i>) for <i>X</i> a scheme the pair (Spec(<b>A</b>),O<sub><b>A</b></sub>) is naturally isomorphic to the scheme (<i>X</i><sup>Zar</sup>,O<sub><i>X</i></sub>) using the equivalence of categories <b>QCoh</b>(Spec(<i>R</i>)) = <b>Mod</b><sub><i>R</i></sub>. More generally abelian categories or triangulated categories or dg-categories or <i>A</i><sub>∞</sub>-categories should be regarded as categories of quasicoherent sheaves (or complexes of sheaves) on noncommutative schemes. This is a starting point in <a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">noncommutative algebraic geometry</a>. It means that one can think of the category <b>A</b> itself as a space. Since <b>A</b> is abelian it allows to naturally do <a href="/wiki/Homological_algebra" title="Homological algebra">homological algebra</a> on noncommutative schemes and hence <a href="/wiki/Sheaf_cohomology" title="Sheaf cohomology">sheaf cohomology</a>. </td></tr> <tr> <td>1998</td> <td><a href="/wiki/Maxim_Kontsevich" title="Maxim Kontsevich">Maxim Kontsevich</a></td> <td><a href="/w/index.php?title=Calabi%E2%80%93Yau_category&action=edit&redlink=1" class="new" title="Calabi–Yau category (page does not exist)">Calabi–Yau categories</a>: A <a href="/wiki/Preadditive_category#R-linear_categories" title="Preadditive category">linear category</a> with a trace map for each object of the category and an associated symmetric (with respects to objects) nondegenerate pairing to the trace map. If <i>X</i> is a smooth projective <a href="/wiki/Calabi-Yau_variety" class="mw-redirect" title="Calabi-Yau variety">Calabi—Yau variety</a> of dimension <i>d</i> then D<sup>b</sup>(Coh(<i>X</i>)) is a unital Calabi–Yau <a href="/wiki/Fukaya_category" title="Fukaya category"><i>A</i></a><sub><a href="/wiki/Fukaya_category" title="Fukaya category">∞</a></sub><a href="/wiki/Fukaya_category" title="Fukaya category">-category</a> of Calabi–Yau dimension <i>d</i>. A Calabi–Yau category with one object is a <a href="/wiki/Frobenius_algebra" title="Frobenius algebra">Frobenius algebra</a>. </td></tr> <tr> <td>1999</td> <td><a href="/wiki/Joseph_Bernstein" title="Joseph Bernstein">Joseph Bernstein</a>–<a href="/wiki/Igor_Frenkel" title="Igor Frenkel">Igor Frenkel</a>–<a href="/wiki/Mikhail_Khovanov" title="Mikhail Khovanov">Mikhail Khovanov</a></td> <td><a href="/w/index.php?title=Temperley%E2%80%93Lieb_category&action=edit&redlink=1" class="new" title="Temperley–Lieb category (page does not exist)">Temperley–Lieb categories</a>: Objects are enumerated by nonnegative <a href="/wiki/Integer" title="Integer">integers</a>. The set of homomorphisms from object <i>n</i> to object <i>m</i> is a free <i>R</i>-module with a basis over a ring <i>R</i>. <i>R</i> is given by the isotopy classes of systems of (|<i>n</i>| + |<i>m</i>|)/2 simple pairwise disjoint arcs inside a horizontal strip on the plane that connect in pairs |<i>n</i>| points on the bottom and |<i>m</i>| points on the top in some order. Morphisms are composed by concatenating their diagrams. Temperley–Lieb categories are categorized <a href="/wiki/Temperley%E2%80%93Lieb_algebra" title="Temperley–Lieb algebra">Temperley–Lieb algebras</a>. </td></tr> <tr> <td>1999</td> <td>Moira Chas–<a href="/wiki/Dennis_Sullivan" title="Dennis Sullivan">Dennis Sullivan</a></td> <td>Constructs <a href="/wiki/String_topology" title="String topology">string topology</a> by cohomology. This is string theory on general topological manifolds. </td></tr> <tr> <td>1999</td> <td><a href="/wiki/Mikhail_Khovanov" title="Mikhail Khovanov">Mikhail Khovanov</a></td> <td><a href="/wiki/Khovanov_homology" title="Khovanov homology">Khovanov homology</a>: A homology theory for knots such that the dimensions of the homology groups are the coefficients of the <a href="/wiki/Jones_polynomial" title="Jones polynomial">Jones polynomial</a> of the knot. </td></tr> <tr> <td>1999</td> <td><a href="/wiki/Vladimir_Turaev" title="Vladimir Turaev">Vladimir Turaev</a></td> <td>Homotopy quantum field theory <a href="/w/index.php?title=HQFT&action=edit&redlink=1" class="new" title="HQFT (page does not exist)">HQFT</a> </td></tr> <tr> <td>1999</td> <td><a href="/wiki/Vladimir_Voevodsky" title="Vladimir Voevodsky">Vladimir Voevodsky</a>–Fabien Morel</td> <td>Constructs the <a href="/wiki/A%C2%B9_homotopy_theory" title="A¹ homotopy theory">homotopy category of schemes</a>. </td></tr> <tr> <td>1999</td> <td><a href="/wiki/Ronald_Brown_(mathematician)" title="Ronald Brown (mathematician)">Ronald Brown</a>–George Janelidze</td> <td>2-dimensional Galois theory </td></tr> <tr> <td>2000</td> <td><a href="/wiki/Vladimir_Voevodsky" title="Vladimir Voevodsky">Vladimir Voevodsky</a></td> <td>Gives two constructions of <a href="/wiki/Motivic_cohomology" title="Motivic cohomology">motivic cohomology</a> of varieties, by model categories in homotopy theory and by a triangulated category of DM-motives. </td></tr> <tr> <td>2000</td> <td><a href="/wiki/Yakov_Eliashberg" title="Yakov Eliashberg">Yasha Eliashberg</a>–<a href="/wiki/Alexander_Givental" title="Alexander Givental">Alexander Givental</a>–<a href="/wiki/Helmut_Hofer_(mathematician)" class="mw-redirect" title="Helmut Hofer (mathematician)">Helmut Hofer</a></td> <td><a href="/wiki/Floer_homology#Symplectic_field_theory_(SFT)" title="Floer homology">Symplectic field theory SFT</a>: A functor <i>Z</i> from a geometric category of framed Hamiltonian structures and framed cobordisms between them to an algebraic category of certain differential D-modules and Fourier integral operators between them and satisfying some axioms. </td></tr> <tr> <td>2000</td> <td>Paul Taylor<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></td> <td>ASD (Abstract Stone duality): A reaxiomatisation of the space and maps in general topology in terms of <a href="/wiki/Lambda_calculus" title="Lambda calculus">λ-calculus</a> of computable continuous functions and predicates that is both constructive and computable. The topology on a space is treated not as a lattice, but as an <a href="/wiki/Exponential_object" title="Exponential object">exponential object</a> of the same category as the original space, with an associated <span class="nowrap">λ-calculus</span>. Every expression in the <span class="nowrap">λ-calculus</span> denotes both a continuous function and a program. ASD does not use the <a href="/wiki/Category_of_sets" title="Category of sets">category of sets</a>, but the full subcategory of overt discrete objects plays this role (an overt object is the dual to a compact object), forming an <a href="/w/index.php?title=Arithmetic_universe&action=edit&redlink=1" class="new" title="Arithmetic universe (page does not exist)">arithmetic universe</a> (pretopos with lists) with general recursion. </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="2001–present"><span id="2001.E2.80.93present"></span>2001–present</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Timeline_of_category_theory_and_related_mathematics&action=edit&section=6" title="Edit section: 2001–present"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="plainlinks metadata ambox mbox-small-left ambox-notice" role="presentation" style="width: auto;"><tbody><tr><td class="mbox-image"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/1/1d/Information_icon4.svg/20px-Information_icon4.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/1/1d/Information_icon4.svg/30px-Information_icon4.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/1/1d/Information_icon4.svg/40px-Information_icon4.svg.png 2x" data-file-width="620" data-file-height="620" /></span></span></td><td class="mbox-text" style="width: auto;"><div class="mbox-text-span">This list is <a href="/wiki/Wikipedia:WikiProject_Lists#Incomplete_lists" title="Wikipedia:WikiProject Lists">incomplete</a>; you can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Timeline_of_category_theory_and_related_mathematics&action=edit">adding missing items</a>. <span class="date-container"><i>(<span class="date">January 2020</span>)</i></span></div></td></tr></tbody></table> <table class="wikitable sortable" width="100%"> <tbody><tr> <th>Year </th> <th style="width:22%">Contributors </th> <th>Event </th></tr> <tr> <td>2001</td> <td>Charles Rezk</td> <td>Constructs a <a href="/wiki/Model_category" title="Model category">model category</a> with certain generalized <a href="/wiki/Segal_category" title="Segal category">Segal categories</a> as the fibrant objects, thus obtaining a model for a homotopy theory of homotopy theories. <a href="/wiki/Complete_Segal_space" class="mw-redirect" title="Complete Segal space">Complete Segal spaces</a> are introduced at the same time. </td></tr> <tr> <td>2001</td> <td>Charles Rezk</td> <td><a href="/w/index.php?title=Model_topos&action=edit&redlink=1" class="new" title="Model topos (page does not exist)">Model toposes</a> and their generalization <a href="/w/index.php?title=Homotopy_topos&action=edit&redlink=1" class="new" title="Homotopy topos (page does not exist)">homotopy toposes</a> (a model topos without the t-completeness assumption). </td></tr> <tr> <td>2002</td> <td><a href="/wiki/Bertrand_To%C3%ABn" title="Bertrand Toën">Bertrand Toën</a>-<a href="/w/index.php?title=Gcabriele_Vezzosi&action=edit&redlink=1" class="new" title="Gcabriele Vezzosi (page does not exist)">Gcabriele Vezzosi</a></td> <td><a href="/w/index.php?title=Segal_topos&action=edit&redlink=1" class="new" title="Segal topos (page does not exist)">Segal toposes</a> coming from <a href="/w/index.php?title=Segal_topology&action=edit&redlink=1" class="new" title="Segal topology (page does not exist)">Segal topologies</a>, <a href="/w/index.php?title=Segal_site&action=edit&redlink=1" class="new" title="Segal site (page does not exist)">Segal sites</a> and stacks over them. </td></tr> <tr> <td>2002</td> <td>Bertrand Toën-Gabriele Vezzosi</td> <td><a href="/wiki/Homotopical_algebraic_geometry" class="mw-redirect" title="Homotopical algebraic geometry">Homotopical algebraic geometry</a>: The main idea is to extend <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> by formally replacing the rings with any kind of "homotopy-ring-like object". More precisely this object is a commutative monoid in a <a href="/wiki/Symmetric_monoidal_category" title="Symmetric monoidal category">symmetric monoidal category</a> endowed with a notion of equivalences which are understood as "up-to-homotopy monoid" (e.g. <a href="/wiki/E-infinity_ring" class="mw-redirect" title="E-infinity ring">E<sub>∞</sub>-rings</a>). </td></tr> <tr> <td>2002</td> <td><a href="/wiki/Peter_Johnstone_(mathematician)" title="Peter Johnstone (mathematician)">Peter Johnstone</a></td> <td>Influential book: sketches of an elephant – a topos theory compendium. It serves as an encyclopedia of <a href="/wiki/Topos" title="Topos">topos</a> theory (two out of three volumes published as of 2008). </td></tr> <tr> <td>2003</td> <td><a href="/wiki/Denis-Charles_Cisinski" title="Denis-Charles Cisinski">Denis-Charles Cisinski</a></td> <td>Makes further work on <a href="/w/index.php?title=ABC_model_category&action=edit&redlink=1" class="new" title="ABC model category (page does not exist)">ABC model categories</a> and brings them back into light. From then they are called ABC model categories after their contributors. </td></tr> <tr> <td>2004</td> <td>Mario Caccamo</td> <td>Formal <a href="/w/index.php?title=Category_theoretical_lambda_calculus&action=edit&redlink=1" class="new" title="Category theoretical lambda calculus (page does not exist)">category theoretical expanded λ-calculus</a> for categories. </td></tr> <tr> <td>2004</td> <td>Francis Borceux-Dominique Bourn</td> <td><a href="/w/index.php?title=Homological_category&action=edit&redlink=1" class="new" title="Homological category (page does not exist)">Homological categories</a> </td></tr> <tr> <td>2004 </td> <td><a href="/wiki/Samson_Abramsky" title="Samson Abramsky">Samson Abramsky</a> and <a href="/wiki/Bob_Coecke" title="Bob Coecke">Bob Coecke</a> </td> <td>Paper <i>A categorical semantics of quantum protocols</i> published that starts the Oxford school of <a href="/wiki/Categorical_quantum_mechanics" title="Categorical quantum mechanics">Categorical Quantum Mechanics</a>, based on the theory of <a href="/wiki/Compact_closed_category" title="Compact closed category">compact closed</a> categories. </td></tr> <tr> <td>2004</td> <td>William Dwyer-Philips Hirschhorn-<a href="/wiki/Daniel_Kan" title="Daniel Kan">Daniel Kan</a>-Jeffrey Smith</td> <td>Introduces in the book <i>Homotopy limit functors on model categories and homotopical categories</i> a formalism of <a href="/w/index.php?title=Homotopical_category&action=edit&redlink=1" class="new" title="Homotopical category (page does not exist)">homotopical categories</a> and <a href="/w/index.php?title=Homotopical_functor&action=edit&redlink=1" class="new" title="Homotopical functor (page does not exist)">homotopical functors</a> (weak equivalence preserving functors) that generalize the <a href="/wiki/Model_category" title="Model category">model category</a> formalism of <a href="/wiki/Daniel_Quillen" title="Daniel Quillen">Daniel Quillen</a>. A homotopical category has only a distinguished class of morphisms (containing all isomorphisms) called weak equivalences and satisfy the two out of six axiom. This allows to define homotopical versions of <a href="/wiki/Initial_and_terminal_objects" title="Initial and terminal objects">initial and terminal objects</a>, <a href="/wiki/Limit_(category_theory)" title="Limit (category theory)">limit and colimit</a> functors (that are computed by local constructions in the book), <a href="/wiki/Complete_category" title="Complete category">completeness and cocompleteness</a>, <a href="/wiki/Adjoint_functors" title="Adjoint functors">adjunctions</a>, <a href="/wiki/Kan_extension" title="Kan extension">Kan extensions</a> and <a href="/wiki/Universal_property" title="Universal property">universal properties</a>. </td></tr> <tr> <td>2004</td> <td><a href="/w/index.php?title=Dominic_Verity&action=edit&redlink=1" class="new" title="Dominic Verity (page does not exist)">Dominic Verity</a></td> <td>Proves the <a href="/w/index.php?title=Street-Roberts_conjecture&action=edit&redlink=1" class="new" title="Street-Roberts conjecture (page does not exist)">Street-Roberts conjecture</a>. </td></tr> <tr> <td>2004</td> <td><a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td>Definition of the descent weak ω-category of a cosimplicial weak ω-category. </td></tr> <tr> <td>2004</td> <td><a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a></td> <td><a href="/wiki/Cosmos_(mathematics)" class="mw-redirect" title="Cosmos (mathematics)">Characterization theorem for cosmoses</a>: A bicategory <b>M</b> is a <a href="/wiki/Cosmos_(mathematics)" class="mw-redirect" title="Cosmos (mathematics)">cosmos</a> iff there exists a base bicategory <b>W</b> such that <b>M</b> is biequivalent to <b>Mod</b><sub><b>W</b></sub>. <b>W</b> can be taken to be any full subbicategory of <b>M</b> whose objects form a small <a href="/wiki/Generator_(category_theory)" title="Generator (category theory)">Cauchy generator</a>. </td></tr> <tr> <td>2004</td> <td><a href="/wiki/Ross_Street" title="Ross Street">Ross Street</a>-Brian Day</td> <td><a href="/w/index.php?title=Quantum_category&action=edit&redlink=1" class="new" title="Quantum category (page does not exist)">Quantum categories</a> and <a href="/wiki/Quantum_groupoid" title="Quantum groupoid">quantum groupoids</a>: A quantum category over a <a href="/wiki/Braided_monoidal_category" title="Braided monoidal category">braided monoidal category</a> <b>V</b> is an object R with an <a href="/w/index.php?title=Opmorphism&action=edit&redlink=1" class="new" title="Opmorphism (page does not exist)">opmorphism</a> <i>h</i>: R<sup>op</sup> ⊗ R → A into a pseudomonoid A such that <i>h</i><sup>*</sup> is strong monoidal (preserves tensor product and unit up to coherent natural isomorphisms) and all R, <i>h</i> and A lie in the autonomous monoidal bicategory Comod(<b>V</b>)<sup>co</sup> of comonoids. Comod(<b>V</b>) = Mod(<b>V</b><sup>op</sup>)<sup>coop</sup>. Quantum categories were introduced to generalize <a href="/wiki/Hopf_algebroid" title="Hopf algebroid">Hopf algebroids</a> and groupoids. A quantum groupoid is a <a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebra</a> with several objects. </td></tr> <tr> <td>2004</td> <td><a href="/w/index.php?title=Stephan_Stolz&action=edit&redlink=1" class="new" title="Stephan Stolz (page does not exist)">Stephan Stolz</a>-<a href="/wiki/Peter_Teichner" title="Peter Teichner">Peter Teichner</a></td> <td>Definition of nD <a href="/wiki/Quantum_field_theory" title="Quantum field theory">QFT</a> of degree <i>p</i> parametrized by a manifold. </td></tr> <tr> <td>2004</td> <td><a href="/w/index.php?title=Stephan_Stolz&action=edit&redlink=1" class="new" title="Stephan Stolz (page does not exist)">Stephan Stolz</a>-<a href="/wiki/Peter_Teichner" title="Peter Teichner">Peter Teichner</a></td> <td><a href="/wiki/Graeme_Segal" title="Graeme Segal">Graeme Segal</a> proposed in the 1980s to provide a geometric construction of <a href="/wiki/Elliptic_cohomology" title="Elliptic cohomology">elliptic cohomology</a> (the precursor to <a href="/wiki/Topological_modular_forms" title="Topological modular forms">tmf</a>) as some kind of moduli space of CFTs. Stephan Stolz and Peter Teichner continued and expanded these ideas in a program to construct <a href="/wiki/Topological_modular_forms" title="Topological modular forms">TMF</a> as a moduli space of supersymmetric Euclidean field theories. They conjectured a <a href="/w/index.php?title=Stolz-Teichner_picture&action=edit&redlink=1" class="new" title="Stolz-Teichner picture (page does not exist)">Stolz-Teichner picture</a> (analogy) between <a href="/wiki/Classifying_space" title="Classifying space">classifying spaces</a> of cohomology theories in the <a href="/wiki/Chromatic_filtration" class="mw-redirect" title="Chromatic filtration">chromatic filtration</a> (de Rham cohomology, K-theory, Morava K-theories) and moduli spaces of supersymmetric QFTs parametrized by a manifold (proved in 0D and 1D). </td></tr> <tr> <td>2005</td> <td>Peter Selinger</td> <td>Coined the term <a href="/wiki/Dagger_category" title="Dagger category">Dagger categories</a> and <a href="/wiki/Dagger_functor" class="mw-redirect" title="Dagger functor">dagger functors</a>. Dagger categories seem to be part of a larger framework involving <a href="/w/index.php?title=N-category_with_duals&action=edit&redlink=1" class="new" title="N-category with duals (page does not exist)"><i>n</i>-categories with duals</a>. </td></tr> <tr> <td>2005</td> <td><a href="/wiki/Peter_Ozsv%C3%A1th" title="Peter Ozsváth">Peter Ozsváth</a>-<a href="/wiki/Zolt%C3%A1n_Szab%C3%B3_(mathematician)" title="Zoltán Szabó (mathematician)">Zoltán Szabó</a></td> <td><a href="/wiki/Khovanov_homology#Related_theories" title="Khovanov homology">Knot Floer homology</a> </td></tr> <tr> <td>2006</td> <td>P. Carrasco-A.R. Garzon-E.M. Vitale</td> <td><a href="/w/index.php?title=Categorical_crossed_module&action=edit&redlink=1" class="new" title="Categorical crossed module (page does not exist)">Categorical crossed modules</a> </td></tr> <tr> <td>2006</td> <td>Aslak Bakke Buan–Robert Marsh–Markus Reineke–<a href="/wiki/Idun_Reiten" title="Idun Reiten">Idun Reiten</a>–<a href="/wiki/Gordana_Todorov" title="Gordana Todorov">Gordana Todorov</a></td> <td><a href="/w/index.php?title=Cluster_category&action=edit&redlink=1" class="new" title="Cluster category (page does not exist)">Cluster categories</a>: Cluster categories are a special case of triangulated <a href="/w/index.php?title=Calabi%E2%80%93Yau_category&action=edit&redlink=1" class="new" title="Calabi–Yau category (page does not exist)">Calabi–Yau categories</a> of Calabi–Yau dimension 2 and a generalization of <a href="/wiki/Cluster_algebra" title="Cluster algebra">cluster algebras</a>. </td></tr> <tr> <td>2006</td> <td><a href="/wiki/Jacob_Lurie" title="Jacob Lurie">Jacob Lurie</a></td> <td>Monumental book: <a href="/wiki/Higher_topos_theory" class="mw-redirect" title="Higher topos theory">Higher topos theory</a>: In its 940 pages Jacob Lurie generalizes the common concepts of category theory to higher categories and defines <a href="/w/index.php?title=N-topos&action=edit&redlink=1" class="new" title="N-topos (page does not exist)"><i>n</i>-toposes</a>, <a href="/wiki/%E2%88%9E-topos" title="∞-topos">∞-toposes</a>, <a href="/w/index.php?title=Sheaves_of_n-types&action=edit&redlink=1" class="new" title="Sheaves of n-types (page does not exist)">sheaves of <i>n</i>-types</a>, <a href="/w/index.php?title=%E2%88%9E-site&action=edit&redlink=1" class="new" title="∞-site (page does not exist)">∞-sites</a>, ∞-<a href="/wiki/Yoneda_lemma" title="Yoneda lemma">Yoneda lemma</a> and proves <a href="/wiki/Lurie_characterization_theorem" class="mw-redirect" title="Lurie characterization theorem">Lurie characterization theorem</a> for higher-dimensional toposes. Lurie's theory of higher toposes can be interpreted as giving a good theory of sheaves taking values in ∞-categories. Roughly an ∞-topos is an ∞-category which looks like the ∞-category of all <a href="/wiki/Homotopy_type" class="mw-redirect" title="Homotopy type">homotopy types</a>. In a topos mathematics can be done. In a higher topos not only mathematics can be done but also "<i>n</i>-geometry", which is <a href="/w/index.php?title=Higher_homotopy_theory&action=edit&redlink=1" class="new" title="Higher homotopy theory (page does not exist)">higher homotopy theory</a>. The <a href="/w/index.php?title=Topos_hypothesis&action=edit&redlink=1" class="new" title="Topos hypothesis (page does not exist)">topos hypothesis</a> is that the (<i>n</i>+1)-category <b><i>n</i>Cat</b> is a Grothendieck (<i>n</i>+1)-topos. Higher topos theory can also be used in a purely algebro-geometric way to solve various moduli problems in this setting. An introduction into this circle of ideas can be found in the <a rel="nofollow" class="external text" href="https://kerodon.net/">Kerodon project</a>. </td></tr> <tr> <td>2007</td> <td><a href="/wiki/Bernhard_Keller" title="Bernhard Keller">Bernhard Keller</a>-Hugh Thomas</td> <td><a href="/w/index.php?title=Cluster_category&action=edit&redlink=1" class="new" title="Cluster category (page does not exist)">d-cluster categories</a> </td></tr> <tr> <td>2007</td> <td><a href="/wiki/Dennis_Gaitsgory" title="Dennis Gaitsgory">Dennis Gaitsgory</a>-<a href="/wiki/Jacob_Lurie" title="Jacob Lurie">Jacob Lurie</a></td> <td>Presents a derived version of the geometric <a href="/wiki/Satake_equivalence" class="mw-redirect" title="Satake equivalence">Satake equivalence</a> and formulates a geometric <a href="/wiki/Langlands_duality" class="mw-redirect" title="Langlands duality">Langlands duality</a> for <a href="/wiki/Quantum_group" title="Quantum group">quantum groups</a>. <p>The geometric Satake equivalence realized the category of representations of the <a href="/wiki/Langlands_dual_group" title="Langlands dual group">Langlands dual group</a> <sup>L</sup><i>G</i> in terms of spherical <a href="/wiki/Perverse_sheaves" class="mw-redirect" title="Perverse sheaves">perverse sheaves</a> (or <a href="/wiki/D-module" title="D-module">D-modules</a>) on the <a href="/wiki/Affine_Grassmannian" title="Affine Grassmannian">affine Grassmannian</a> Gr<sub><i>G</i></sub> = <i>G</i>((<i>t</i>))/<i>G</i>[[t]] of the original group <i>G</i>. </p> </td></tr> <tr> <td>2008</td> <td><a href="/wiki/Ieke_Moerdijk" title="Ieke Moerdijk">Ieke Moerdijk</a>-Clemens Berger</td> <td>Extends and improved the definition of <a href="/w/index.php?title=Reedy_category&action=edit&redlink=1" class="new" title="Reedy category (page does not exist)">Reedy category</a> to become invariant under <a href="/wiki/Equivalence_of_categories" title="Equivalence of categories">equivalence of categories</a>. </td></tr> <tr> <td>2008</td> <td><a href="/wiki/Michael_J._Hopkins" title="Michael J. Hopkins">Michael J. Hopkins</a>–<a href="/wiki/Jacob_Lurie" title="Jacob Lurie">Jacob Lurie</a></td> <td>Sketch of proof of Baez-Dolan <a href="/wiki/Tangle_hypothesis" class="mw-redirect" title="Tangle hypothesis">tangle hypothesis</a> and Baez-Dolan <a href="/wiki/Cobordism_hypothesis" title="Cobordism hypothesis">cobordism hypothesis</a> which classify <a href="/w/index.php?title=Extended_topological_quantum_field_theory&action=edit&redlink=1" class="new" title="Extended topological quantum field theory (page does not exist)">extended TQFT</a> in all dimensions. Jacob Lurie later publishes the complete proof of the cobordism hypothesis (2010). </td></tr> <tr> <td>2019</td> <td><a href="/w/index.php?title=Brendan_Fong&action=edit&redlink=1" class="new" title="Brendan Fong (page does not exist)">Brendan Fong</a>–<a href="/wiki/David_Spivak" title="David Spivak">David Spivak</a></td> <td>First textbook for the emerging field identifying itself as <a href="/wiki/Applied_category_theory" title="Applied category theory">applied category theory</a>, in which category theory is applied outside pure mathematics: <i>An Invitation to Applied Category Theory: Seven Sketches in Compositionality</i> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Timeline_of_category_theory_and_related_mathematics&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%C3%89l%C3%A9ments_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique" title="Éléments de géométrie algébrique">EGA</a></li> <li><a href="/wiki/Fondements_de_la_G%C3%A9ometrie_Alg%C3%A9brique" title="Fondements de la Géometrie Algébrique">FGA</a></li> <li><a href="/wiki/Grothendieck%27s_S%C3%A9minaire_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique" class="mw-redirect" title="Grothendieck's Séminaire de géométrie algébrique">SGA</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Timeline_of_category_theory_and_related_mathematics&action=edit&section=8" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.PaulTaylor.EU/ASD/">Abstract Stone Duality</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Timeline_of_category_theory_and_related_mathematics&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://ncatlab.org/nlab/list">nLab</a>, just as a higher-dimensional Wikipedia, started in late 2008; see <a href="/wiki/NLab" title="NLab">nLab</a></li> <li>Zhaohua Luo; <a rel="nofollow" class="external text" href="http://www.geometry.net/cg/index.html">Categorical geometry homepage</a></li> <li>John Baez, Aaron Lauda; <a rel="nofollow" class="external text" href="http://math.ucr.edu/home/baez/history.pdf">A prehistory of n-categorical physics</a></li> <li>Ross Street; <a rel="nofollow" class="external text" href="http://www.maths.mq.edu.au/~street/Minneapolis.pdf">An Australian conspectus of higher categories</a></li> <li>Elaine Landry, Jean-Pierre Marquis; <a rel="nofollow" class="external text" href="http://philmat.oxfordjournals.org/cgi/reprint/13/1/1">Categories in context: historical, foundational, and philosophical</a></li> <li>Jim Stasheff; <a rel="nofollow" class="external text" href="http://www.math.unc.edu/Faculty/jds/survey.pdf">A survey of cohomological physics</a></li> <li>John Bell; <a rel="nofollow" class="external text" href="http://publish.uwo.ca/~jbell/catlogprime.pdf">The development of categorical logic</a></li> <li>Jean Dieudonné; <a rel="nofollow" class="external text" href="https://web.archive.org/web/20090319200302/http://www.joma.org/images/upload_library/22/Ford/Dieudonne.pdf">The historical development of algebraic geometry</a></li> <li>Charles Weibel; <a rel="nofollow" class="external text" href="http://www.math.uiuc.edu/K-theory/0245/survey.pdf">History of homological algebra</a></li> <li>Peter Johnstone; <a rel="nofollow" class="external text" href="http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.bams/1183550014&page=record">The point of pointless topology</a></li> <li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFStasheff1996" class="citation conference cs1"><a href="/wiki/Jim_Stasheff" title="Jim Stasheff">Stasheff, Jim</a> (January 21, 1996). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/2485648">"The Pre-History Of Operads"</a>. In <a href="/wiki/Jean-Louis_Loday" title="Jean-Louis Loday">Loday, Jean-Louis</a>; <a href="/wiki/Jim_Stasheff" title="Jim Stasheff">Stasheff, James D.</a>; <a href="/wiki/Alexander_A._Voronov" title="Alexander A. Voronov">Voronov, Alexander A.</a> (eds.). <i>Operads: Proceedings of Renaissance Conferences</i>. Contemporary Mathematics. Vol. 202. <a href="/wiki/Providence,_Rhode_Island" title="Providence, Rhode Island">Providence, Rhode Island</a>: <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. pp. 9–14. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.5089">10.1.1.25.5089</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fconm%2F202%2F02592">10.1090/conm/202/02592</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-0513-4" title="Special:BookSources/0-8218-0513-4"><bdi>0-8218-0513-4</bdi></a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0271-4132">0271-4132</a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/96-37049">96-37049</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1436913">1436913</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2021-12-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=The+Pre-History+Of+Operads&rft.btitle=Operads%3A+Proceedings+of+Renaissance+Conferences&rft.place=Providence%2C+Rhode+Island&rft.series=Contemporary+Mathematics&rft.pages=9-14&rft.pub=American+Mathematical+Society&rft.date=1996-01-21&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1436913%23id-name%3DMR&rft_id=info%3Adoi%2F10.1090%2Fconm%2F202%2F02592&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.25.5089%23id-name%3DCiteSeerX&rft.issn=0271-4132&rft_id=info%3Alccn%2F96-37049&rft.isbn=0-8218-0513-4&rft.aulast=Stasheff&rft.aufirst=Jim&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F2485648&rfr_id=info%3Asid%2Fen.wikipedia.org%3ATimeline+of+category+theory+and+related+mathematics" class="Z3988"></span></li> <li>George Whitehead; <a rel="nofollow" class="external text" href="http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183550012">Fifty years of homotopy theory</a></li> <li>Haynes Miller; <a rel="nofollow" class="external text" href="http://www-math.mit.edu/~hrm/papers/ss.ps">The origin of sheaf theory</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul 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<ul><li><a href="/wiki/Timeline_of_numerals_and_arithmetic" title="Timeline of numerals and arithmetic">timeline</a></li></ul></li> <li><a href="/wiki/History_of_calculus" title="History of calculus">Calculus</a> <ul><li><a href="/wiki/Timeline_of_calculus_and_mathematical_analysis" title="Timeline of calculus and mathematical analysis">timeline</a></li> <li><a href="/wiki/History_of_Grandi%27s_series" title="History of Grandi's series">Grandi's series</a></li></ul></li> <li>Category theory <ul><li><a class="mw-selflink selflink">timeline</a></li> <li><a href="/wiki/History_of_topos_theory" title="History of topos theory">Topos theory</a></li></ul></li> <li><a href="/wiki/History_of_combinatorics" title="History of combinatorics">Combinatorics</a></li> <li><a href="/wiki/History_of_the_function_concept" title="History of the function concept">Functions</a> <ul><li><a href="/wiki/History_of_logarithms" title="History of logarithms">Logarithms</a></li></ul></li> <li><a href="/wiki/History_of_geometry" title="History of geometry">Geometry</a> <ul><li><a href="/wiki/History_of_trigonometry" title="History of trigonometry">Trigonometry</a></li> <li><a href="/wiki/Timeline_of_geometry" title="Timeline of geometry">timeline</a></li></ul></li> <li><a href="/wiki/History_of_group_theory" title="History of group theory">Group theory</a></li> <li><a href="/wiki/History_of_information_theory" title="History of information theory">Information theory</a> <ul><li><a href="/wiki/Timeline_of_information_theory" title="Timeline of information theory">timeline</a></li></ul></li> <li><a href="/wiki/History_of_logic" title="History of logic">Logic</a> <ul><li><a href="/wiki/Timeline_of_mathematical_logic" title="Timeline of mathematical logic">timeline</a></li></ul></li> <li><a href="/wiki/History_of_mathematical_notation" title="History of mathematical notation">Math notation</a></li> <li>Number theory <ul><li><a href="/wiki/Timeline_of_number_theory" title="Timeline of number theory">timeline</a></li></ul></li> <li><a href="/wiki/History_of_statistics" title="History of statistics">Statistics</a> <ul><li><a href="/wiki/Timeline_of_probability_and_statistics" title="Timeline of probability and statistics">timeline</a></li> <li><a href="/wiki/History_of_probability" title="History of probability">Probability</a></li></ul></li> <li>Topology <ul><li><a href="/wiki/History_of_manifolds_and_varieties" title="History of manifolds and varieties">Manifolds</a> <ul><li><a href="/wiki/Timeline_of_manifolds" title="Timeline of manifolds">timeline</a></li></ul></li> <li><a href="/wiki/History_of_the_separation_axioms" title="History of the separation axioms">Separation axioms</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Numeral systems</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Prehistoric_counting" title="Prehistoric counting">Prehistoric</a></li> <li><a href="/wiki/History_of_ancient_numeral_systems" title="History of ancient numeral systems">Ancient</a></li> <li><a href="/wiki/History_of_the_Hindu%E2%80%93Arabic_numeral_system" title="History of the Hindu–Arabic numeral system">Hindu-Arabic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By ancient cultures</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Babylonian_mathematics" title="Babylonian mathematics">Mesopotamia</a></li> <li><a href="/wiki/Ancient_Egyptian_mathematics" title="Ancient Egyptian mathematics">Ancient Egypt</a></li> <li><a href="/wiki/Greek_mathematics" title="Greek mathematics">Ancient Greece</a></li> <li><a href="/wiki/Chinese_mathematics" title="Chinese mathematics">China</a></li> <li><a href="/wiki/Indian_mathematics" title="Indian mathematics">India</a></li> <li><a href="/wiki/Mathematics_in_the_medieval_Islamic_world" title="Mathematics in the medieval Islamic world">Medieval Islamic world</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Controversies</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Brouwer%E2%80%93Hilbert_controversy" title="Brouwer–Hilbert controversy">Brouwer–Hilbert</a></li> <li><a href="/wiki/Controversy_over_Cantor%27s_theory" title="Controversy over Cantor's theory">Over Cantor's theory</a></li> <li><a href="/wiki/Leibniz%E2%80%93Newton_calculus_controversy" title="Leibniz–Newton calculus controversy">Leibniz–Newton</a></li> <li><a href="/wiki/Hobbes%E2%80%93Wallis_controversy" title="Hobbes–Wallis controversy">Hobbes–Wallis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li>Women in mathematics <ul><li><a href="/wiki/Timeline_of_women_in_mathematics" title="Timeline of women in mathematics">timeline</a></li></ul></li> <li><a href="/wiki/Approximations_of_%CF%80" title="Approximations of π">Approximations of π</a> <ul><li><a href="/wiki/Chronology_of_computation_of_%CF%80" title="Chronology of computation of π">timeline</a></li></ul></li> <li><a href="/wiki/Future_of_mathematics" title="Future of mathematics">Future of mathematics</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, 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