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xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown"> <h3 id="context">Context</h3> <h4 id="foundations">Foundations</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a> <h2 id="the_basis_of_it_all">The basis of it all</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mathematical+logic">mathematical logic</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deduction+system">deduction system</a>, <a class="existingWikiWord" href="/nlab/show/natural+deduction">natural deduction</a>, <a class="existingWikiWord" href="/nlab/show/sequent+calculus">sequent calculus</a>, <a class="existingWikiWord" href="/nlab/show/lambda-calculus">lambda-calculus</a>, <a class="existingWikiWord" href="/nlab/show/judgment">judgment</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/type+theory">type theory</a>, <a class="existingWikiWord" href="/nlab/show/simple+type+theory">simple type theory</a>, <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/collection">collection</a>, <a class="existingWikiWord" href="/nlab/show/object">object</a>, <a class="existingWikiWord" href="/nlab/show/type">type</a>, <a class="existingWikiWord" href="/nlab/show/term">term</a>, <a class="existingWikiWord" href="/nlab/show/set">set</a>, <a class="existingWikiWord" href="/nlab/show/element">element</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/equality">equality</a>, <a class="existingWikiWord" href="/nlab/show/judgmental+equality">judgmental equality</a>, <a class="existingWikiWord" href="/nlab/show/typal+equality">typal equality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/universe">universe</a>, <a class="existingWikiWord" href="/nlab/show/size+issues">size issues</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher-order+logic">higher-order logic</a> </li> </ul> <h2 id="set_theory"> Set theory</h2> <div> <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a> <ul> <li>fundamentals of set theory <ul> <li><a class="existingWikiWord" href="/nlab/show/propositional+logic">propositional logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+logic">first-order logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/typed+predicate+logic">typed predicate logic</a></li> <li><a class="existingWikiWord" href="/nlab/show/membership+relation">membership relation</a></li> <li><a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional equality</a></li> <li><a class="existingWikiWord" href="/nlab/show/set">set</a>, <a class="existingWikiWord" href="/nlab/show/element">element</a>, <a class="existingWikiWord" href="/nlab/show/function">function</a>, <a class="existingWikiWord" href="/nlab/show/relation">relation</a></li> <li><a class="existingWikiWord" href="/nlab/show/universe">universe</a>, <a class="existingWikiWord" href="/nlab/show/small+set">small set</a>, <a class="existingWikiWord" href="/nlab/show/large+set">large set</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/membership+relation">membership relation</a>, <a class="existingWikiWord" href="/nlab/show/propositional+equality">propositional equality</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+extensionality">axiom of extensionality</a></li> <li><a class="existingWikiWord" href="/nlab/show/pairing+structure">pairing structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+pairing">axiom of pairing</a></li> <li><a class="existingWikiWord" href="/nlab/show/union+structure">union structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+union">axiom of union</a></li> <li><a class="existingWikiWord" href="/nlab/show/powerset+structure">powerset structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+power+sets">axiom of power sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/natural+numbers+structure">natural numbers structure</a>, <a class="existingWikiWord" href="/nlab/show/axiom+of+infinity">axiom of infinity</a></li> </ul> </li> <li>presentations of set theory <ul> <li><a class="existingWikiWord" href="/nlab/show/first-order+set+theory">first-order set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/unsorted+set+theory">unsorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/simply+sorted+set+theory">simply sorted set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/one-sorted+set+theory">one-sorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/two-sorted+set+theory">two-sorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/three-sorted+set+theory">three-sorted set theory</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/dependently+sorted+set+theory">dependently sorted set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/structurally+presented+set+theory">structurally presented set theory</a></li> </ul> </li> <li>structuralism in set theory <ul> <li><a class="existingWikiWord" href="/nlab/show/material+set+theory">material set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/ZFC">ZFC</a></li> <li><a class="existingWikiWord" href="/nlab/show/ZFA">ZFA</a></li> <li><a class="existingWikiWord" href="/nlab/show/Mostowski+set+theory">Mostowski set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/New+Foundations">New Foundations</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/categorical+set+theory">categorical set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/ETCS">ETCS</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/fully+formal+ETCS">fully formal ETCS</a></li> <li><a class="existingWikiWord" href="/nlab/show/ETCS+with+elements">ETCS with elements</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+on+ETCS+I">Trimble on ETCS I</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+on+ETCS+II">Trimble on ETCS II</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+on+ETCS+III">Trimble on ETCS III</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/structural+ZFC">structural ZFC</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/allegorical+set+theory">allegorical set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/SEAR">SEAR</a></li> </ul> </li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/class-set+theory">class-set theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/class">class</a>, <a class="existingWikiWord" href="/nlab/show/proper+class">proper class</a></li> <li><a class="existingWikiWord" href="/nlab/show/universal+class">universal class</a>, <a class="existingWikiWord" href="/nlab/show/universe">universe</a></li> <li><a class="existingWikiWord" href="/nlab/show/category+of+classes">category of classes</a></li> <li><a class="existingWikiWord" href="/nlab/show/category+with+class+structure">category with class structure</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/constructive+set+theory">constructive set theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+set+theory">algebraic set theory</a></li> </ul> </div> <h2 id="foundational_axioms">Foundational axioms</h2> <div> <a class="existingWikiWord" href="/nlab/show/foundational+axiom">foundational</a> <a class="existingWikiWord" href="/nlab/show/axioms">axioms</a> <ul> <li> basic constructions: <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+cartesian+products">axiom of cartesian products</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+disjoint+unions">axiom of disjoint unions</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+the+empty+set">axiom of the empty set</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+fullness">axiom of fullness</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+function+sets">axiom of function sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+power+sets">axiom of power sets</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+quotient+sets">axiom of quotient sets</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/material+set+theory">material axioms</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+extensionality">axiom of extensionality</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+foundation">axiom of foundation</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+anti-foundation">axiom of anti-foundation</a></li> <li><a class="existingWikiWord" href="/nlab/show/Mostowski%27s+axiom">Mostowski's axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+pairing">axiom of pairing</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+transitive+closure">axiom of transitive closure</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+union">axiom of union</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural axioms</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+materialization">axiom of materialization</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/type+theory">type theoretic axioms</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/axioms+of+set+truncation">axioms of set truncation</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/uniqueness+of+identity+proofs">uniqueness of identity proofs</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+K">axiom K</a></li> <li><a class="existingWikiWord" href="/nlab/show/boundary+separation">boundary separation</a></li> <li><a class="existingWikiWord" href="/nlab/show/equality+reflection">equality reflection</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+circle+type+localization">axiom of circle type localization</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theoretic axioms</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/univalence+axiom">univalence axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+principle">Whitehead's principle</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/axioms+of+choice">axioms of choice</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+countable+choice">axiom of countable choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+dependent+choice">axiom of dependent choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+excluded+middle">axiom of excluded middle</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+existence">axiom of existence</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+multiple+choice">axiom of multiple choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/Markov%27s+axiom">Markov's axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/presentation+axiom">presentation axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/small+cardinality+selection+axiom">small cardinality selection axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+small+violations+of+choice">axiom of small violations of choice</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+weakly+initial+sets+of+covers">axiom of weakly initial sets of covers</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/large+cardinal+axioms">large cardinal axioms</a>: <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+infinity">axiom of infinity</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+universes">axiom of universes</a></li> <li><a class="existingWikiWord" href="/nlab/show/regular+extension+axiom">regular extension axiom</a></li> <li><a class="existingWikiWord" href="/nlab/show/inaccessible+cardinal">inaccessible cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/measurable+cardinal">measurable cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/elementary+embedding">elementary embedding</a></li> <li><a class="existingWikiWord" href="/nlab/show/supercompact+cardinal">supercompact cardinal</a></li> <li><a class="existingWikiWord" href="/nlab/show/Vop%C4%9Bnka%27s+principle">Vopěnka's principle</a></li> </ul> </li> <li> strong axioms <ul> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+separation">axiom of separation</a></li> <li><a class="existingWikiWord" href="/nlab/show/axiom+of+replacement">axiom of replacement</a></li> </ul> </li> <li> further <ul> <li><a class="existingWikiWord" href="/nlab/show/reflection+principle">reflection principle</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/axiom+of+tight+apartness">axiom of tight apartness</a> </li> </ul> </div> <h2 id="removing_axioms">Removing axioms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/constructive+mathematics">constructive mathematics</a></li> <li><a class="existingWikiWord" href="/nlab/show/predicative+mathematics">predicative mathematics</a></li> </ul> <div> <a href="/nlab/edit/foundations+-+contents">Edit this sidebar</a> </div></div></div> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/category">category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/functor">functor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </div></div></div> <h4 id="topos_theory">Topos Theory</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Toposes">Toposes</a></li> </ul> <h2 id="background">Background</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/category">category</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/functor">functor</a> </li> </ul> </li> </ul> <h2 id="toposes">Toposes</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topos">topos</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/site">site</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/sieve">sieve</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+pretopology">pretopology</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">topology</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, <a class="existingWikiWord" href="/nlab/show/indexed+topos">indexed topos</a> </li> </ul> <h2 id="toc_internal_logic">Internal Logic</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a> </li> </ul> </li> </ul> <h2 id="topos_morphisms">Topos morphisms</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a>/<a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/global+section">global sections</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective geometric morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/connected+geometric+morphism">connected geometric morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/totally+connected+geometric+morphism">totally connected geometric morphism</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">étale geometric morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/separated+geometric+morphism">separated geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+topos">Hausdorff topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/bounded+geometric+morphism">bounded geometric morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/localic+geometric+morphism">localic geometric morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/hyperconnected+geometric+morphism">hyperconnected geometric morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/atomic+geometric+morphism">atomic geometric morphism</a> </li> </ul> </li> </ul> <h2 id="extra_stuff_structure_properties">Extra stuff, structure, properties</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/topological+locale">topological locale</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/localic+topos">localic topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos/gros topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a> </li> </ul> <h2 id="cohomology_and_homotopy">Cohomology and homotopy</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos">homotopy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> </li> </ul> <h2 id="in_higher_category_theory">In higher category theory</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2-site">2-site</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a> </li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Barr%27s+theorem">Barr's theorem</a> </li> </ul> <div> <a href="/nlab/edit/topos+theory+-+contents">Edit this sidebar</a> </div></div></div> <h4 id="synthetic_differential_geometry">Synthetic differential geometry</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic</a> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> Introductions <a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">from point-set topology to differentiable manifolds</a> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+coordinate+systems">coordinate systems</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+spaces">smooth spaces</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds">manifolds</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">supergeometry</a> Differentials <ul> <li> <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>, <a class="existingWikiWord" href="/nlab/show/amazing+right+adjoint">amazing right adjoint</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/V-manifolds">V-manifolds</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/analytic+manifold">analytic manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/formal+smooth+manifold">formal smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a> </li> </ul> Tangency <ul> <li> <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>, <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a>, <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a>; </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+forms+in+synthetic+differential+geometry">differential forms</a>, <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>, <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a>, <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant differential form</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a>, <a class="existingWikiWord" href="/nlab/show/horizontal+differential+form">horizontal differential form</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/cogerm+differential+form">cogerm differential form</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/immersion">immersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a>, <a class="existingWikiWord" href="/nlab/show/crystal">crystal</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a> </li> </ul> The magic algebraic facts <ul> <li> <a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">derivations of smooth functions are vector fields</a> </li> </ul> Theorems <ul> <li> <a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Borel%27s+theorem">Borel's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Boman%27s+theorem">Boman's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitney+extension+theorem">Whitney extension theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Steenrod-Wockel+approximation+theorem">Steenrod-Wockel approximation theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hochschild-Kostant-Rosenberg+theorem">Hochschild-Kostant-Rosenberg theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a> </li> </ul> Axiomatics <ul> <li> <a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, <a class="existingWikiWord" href="/nlab/show/super+smooth+topos">super smooth topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/microlinear+space">microlinear space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integration+axiom">integration axiom</a> </li> </ul> </li> </ul> <div> <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a> <ul> <li> (<a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math> <ul> <li> <a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+object">codiscrete object</a>, <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures">structures in cohesion</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/dR-shape+modality">dR-shape modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dR-flat+modality">dR-flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">ʃ</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math> </li> </ul> <a class="existingWikiWord" href="/nlab/show/infinitesimal+cohesion">infinitesimal cohesion</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+modality">classical modality</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesion</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+cohomology+diagram">differential cohomology diagram</a></li> </ul> <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> <ul> <li> (<a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+flat+modality">infinitesimal flat modality</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Re \dashv \Im \dashv \&)</annotation></semantics></math> <ul> <li> <a class="existingWikiWord" href="/nlab/show/reduced+object">reduced object</a>, <a class="existingWikiWord" href="/nlab/show/coreduced+object">coreduced object</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+object">formally smooth object</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+map">formally étale map</a> </li> <li> <a href="cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion">structures in differential cohesion</a> </li> </ul> </li> </ul> <a class="existingWikiWord" href="/nlab/show/super+smooth+infinity-groupoid">graded differential cohesion</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/fermionic+modality">fermionic modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/rheonomy+modality">rheonomy modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math> </li> </ul> <a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">singular cohesion</a> <div id="Diagram" class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant="normal">R</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi mathvariant="normal">h</mi></mtd> <mtd><mover><mrow></mrow><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow></mrow><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>cohesive</mi></mover></mtd> <mtd><mo lspace="0em" rspace="thinmathspace">ʃ</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow></mrow><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div></div> Models <ul> <li> <a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-ring) </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Fermat+theory">Fermat theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/formal+smooth+%E2%88%9E-groupoid">formal smooth ∞-groupoid</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a>, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>, <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+complex">Euler-Lagrange complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a>, </li> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a>, <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>, <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Wilson+line">Wilson line</a>, <a class="existingWikiWord" href="/nlab/show/Wilson+surface">Wilson surface</a> </li> </ul> <a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> (<a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super</a>, <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher</a>) <ul> <li> <a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>, (<a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a>, <a class="existingWikiWord" href="/nlab/show/hyperbolic+geometry">hyperbolic geometry</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+geometry">elliptic geometry</a> </li> <li> (<a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/Killing+vector+field">Killing vector field</a>, <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, <a class="existingWikiWord" href="/nlab/show/super-spacetime">super-spacetime</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a> </li> </ul> </div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a> </li> </ul> <hr /> <a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a> <a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity">gravity</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a> <ul> <li> Axiomatizations <ul> <li> <a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/local+net">local net</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem </li> <li> <a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a> </li> </ul> </li> </ul> </li> <li> Tools <ul> <li> <a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> Structural phenomena <ul> <li> <a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/instanton">instanton</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a> </li> </ul> </li> <li> Types of quantum field thories <ul> <li> <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/CFT">CFT</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a> </li> <li> examples <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> <ul> <li> <a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/membrane">membrane</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a> </li> </ul> </li> </ul> </li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#overview'>Overview</a></li> <li><a href='#topics'>Topics</a></li> <ul> <li><a href='#main_contributions'>Main contributions</a></li> <li><a href='#MotivationFromFoundationsOfPhysics'>Mathematics relating to Physics</a></li> <li><a href='#RelationToPhilosophy'>Mathematics relating to Philosophy</a></li> </ul> <li><a href='#Talks'>Talks, Lecture notes and Publications</a></li> </ul> </div> <h2 id="overview">Overview</h2> <blockquote> Talking with Bill, I often feel like a fly buzzing around a cow. (It seems to me I can liken Bill to a cow, if I’m just a fly myself.) On any easy question, I’ll probably see the answer first. But his thoughts seem to move on a level where I don’t function, I can barely see down there. (From an <a href="http://at.yorku.ca/t/o/p/c/06.htm">interview</a> with <a class="existingWikiWord" href="/nlab/show/John+Isbell">John Isbell</a>.) </blockquote> F. William Lawvere (February 9, 1937 – January 23, 2023) <ul> <li>(<a href="http://www.acsu.buffalo.edu/~wlawvere/">web site</a>, <a href="http://en.wikipedia.org/wiki/William_Lawvere">Wikipedia entry</a>)</li> </ul> was an influential <a class="existingWikiWord" href="/nlab/show/category+theory">category theorist</a>. See <ul> <li id="Interview07">An interview with F. William Lawvere, <a href="http://www.cim.pt/?q=publications">Bulletin of the International Center for Mathematics</a> (<a href="http://www.cim.pt/files/publications/b23www.pdf">part 1, December 2007</a>, <a href="http://www.cim.pt/files/publications/b24www.pdf">part 2, June 2008</a>, <a href="http://www.mat.uc.pt/~picado/lawvere/interview.pdf">full pdf</a>, <a class="existingWikiWord" href="/nlab/files/LawvereInterview.pdf" title="pdf">pdf</a>)</li> </ul> for a survey of his academic path and work. See also <ul> <li>Interview by Felice Cardone, March 2013 (<a href="https://dl.dropboxusercontent.com/u/92056191/Archive/temporary_new_material/lawvere-cardone-2000.html">link</a>)</li> </ul> for an interview on his contributions to <a class="existingWikiWord" href="/nlab/show/categorical+logic">categorical logic</a>. A recording of the commemorative special session in honour of Bill Lawvere from CT2023 can be found here: (<a href="https://www.youtube.com/watch?v=x23KS3VOpRg&list=PLu4STGsfbix9l6rPxGsjG6Gl822k1hj2X&index=1&pp=iAQB">video</a>) For a (somewhat random) list of further links see also at “<a href="http://conceptualmathematics.wordpress.com/2013/02/09/happy-birthday-professor-f-william-lawvere/">conceptual mathematics</a>”. <h2 id="topics">Topics</h2> <h3 id="main_contributions">Main contributions</h3> Lawvere invented <a class="existingWikiWord" href="/nlab/show/categorical+logic">categorical logic</a> and introduced the eponymous <a class="existingWikiWord" href="/nlab/show/Lawvere+theories">Lawvere theories</a> as a <a class="existingWikiWord" href="/nlab/show/category+theory">category-theoretic</a> way to describe finitary <a class="existingWikiWord" href="/nlab/show/algebraic+theories">algebraic theories</a>. He generalised <a class="existingWikiWord" href="/nlab/show/Grothendieck+toposes">Grothendieck toposes</a> to <a class="existingWikiWord" href="/nlab/show/elementary+toposes">elementary toposes</a>, revolutionising the <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a> of <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a>; in this vein, he developed the <a class="existingWikiWord" href="/nlab/show/foundation">foundation</a> in <a class="existingWikiWord" href="/nlab/show/structural+set+theory">structural set theory</a> called <a class="existingWikiWord" href="/nlab/show/ETCS">ETCS</a>. He also introduced and worked on <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> as a foundation for <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> and <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> in <a class="existingWikiWord" href="/nlab/show/continuum+physics">continuum physics</a>. Later he introduced the notion of <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> as a more general foundation of <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>. <h3 id="MotivationFromFoundationsOfPhysics">Mathematics relating to Physics</h3> A central motivation for Lawvere’s work is the search for a good mathematical <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a> of <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, specifically of (<a class="existingWikiWord" href="/nlab/show/classical+physics">classical</a>) <a class="existingWikiWord" href="/nlab/show/continuum+mechanics">continuum mechanics</a> (or at least some <a class="existingWikiWord" href="/nlab/show/kinematics">kinematical</a> aspects thereof, Lawvere does not seem to mention <a class="existingWikiWord" href="/nlab/show/Hamiltonians">Hamiltonians</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangians">Lagrangians</a> or <a class="existingWikiWord" href="/nlab/show/action+functionals">action functionals</a>). In (<a href="http://www.mat.uc.pt/~picado/lawvere/interview.pdf">interview, p. 8</a>) he recalls: <blockquote> I had been a student at Indiana University from 1955 to January 1960. I liked <a class="existingWikiWord" href="/nlab/show/experiment">experimental</a> <a class="existingWikiWord" href="/nlab/show/physics">physics</a> but did not appreciate the imprecise reasoning in some theoretical courses. So I decided to study <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a> first. <a class="existingWikiWord" href="/nlab/show/Clifford+Truesdell">Truesdell</a> was at the Mathematics Department but he had a great knowledge in Engineering Physics. He took charge of my education there. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math> in 1955 (and subsequently) had advised me on pursuing the study of <a class="existingWikiWord" href="/nlab/show/continuum+mechanics">continuum mechanics</a> and <a class="existingWikiWord" href="/nlab/show/kinetic+theory">kinetic theory</a>. In Summer 1958 I studied <a class="existingWikiWord" href="/nlab/show/topological+dynamics">Topological Dynamics</a> with George Whaples, with the agenda of understanding as much as possible in categorical terms. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math> Categories would clearly be important for simplifying the foundations of <a class="existingWikiWord" href="/nlab/show/continuum+physics">continuum physics</a>. I concluded that I would make category theory a central line of my study. </blockquote> Then in (<a href="http://www.mat.uc.pt/~picado/lawvere/interview.pdf">interview, p. 11</a>) about the early 1960s: <blockquote> I felt a strong need to learn more <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a> and <a class="existingWikiWord" href="/nlab/show/logic">logic</a> from experts in that field, still of course with the aim of clarifying the foundations of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> and of <a class="existingWikiWord" href="/nlab/show/physics">physics</a>. </blockquote> The title of the early text <a class="existingWikiWord" href="/nlab/show/Toposes+of+laws+of+motion">Toposes of laws of motion</a>, which is often cited as the text introducing <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>, clearly witnesses the origin and motivation of these ideas in <a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a>. On this, in (<a href="http://www.mat.uc.pt/~picado/lawvere/interview.pdf">interview, p. 15</a>): <blockquote> Q: As an assistant professor in Chicago, in 1967, you taught with Mac Lane a course on <a class="existingWikiWord" href="/nlab/show/mechanics">Mechanics</a>, where you “started to think about the justification of older intuitive methods in geometry” You called it “<a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>”. How did you arrive at the program of <a class="existingWikiWord" href="/nlab/show/Categorical+dynamics">Categorical Dynamics</a> and <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">Synthetic Differential Geometry</a>? A: From January 1967 to August 1967 I was Assistant Professor at the University of Chicago. Mac Lane and I soon organized to teach a joint course based on Mackey’s book “Mathematical Foundations of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a>”. Q: So, Mackey, a functional analyst from Harvard mainly concerned with the relationship between quantum mechanics and representation theory, had some relation to category theory. </blockquote> Then (<a href="http://www.mat.uc.pt/~picado/lawvere/interview.pdf">interview, p. 16</a>): <blockquote> Q: Back to the origins of Synthetic Differential Geometry, where did the idea of organizing such a joint course on Mechanics originate ? Apparently, Chandra had suggested that Saunders give some courses relevant to physics, and our joint course was the first of a sequence. Eventually Mac Lane gave a talk about the <a class="existingWikiWord" href="/nlab/show/Hamilton-Jacobi+equation">Hamilton-Jacobi equation</a> at the Naval Academy in summer 1970 that was published in the American Mathematical Monthly A: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math> I began to apply the Grothendieck topos theory that I had learned from Gabriel to the problem of simplied foundations of <a class="existingWikiWord" href="/nlab/show/continuum+mechanics">continuum mechanics</a> as it had been inspired by <a class="existingWikiWord" href="/nlab/show/Clifford+Truesdell">Truesdell</a>‘s teachings, <a class="existingWikiWord" href="/nlab/show/Walter+Noll">Noll</a>’s <a class="existingWikiWord" href="/nlab/show/axiom">axiomatizations</a>, and by my 1958 efforts to render categorical the subject of <a class="existingWikiWord" href="/nlab/show/topological+dynamics">topological dynamics</a>. </blockquote> A review of this with more comments on more relations to physics is in the introduction to the book collection <a class="existingWikiWord" href="/nlab/show/Categories+in+Continuum+Physics">Categories in Continuum Physics</a>, which is the proceedings of a meeting organized by Lawvere in 1982. In the talk <a class="existingWikiWord" href="/nlab/show/Toposes+of+laws+of+motion">Toposes of laws of motion</a> in 1997, Lawvere starts with the following remark <blockquote> I read somewhere recently that the basic program of infinitesimal <a class="existingWikiWord" href="/nlab/show/calculus">calculus</a>, <a class="existingWikiWord" href="/nlab/show/continuum+mechanics">continuum mechanics</a>, and <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> is that all the world can be reconstructed from the <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitely small</a>. One may think this is not possible, but nonetheless it’s certainly a program that has been very fruitful over the last 300 years. I think we are now finally in a position to actually make more explicit what that program amounts to. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math> I think that on the basis of these developments we can focus on this question of making very explicit how <a class="existingWikiWord" href="/nlab/show/continuum+physics">continuum physics</a> etc. can be built up mathematically from very simple ingredients. </blockquote> In the same talk, a few lines later after discussion of <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+points">infinitesimally thickened points</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, it says: <blockquote> The basic <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> which are needed for <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a> and theories of <a class="existingWikiWord" href="/nlab/show/physical+fields">physical fields</a> are thus in some sense available in any <a class="existingWikiWord" href="/nlab/show/topos">topos</a> with a suitable <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>. </blockquote> In 2000 in the article <a class="existingWikiWord" href="/nlab/show/Comments+on+the+development+of+topos+theory">Comments on the development of topos theory</a> Lawvere writes in the closing section 7 titled “From and to continuum physics”: <blockquote> What was the impetus which demanded the simplification and generalization of <a class="existingWikiWord" href="/nlab/show/Grothendieck">Grothendieck</a>‘s concept of <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, if indeed the application to <a class="existingWikiWord" href="/nlab/show/logic">logic</a> and <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a> were not decisive. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math> My own motivation came from my earlier study of <a class="existingWikiWord" href="/nlab/show/physics">physics</a>. The <a class="existingWikiWord" href="/nlab/show/foundation">foundation</a> of the <a class="existingWikiWord" href="/nlab/show/continuum+physics">continuum physics</a> of general materials, in the spirit of <a class="existingWikiWord" href="/nlab/show/Clifford+Truesdell">Truesdell</a>, <a class="existingWikiWord" href="/nlab/show/Walter+Noll">Noll</a> and others, involves powerful and clear physical ideas, which unfortunately have been submerged under a mathematical apparatus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math>. But, as Fichera <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>25</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[25]</annotation></semantics></math> lamented, all this apparatus gives often a very uncertain fit to the phenomena. The apparatus may well be helpful in the solution of certain problems, but can the problems themselves and the needed axioms be stated in a direct and clear manner? And might this not lead to a simpler, equally rigorous account? These were the questions to which I began to apply the <a class="existingWikiWord" href="/nlab/show/topos">topos</a> method in my 1967 Chicago lectures <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Categorical+dynamics">Categorical dynamics</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math>. It was clear that work on the notion of topos itself would be needed to achieve the goal. I had spent 1961-62 with the Berkeley logicians, believing that listening to experts on <a class="existingWikiWord" href="/nlab/show/foundations">foundations</a> might be the road to clarifying foundational questions. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math> Several books treating the simplified <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a> (<a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">MacLane-Moerdijk</a> being the most recent and readable text), together with the three excellent books on <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[...]</annotation></semantics></math> provide a solid basis on which further treatment of <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a> and the needed development of <a class="existingWikiWord" href="/nlab/show/continuum+physics">continuum physics</a> can be based. </blockquote> The <a href="http://en.wikipedia.org/wiki/William_Lawvere">Wikipedia entry</a> concludes: <blockquote> Lawvere continues to work on his 50-year quest for a rigorous flexible base for physical ideas, free of unnecessary analytic complications. </blockquote> See also at <a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a> for more on this. <h3 id="RelationToPhilosophy">Mathematics relating to Philosophy</h3> Lawvere has proposed formalizations in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, <a class="existingWikiWord" href="/nlab/show/categorical+logic">categorical logic</a> and <a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a> of concepts which are motivated from <a class="existingWikiWord" href="/nlab/show/philosophy">philosophy</a>, notably in <a class="existingWikiWord" href="/nlab/show/Georg+Hegel">Georg Hegel</a>‘s <a class="existingWikiWord" href="/nlab/show/Science+of+Logic">Science of Logic</a> (see there for more). This includes for instance definitions of concepts found there such as: <ul> <li><a class="existingWikiWord" href="/nlab/show/objective+and+subjective+logic">objective and subjective logic</a>, <a class="existingWikiWord" href="/nlab/show/abstract+general">abstract general</a>, <a class="existingWikiWord" href="/nlab/show/concrete+general">concrete general</a>, <a class="existingWikiWord" href="/nlab/show/concrete+particular">concrete particular</a>, <a class="existingWikiWord" href="/nlab/show/unity+of+opposites">unity of opposites</a>, <a class="existingWikiWord" href="/nlab/show/Aufhebung">Aufhebung</a>, <a class="existingWikiWord" href="/nlab/show/being">being</a>, <a class="existingWikiWord" href="/nlab/show/becoming">becoming</a>, <a class="existingWikiWord" href="/nlab/show/space+and+quantity">space and quantity</a>, <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a>, <a class="existingWikiWord" href="/nlab/show/intensive+and+extensive+quantity">intensive and extensive quantity</a></li> </ul> (see the references in these entries for pointers). In (<a href="#Lawvere92">Lawvere 92</a>) it says: <blockquote> It is my belief that in the next decade and in the next century the technical advances forged by category theorists will be of value to dialectical philosophy, lending precise form with disputable mathematical models to ancient philosophical distinctions such as general vs. particular, objective vs. subjective, being vs. becoming, space vs. quantity, equality vs. difference, quantitative vs. qualitative etc. In turn the explicit attention by mathematicians to such philosophical questions is necessary to achieve the goal of making mathematics (and hence other sciences) more widely learnable and useable. Of course this will require that philosophers learn mathematics and that mathematicians learn philosophy. </blockquote> A precursor to this undertaking is <a class="existingWikiWord" href="/nlab/show/Hermann+Grassmann">Hermann Grassmann</a> with his <a class="existingWikiWord" href="/nlab/show/Ausdehnungslehre">Ausdehnungslehre</a> (<a href="#Lawvere95">Lawvere 95</a>), see there for more. <h2 id="Talks">Talks, Lecture notes and Publications</h2> The following is a list of texts by Lawvere, equipped with brief comments and hyperlinks to further material on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>Lab. See also the <ul> <li> <a href="https://lawverearchives.com/">The Lawvere Archives</a> </li> <li> <a class="existingWikiWord" href="/nlab/show/Matt+Earnshaw">Matt Earnshaw</a>: <a href="https://github.com/mattearnshaw/lawvere">repository of Lawvere’s publications</a> </li> </ul> (Some of Lawvere’s writings don’t exist as published articles, but circulate in some form or other. Notably the “<a class="existingWikiWord" href="/nlab/show/Archive+for+Mathematical+Sciences+%26+Philosophy">Archive for Mathematical Sciences Philosophy</a>” run by <a class="existingWikiWord" href="/nlab/show/Michael+Wright">Michael Wright</a> has a lot of recordings or lectures by Lawvere, though presently few or none of the files in the archive are available online.) <ul> <li> <a class="existingWikiWord" href="/nlab/show/Functorial+Semantics+of+Algebraic+Theories">Functorial Semantics of Algebraic Theories</a> Originally published as: Ph.D. thesis, Columbia University, 1963 and in Reports of the Midwest Category Seminar II, 1968, 41-61, Republished in: Reprints in Theory and Applications of Categories, No. 5 (2004) pp 1-121 (<a href="http://www.tac.mta.ca/tac/reprints/articles/5/tr5abs.html">tac</a>) (on <a class="existingWikiWord" href="/nlab/show/algebraic+theories">algebraic theories</a>, now also called <a class="existingWikiWord" href="/nlab/show/Lawvere+theories">Lawvere theories</a>) </li> <li> An elementary theory of the category of sets, 1964, Proceedings of the National Academy of Science of the U.S.A 52, 1506-1511. Expanded version with commentary by <a class="existingWikiWord" href="/nlab/show/Colin+McLarty">Colin McLarty</a> and the author in: Reprints in Theory and Applications of Categories, No. 11 (2005) pp. 1-35 (<a href="http://www.tac.mta.ca/tac/reprints/articles/11/tr11abs.html">tac</a>) <blockquote> (axiomatizing the category <a class="existingWikiWord" href="/nlab/show/Set">Set</a> of sets; see at <a class="existingWikiWord" href="/nlab/show/ETCS">ETCS</a>) </blockquote> </li> <li> The Category of Categories as a Foundation for Mathematics, pp. 1-20 in: <a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">S. Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/D.+K.+Harrison">D. K. Harrison</a>, <a class="existingWikiWord" href="/nlab/show/S.+MacLane">S. MacLane</a>, <a class="existingWikiWord" href="/nlab/show/H.+R%C3%B6hrl">H. Röhrl</a> (eds.): <a class="existingWikiWord" href="/nlab/show/Proceedings+of+the+Conference+on+Categorical+Algebra+-+La+Jolla+1965">Proceedings of the Conference on Categorical Algebra - La Jolla 1965</a>, Springer (1966) [<a href="https://doi.org/10.1007/978-3-642-99902-4">doi:10.1007/978-3-642-99902-4</a>] <blockquote> (axiomatizing the <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> of categories; see at <a class="existingWikiWord" href="/nlab/show/ETCC">ETCC</a>) </blockquote> </li> <li> <a class="existingWikiWord" href="/nlab/show/Categorical+dynamics">Categorical dynamics</a>, 1967 Chicago lectures (<a href="http://www.mat.uc.pt/~ct2011/abstracts/lawvere_w.pdf">pdf</a>) Categorical Dynamics Revisited, talk at Sets Within Geometry, Nancy, France 26-29 July 2011 (<a href="https://www.youtube.com/watch?v=sXATg9_6iWE">recording</a>) (on <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> as a context for axiomatizing <a class="existingWikiWord" href="/nlab/show/ordinary+differential+equations">ordinary differential equations</a> as they appear in <a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a>) </li> <li id="AdjointnessInFoundations"> <a class="existingWikiWord" href="/nlab/show/Adjointness+in+Foundations">Adjointness in Foundations</a>, (<a href="http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html">TAC</a>), Dialectica 23 (1969), 281-296 (on the interpretation of <a class="existingWikiWord" href="/nlab/show/quantifiers">quantifiers</a> in <a class="existingWikiWord" href="/nlab/show/categorical+logic">categorical logic</a> as <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> <a class="existingWikiWord" href="/nlab/show/adjoints">adjoints</a>) </li> <li id="Lawvere69"> <a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a>, Diagonal Arguments and Cartesian Closed Categories, Lecture Notes in Mathematics, 92 (1969) 134-145 [<a href="http://tac.mta.ca/tac/reprints/articles/15/tr15abs.html">tac:15</a>] (On deriving <a class="existingWikiWord" href="/nlab/show/G%C3%B6del%27s+incompleteness+theorem">Gödel's incompleteness theorem</a> etc. from <a class="existingWikiWord" href="/nlab/show/Lawvere%27s+fixed+point+theorem">Lawvere's fixed point theorem</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/H.+Applegate">H. Applegate</a>, <a class="existingWikiWord" href="/nlab/show/M.+Barr">M. Barr</a>, <a class="existingWikiWord" href="/nlab/show/J.+Beck">J. Beck</a>, <a class="existingWikiWord" href="/nlab/show/F.+W.+Lawvere">F. W. Lawvere</a>, <a class="existingWikiWord" href="/nlab/show/F.+E.+J.+Linton">F. E. J. Linton</a>, <a class="existingWikiWord" href="/nlab/show/E.+Manes">E. Manes</a>, <a class="existingWikiWord" href="/nlab/show/M.+Tierney">M. Tierney</a>, <a class="existingWikiWord" href="/nlab/show/F.+Ulmer">F. Ulmer</a>: <a class="existingWikiWord" href="/nlab/show/Seminar+on+Triples+and+Categorical+Homology+Theory">Seminar on Triples and Categorical Homology Theory</a>, ETH 1966/67, edited by <a class="existingWikiWord" href="/nlab/show/Beno+Eckmann">Beno Eckmann</a> and <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a>, <a class="existingWikiWord" href="/nlab/show/LNM+80">LNM 80</a>, Springer (1969), reprinted as: Reprints in Theory and Applications of Categories 18 (2008) 1-303 [<a href="http://www.tac.mta.ca/tac/reprints/articles/18/tr18abs.html">TAC:18</a>, <a href="http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf">pdf</a>] (on <a class="existingWikiWord" href="/nlab/show/monads">monads</a> in <a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a> and (<a class="existingWikiWord" href="/nlab/show/cohomology">co-</a>)<a class="existingWikiWord" href="/nlab/show/homology">homology</a>-theory: ) </li> <li> Quantifiers and sheaves, Actes Congrès intern. Math. 1 (1970) 329-334 [<a href="https://raw.githubusercontent.com/mattearnshaw/lawvere/master/pdfs/1970-quantifiers-and-sheaves.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Lawvere-QuantifiersAndSheaves.pdf" title="pdf">pdf</a>] on <a class="existingWikiWord" href="/nlab/show/geometric+modalities">geometric modalities</a>/<a class="existingWikiWord" href="/nlab/show/Lawvere-Tierney+topology">Lawvere-Tierney topology</a> and <a class="existingWikiWord" href="/nlab/show/universal+quantifier">universal</a>/<a class="existingWikiWord" href="/nlab/show/existential+quantifier">existential</a> <a class="existingWikiWord" href="/nlab/show/quantifiers">quantifiers</a> related to <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a>/<a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> further developed in: </li> <li> Equality in hyperdoctrines and comprehension schema as an adjoint functor. In A. Heller, ed., Proc. New York Symp. on Applications of Categorical Algebra, pp. 1–14. AMS, 1970. (<a class="existingWikiWord" href="/nlab/files/LawvereComprehension.pdf" title="pdf">pdf</a>) (on <a class="existingWikiWord" href="/nlab/show/comprehension">comprehension</a> in <a class="existingWikiWord" href="/nlab/show/hyperdoctrines">hyperdoctrines</a>) </li> <li id="PerugiaNotes"> Teoria delle categorie sopra un topos di base, lecture notes from Perugia (1972–73) [<a href="https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1972-perugia-lecture-notes.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/LawverePerugia1972.pdf" title="pdf">pdf</a>] (on basic category theory, the <a class="existingWikiWord" href="/nlab/show/category+of+sets">category of sets</a> and <a class="existingWikiWord" href="/nlab/show/elementary+toposes">elementary toposes</a>, and an approach to <a class="existingWikiWord" href="/nlab/show/indexed+categories">indexed categories</a>) </li> <li> Metric spaces, generalized logic and closed categories Reprints in Theory and Applications of Categories, No. 1 (2002) pp 1-37 (<a href="http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html">tac</a>) (A classic text from 1973 on <a class="existingWikiWord" href="/nlab/show/Cauchy+complete+categories">Cauchy complete categories</a> and <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a>) </li> <li> Variable sets etendu and variable structure in topoi , Lecture notes University of Chicago 1975. (a review of topos theory: on structural sets, role of <a class="existingWikiWord" href="/nlab/show/forcing">forcing</a> and non-standard analysis and <a class="existingWikiWord" href="/nlab/show/%C3%A9tendue">étendue</a> toposes) </li> <li> Variable quantities and variable structures in topoi, in <a class="existingWikiWord" href="/nlab/show/Alex+Heller">Alex Heller</a>, <a class="existingWikiWord" href="/nlab/show/Myles+Tierney">Myles Tierney</a> (eds.), Algebra, Topology and Category Theory – A Collection of Papers in Honor of <a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, Academic Press New York (1976) 101-131 [<a href="https://doi.org/10.1016/C2013-0-10841-0">doi:10.1016/C2013-0-10841-0</a>] (reviews toposes in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>, first attention to <a class="existingWikiWord" href="/nlab/show/gros+topos">gros topos</a>) </li> <li> Topos-theoretic methods in geometry, Aarhus 1997 </li> <li> Extensive and intensive quantities Workshop on Categorical Methods in Geometry, Aarhus 1983 (a proposal for a general axiomatization of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>/<a class="existingWikiWord" href="/nlab/show/homology">homology</a>-like “<a class="existingWikiWord" href="/nlab/show/extensive">extensive</a> quantities” and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>-like “<a class="existingWikiWord" href="/nlab/show/intensive">intensive</a> quantities”) </li> <li> State Categories, Closed Categories, and the Existence Semi-Continuous Entropy Functions , IMA reprint 86, 1984 (<a href="https://conservancy.umn.edu/bitstream/handle/11299/4672/86s.pdf?sequence=1&isAllowed=y">pdf</a>) </li> <li> Functional Remarks on the General Concept of Chaos , IMA reprint 87, 1984 (<a href="http://www.ima.umn.edu/sites/default/files/87s.pdf">pdf</a>) (on a formalization of the concept of <a class="existingWikiWord" href="/nlab/show/chaos">chaos</a>) </li> <li> Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body, Cah.top.géo.diff.cat, 21 nr.4, 1980, pp.377-392 (<a href="http://www.numdam.org/item?id=CTGDC_1980__21_4_377_0">Numdam</a>) (contains a section with some remarks on Lagrangian mechanics) </li> <li> with <a class="existingWikiWord" href="/nlab/show/Stephen+Schanuel">Stephen Schanuel</a>, <a class="existingWikiWord" href="/nlab/show/Categories+in+Continuum+Physics">Categories in Continuum Physics</a>, Lectures given at a Workshop held at SUNY, Buffalo 1982, Lecture Notes in Mathematics 1174, 1986 (on <a class="existingWikiWord" href="/nlab/show/continuum+mechanics">continuum mechanics</a>, <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a> and <a class="existingWikiWord" href="/nlab/show/laws+of+motion">laws of motion</a> in <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> and in <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> and <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+spaces">Frölicher spaces</a>, and on <a class="existingWikiWord" href="/nlab/show/intensive+and+extensive">intensive and extensive</a> properties in terms of <a class="existingWikiWord" href="/nlab/show/ring+objects">ring objects</a> and <a class="existingWikiWord" href="/nlab/show/modules">modules</a> in a <a class="existingWikiWord" href="/nlab/show/topos">topos</a>.) </li> <li id="TakingCategoriesSeriously"> Taking categories seriously, Revista Colombiana de Matemáticas XX (1986) 147-178, Reprints in Theory and Applications of Categories, 8 (2005) 1-24. [<a href="http://www.tac.mta.ca/tac/reprints/articles/8/tr8abs.html">tac:tr8</a>, <a href="http://www.emis.de/journals/TAC/reprints/articles/8/tr8.pdf">pdf</a>] (on the relevance of fundamental concepts in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, such as <a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a>) </li> <li> Categories of spaces may not be generalized spaces, as exemplified by directed graphs, Revista Colombiana de Matemáticas XX (1986) 179-186, reprinted as: Reprints in Theory and Applications of Categories, 9 (2005) 1-7 [<a href="http://www.tac.mta.ca/tac/reprints/articles/9/tr9abs.html">tac:tr9</a>] (on the notion of <a class="existingWikiWord" href="/nlab/show/gros+topos">gros toposes</a>, details of his claims are worked out at <a class="existingWikiWord" href="/nlab/show/sufficiently+cohesive+topos">sufficiently cohesive topos</a>) </li> <li> Qualitative Distinctions between some Toposes of Generalized Graphs, in Categories in Computer Science and Logic, Cont. Math. 92 (1989) 261-299 [<a href="http://dx.doi.org/10.1090/conm/092">doi:10.1090/conm/092</a>, <a href="https://github.com/mattearnshaw/lawvere/raw/master/pdfs/1989-qualitative-distinctions-between-some-toposes-of-generalized-graphs.pdf">pdf</a>] (Uses toposes of graphs to introduce his ideas on petit/gros toposes) </li> <li> Display of graphics and their applications, as exemplified by 2-categories and the Hegelian “taco”, Proceedings of the first international Conference on algebraic methodology and software technology University of Iowa, May 22-24 1989, Iowa City pp. 51-74 (on <a class="existingWikiWord" href="/nlab/show/Aufhebung">Aufhebung</a>, <a class="existingWikiWord" href="/nlab/show/graphic+category">graphic categories</a>, the <a class="existingWikiWord" href="/nlab/show/Hegelian+taco">Hegelian taco</a>; the most explicit text with Lawvere’s ideas on <a class="existingWikiWord" href="/nlab/show/Hegel">Hegel</a>) </li> <li> More on graphic toposes, Cah. Top. Géom. Diff. Cat. XXXII no. 1 (1991) pp.5-10. (<a href="http://www.numdam.org/item?id=CTGDC_1991__32_1_5_0">Numdam</a>) (on <a class="existingWikiWord" href="/nlab/show/graphic+category">graphic categories</a> and their presheaf categories) </li> <li> (with <a class="existingWikiWord" href="/nlab/show/Max+Kelly">Max Kelly</a>) On the complete lattice of essential localizations, Bull. Soc. Math. Belgique 41 (1989) 289–319 [<a class="existingWikiWord" href="/nlab/files/Kelly-Lawvere_EssentialLocalizations.pdf" title="pdf">pdf</a>] <blockquote> (on <a class="existingWikiWord" href="/nlab/show/essential+localizations">essential localizations</a>) </blockquote> </li> <li> <a class="existingWikiWord" href="/nlab/show/Some+Thoughts+on+the+Future+of+Category+Theory">Some Thoughts on the Future of Category Theory</a> in A. Carboni, M. Pedicchio, G. Rosolini (eds.), Category Theory , <a class="existingWikiWord" href="/nlab/show/Como">Proceedings of the International Conference held in Como</a>, LNM 1488 Springer Heidelberg 1991. (on – implicitly – <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive toposes</a>); </li> <li> with <a class="existingWikiWord" href="/nlab/show/Stephen+Schanuel">Stephen Schanuel</a>, Conceptual Mathematics - A first introduction to categories , Cambridge UP 1997. (first steps into category theory; the Spanish translation is available <a href="http://www.acsu.buffalo.edu/~wlawvere/concep-3.pdf">here</a>) </li> <li id="Lawvere92"> Categories of space and quantity in: J. Echeverria et al (eds.), The Space of mathematics , de Gruyter, Berlin, New York (1992) (on <a class="existingWikiWord" href="/nlab/show/space+and+quantity">space and quantity</a> and on formalizing concepts from <a class="existingWikiWord" href="/nlab/show/philosophy">philosophy</a> as in the <a class="existingWikiWord" href="/nlab/show/Science+of+Logic">Science of Logic</a> in terms of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> and <a class="existingWikiWord" href="/nlab/show/categorical+logic">categorical logic</a>) </li> <li id="IntroductionToLinearCategories"> <a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a>, Introduction to Linear Categories and Applications, course lecture notes (1992) [<a href="https://github.com/mattearnshaw/lawvere/blob/192dac273e8bf352f307f87b9ec4fe8ef7dc85b9/pdfs/1992-introduction-to-linear-categories-and-applications.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Lawvere-LinearCategories.pdf" title="pdf">pdf</a>] (on <a class="existingWikiWord" href="/nlab/show/additive+categories">additive categories</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Cohesive+Toposes+and+Cantor%27s+%22lauter+Einsen%22">Cohesive Toposes and Cantor's "lauter Einsen"</a> Philosophia Mathematica (3) Vol. 2 (1994), pp. 5-15. (<a class="existingWikiWord" href="/nlab/files/LawvereCohesiveToposes.pdf" title="pdf">pdf</a>) (an early version of the notion of <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive toposes</a>) </li> <li id="Lawvere94b"> <a class="existingWikiWord" href="/nlab/show/Tools+for+the+advancement+of+objective+logic">Tools for the advancement of objective logic</a>: closed categories and toposes, in J. Macnamara and <a class="existingWikiWord" href="/nlab/show/Gonzalo+Reyes">Gonzalo Reyes</a> (Eds.), The Logical Foundations of Cognition, Oxford University Press 1993 (Proceedings of the Febr. 1991 Vancouver Conference “Logic and Cognition”), pages 43-56, 1994. (on formalizing <a class="existingWikiWord" href="/nlab/show/philosophy">philosophy</a> as in the <a class="existingWikiWord" href="/nlab/show/Science+of+Logic">Science of Logic</a> in terms of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> and <a class="existingWikiWord" href="/nlab/show/categorical+logic">categorical logic</a>.) </li> <li id="Lawvere95"> A new branch of mathematics, “The Ausdehnungslehre of 1844,” and other works. Open Court (1995), Translated by Lloyd C. Kannenberg, with foreword by Albert C. Lewis, Historia Mathematica Volume 32, Issue 1, February 2005, Pages 99–106 (<a href="http://www.sciencedirect.com/science/article/pii/S031508600400059X">publisher</a>) (on <a class="existingWikiWord" href="/nlab/show/Hermann+Grassmann">Hermann Grassmann</a>‘s <a class="existingWikiWord" href="/nlab/show/Ausdehnungslehre">Ausdehnungslehre</a> and hence also on <a class="existingWikiWord" href="/nlab/show/extensive+and+intensive+quantity">extensive and intensive quantity</a>) </li> <li> Grassmann’s Dialectics and Category Theory, in Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar, Boston Studies in the Philosophy of Science Volume 187, 1996, pp 255-264 (<a href="http://link.springer.com/chapter/10.1007%2F978-94-015-8753-2_21">publisher</a>) </li> <li> Volterra’s functionals and covariant cohesion of space Perugia Studies in Mathematics (Proceedings of the May 1997 Meeting in Perugia) (<a href="http://www.acsu.buffalo.edu/~wlawvere/Volterra.pdf">pdf</a>) (with first remarks on <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesion</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Toposes+of+laws+of+motion">Toposes of laws of motion</a> , transcript of a talk in Montreal, Sept. 1997 (<a href="http://www.acsu.buffalo.edu/~wlawvere/ToposMotion.pdf">pdf</a>) (on the description of <a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a> in terms of <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> and the notion of <a class="existingWikiWord" href="/nlab/show/toposes+of+laws+of+motion">toposes of laws of motion</a>) </li> <li> Outline of synthetic differential geometry , seminar notes (1998) (<a href="http://www.acsu.buffalo.edu/~wlawvere/SDG_Outline.pdf">pdf</a>) (the origin of the concept of <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>) </li> <li> Kinship and Mathematical Categories , pp.411-425 in Jackendoff, Bloom, Wynn (eds.), Language, Logic, and Concepts , MIT Press Cambridge 1999. (Lawvere’s venture into anthropology; a summary is at <a class="existingWikiWord" href="/nlab/show/kinship">kinship</a>) </li> <li> <a class="existingWikiWord" href="/nlab/show/Comments+on+the+development+of+topos+theory">Comments on the development of topos theory</a> , pp.715-734 in Jean-Paul Pier (ed.), Development of mathematics 1950-2000, Birkhäuser Basel 2000. (review of the history of topos theory) </li> <li> Linearization of graphic toposes via Coxeter groups, JPAA 168 (2002) pp. 425-436. (<a href="http://www.sciencedirect.com/science/article/pii/S0022404901001074/pdfft?md5=a4ca9bc67df6ae63ddf53c559bd71315&pid=1-s2.0-S0022404901001074-main.pdf">pdf</a>) </li> <li> Categorical algebra for continuum micro physics, JPAA 175 (2002) pp. 267-287. (<a href="http://www.sciencedirect.com/science/article/pii/S002240490200138X/pdfft?md5=ebd435105778d597f25141bf065978e3&pid=1-s2.0-S002240490200138X-main.pdf">pdf</a>) </li> <li> Foundations and Applications: Axiomatization and Education, Bulletin of Symbolic Logic 9 no.2 (2003) pp.213-224. (<a href="https://www.math.ucla.edu/~asl/bsl/0902/0902-006.ps">ps-preprint</a>) (a review of (not only set-theoretic) foundations in mathematics) </li> <li id="LawvereRosebrugh"> with <a class="existingWikiWord" href="/nlab/show/Robert+Rosebrugh">Robert Rosebrugh</a>, <a class="existingWikiWord" href="/nlab/show/Sets+for+Mathematics">Sets for Mathematics</a> , Cambridge UP 2003. (<a href="http://books.google.de/books?id=h3_7aZz9ZMoC&pg=PP1&dq=sets+for+mathematics">web</a>) (on <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a> from a practical <a class="existingWikiWord" href="/nlab/show/category+theory">theoretic</a> point of view (informal <a class="existingWikiWord" href="/nlab/show/ETCS">ETCS</a>)) </li> <li> Axiomatic cohesion, Theory and Applications of Categories 19 3 (2007) 41-49 [<a href="http://www.tac.mta.ca/tac/volumes/19/3/19-03abs.html">tac:19-03</a>, <a href="http://www.tac.mta.ca/tac/volumes/19/3/19-03.pdf">pdf</a>] (a more formal definition of <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive toposes</a>, <a class="existingWikiWord" href="/nlab/show/sufficiently+cohesive+topos">sufficiently cohesive topos</a> works out details of his claims concerning ‘sufficient cohesion’) </li> <li> <a class="existingWikiWord" href="/nlab/show/Cohesive+Toposes+--+Combinatorial+and+Infinitesimal+Cases">Cohesive Toposes – Combinatorial and Infinitesimal Cases</a> , lecture in Como (2008) (on <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive toposes</a>) </li> <li> Open problems in topos theory, April 2009 (<a href="https://web.archive.org/web/20170421103453/http://cheng.staff.shef.ac.uk/pssl88/lawvere.pdf">pdf</a>) (problem 7 presented there “The algebra of time”, concerns the characterization of <a class="existingWikiWord" href="/nlab/show/Toposes+of+laws+of+motion">Toposes of laws of motion</a>) </li> <li> The Dialectic of the Continuous and the Discrete in the history of the struggle for a usable guide to mathematical thought, talk at Sets Within Geometry, Nancy, France 26-29 July 2011 (<a href="https://www.youtube.com/watch?v=OE-R3565jrs">recording</a>) </li> <li> What are Foundations of Geometry and Algebra, Keynote lecture at the <a href="http://www.math.union.edu/~niefiels/13conference/Web/">Fifty Years of Functorial Semantics</a> conference, Union College, October 2013 (<a href="https://www.youtube.com/watch?v=ZYGyEPXu8as">recording</a>) </li> <li> Alexander Grothendieck and the Concept of Space, address CT15 Aveiro 2016. (<a href="http://www.acsu.buffalo.edu/~wlawvere/GrothendiekAveiro15.htm">link</a>) (a talk given at CT15 reviewing 50 years of developments since the 1965 La Jolla conference) </li> <li id="Lawvere21"> Toposes generated by codiscrete objects in combinatorial topology and functional analysis, Reprints in Theory and Applications of Categories, No. 27 (2021) pp. 1-11, <a href="http://tac.mta.ca/tac/reprints/articles/27/tr27abs.html">pdf</a>. </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/people">people</a></div></body></html> </div> <div class="revisedby"> Last revised on January 25, 2025 at 18:04:33. 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