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this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> Showing changes from revision #46 to #47: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='homotopy_theory'>Homotopy theory</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a> flavors: <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable</a>, <a class='existingWikiWord' href='/nlab/show/diff/equivariant+homotopy+theory'>equivariant</a>, <a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational</a>, <a class='existingWikiWord' href='/nlab/show/diff/p-adic+homotopy+theory'>p-adic</a>, <a class='existingWikiWord' href='/nlab/show/diff/proper+homotopy+theory'>proper</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+type+theory'>geometric</a>, <a class='existingWikiWord' href='/nlab/show/diff/cohesive+homotopy+theory'>cohesive</a>, <a class='existingWikiWord' href='/nlab/show/diff/directed+homotopy+theory'>directed</a>… models: <a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological</a>, <a class='existingWikiWord' href='/nlab/show/diff/simplicial+homotopy+theory'>simplicial</a>, <a class='existingWikiWord' href='/nlab/show/diff/localic+homotopy+theory'>localic</a>, … see also <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a> Introductions <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+2'>Introduction to Basic Homotopy Theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Homotopy+Theory'>Introduction to Abstract Homotopy Theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+homotopy+types'>geometry of physics -- homotopy types</a> </li> </ul> Definitions <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy'>higher homotopy</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Pi-algebra'>Pi-algebra</a>, <a class='existingWikiWord' href='/nlab/show/diff/spherical+object'>spherical object and Pi(A)-algebra</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+category+theory'>homotopy coherent category theory</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/category+of+fibrant+objects'>category of fibrant objects</a>, <a class='existingWikiWord' href='/nlab/show/diff/cofibration+category'>cofibration category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Waldhausen+category'>Waldhausen category</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Ho%28Top%29'>Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/homotopy+category+of+an+%28infinity%2C1%29-category'>homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> Paths and cylinders <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>left homotopy</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder object</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>right homotopy</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/path+space+object'>path object</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/mapping+cocone'>mapping cocone</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/generalized+universal+bundle'>universal bundle</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/interval+object'>interval object</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/localization+at+geometric+homotopies'>homotopy localization</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/infinitesimal+interval+object'>infinitesimal interval object</a> </li> </ul> </li> </ul> Homotopy groups <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group+of+a+topos'>fundamental group of a topos</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Brown-Grossman+homotopy+group'>Brown-Grossman homotopy group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/categorical+homotopy+groups+in+an+%28infinity%2C1%29-topos'>categorical homotopy groups in an (∞,1)-topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+groups+in+an+%28infinity%2C1%29-topos'>geometric homotopy groups in an (∞,1)-topos</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+groupoid'>fundamental groupoid</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/path+groupoid'>path groupoid</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+in+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid+of+a+locally+infinity-connected+%28infinity%2C1%29-topos'>fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+%28infinity%2C1%29-category'>fundamental (∞,1)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+category'>fundamental category</a></li> </ul> </li> </ul> Basic facts <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group+of+the+circle+is+the+integers'>fundamental group of the circle is the integers</a></li> </ul> Theorems <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+theorem+of+covering+spaces'>fundamental theorem of covering spaces</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Blakers-Massey+theorem'>Blakers-Massey theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy+van+Kampen+theorem'>higher homotopy van Kampen theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/nerve+theorem'>nerve theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead's theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+theorem'>Hurewicz theorem</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Galois+theory'>Galois theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+hypothesis'>homotopy hypothesis</a>-theorem </li> </ul> </div> <h4 id='higher_algebra'>Higher algebra</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebra</a> <a class='existingWikiWord' href='/nlab/show/diff/universal+algebra'>universal algebra</a> <h2 id='algebraic_theories'>Algebraic theories</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/algebraic+theory'>algebraic theory</a> / <a class='existingWikiWord' href='/nlab/show/diff/2-Lawvere+theory'>2-algebraic theory</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-algebraic+theory'>(∞,1)-algebraic theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-monad'>(∞,1)-monad</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/operad'>operad</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-operad'>(∞,1)-operad</a> </li> </ul> <h2 id='algebras_and_modules'>Algebras and modules</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/algebra+over+a+monad'>algebra over a monad</a> <a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-monad'>∞-algebra over an (∞,1)-monad</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/algebra+over+a+Lawvere+theory'>algebra over an algebraic theory</a> <a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-algebraic+theory'>∞-algebra over an (∞,1)-algebraic theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/algebra+over+an+operad'>algebra over an operad</a> <a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-operad'>∞-algebra over an (∞,1)-operad</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/action'>action</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-representation'>∞-representation</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/module'>module</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module'>∞-module</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/associated+bundle'>associated bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/associated+infinity-bundle'>associated ∞-bundle</a> </li> </ul> <h2 id='higher_algebras'>Higher algebras</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/monoidal+%28infinity%2C1%29-category'>monoidal (∞,1)-category</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+%28infinity%2C1%29-category'>symmetric monoidal (∞,1)-category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+%28infinity%2C1%29-category'>monoid in an (∞,1)-category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid+in+a+symmetric+monoidal+%28infinity%2C1%29-category'>commutative monoid in an (∞,1)-category</a> </li> </ul> </li> <li> symmetric monoidal (∞,1)-category of spectra <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/smash+product+of+spectra'>smash product of spectra</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/symmetric+smash+product+of+spectra'>symmetric monoidal smash product of spectra</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>ring spectrum</a>, <a class='existingWikiWord' href='/nlab/show/diff/module+spectrum'>module spectrum</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebra+spectrum'>algebra spectrum</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/A-infinity-algebra'>A-∞ algebra</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-ring'>A-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/diff/A-infinity-space'>A-∞ space</a></li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/C_%E2%88%9E-algebra'>C-∞ algebra</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/diff/E-infinity+algebra'>E-∞ algebra</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module'>∞-module</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module+bundle'>(∞,1)-module bundle</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/multiplicative+cohomology+theory'>multiplicative cohomology theory</a> </li> </ul> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebra</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/deformation+theory'>deformation theory</a></li> </ul> </li> </ul> <h2 id='model_category_presentations'>Model category presentations</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+algebras'>model structure on simplicial T-algebras</a> / <a class='existingWikiWord' href='/nlab/show/diff/homotopy+T-algebra'>homotopy T-algebra</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+operads'>model structure on operads</a> <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+algebras+over+an+operad'>model structure on algebras over an operad</a> </li> </ul> <h2 id='geometry_on_formal_duals_of_algebras'>Geometry on formal duals of algebras</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>Isbell duality</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/derived+geometry'>derived geometry</a> </li> </ul> <h2 id='theorems'>Theorems</h2> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Deligne+conjecture'>Deligne conjecture</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/delooping+hypothesis'>delooping hypothesis</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal Dold-Kan correspondence</a> </li> </ul> <div> <a href='/nlab/edit/higher+algebra+-+contents'>Edit this sidebar</a> </div></div> <h4 id='algebraic_topology'>Algebraic topology</h4> <div class='hide'> <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a> – application of <a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebra</a> and <a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a> to the study of (<a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable</a>) <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy</a> <a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a> <a class='existingWikiWord' href='/nlab/show/diff/homology'>homology</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/spectral+sequence'>spectral sequence</a> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a><ul><li><a href='#the_idea_of_functorial_invariants'>The idea of functorial invariants</a></li></ul></li><li><a href='#overview_of_methods'>Overview of methods</a></li><li><a href='#related_entries'>Related entries</a></li><li><a href='#References'>References</a><ul><li><a href='#prehistory'>Pre-history</a></li><li><a href='#ReferencesTopologicalHomotopyTheory'>Topological homotopy theory</a></li><li><a href='#ReferencesAlegbraicTopology'>Algebraic topology</a></li><li><a href='#ReferencesAbstractHomotopyTheory'>Abstract homotopy theory</a></li><li><a href='#ReferencesSimplicialHomotopyTheory'>Simplicial homotopy theory</a></li><li><a href='#ReferencesBasicInfinityCategoryTheory'>Basic <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-category theory</a></li><li><a href='#ReferencesBasicHomotopyTypeTheory'>Basic homotopy type theory</a></li><li><a href='#ReferencesOutlook'>Outlook</a></li></ul></li></ul></div> <h2 id='idea'>Idea</h2> Algebraic topology refers to the application of methods of <a class='existingWikiWord' href='/nlab/show/diff/algebra'>algebra</a> to problems in <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a>. More specifically, the method of algebraic topology is to assign <a class='existingWikiWord' href='/nlab/show/diff/homeomorphism'>homeomorphism</a>/<a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a>-<a class='existingWikiWord' href='/nlab/show/diff/invariant'>invariants</a> to <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a>, or more systematically, to the construction and applications of <a class='existingWikiWord' href='/nlab/show/diff/functor'>functors</a> from some <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> of topological objects (e.g. <a class='existingWikiWord' href='/nlab/show/diff/Hausdorff+space'>Hausdorff spaces</a>, topological <a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>fibre bundles</a>) to some algebraic category (e.g. <a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian groups</a>, <a class='existingWikiWord' href='/nlab/show/diff/module'>modules</a> over the <a class='existingWikiWord' href='/nlab/show/diff/Steenrod+algebra'>Steenrod algebra</a>). Landing in an algebraic category aids to the computability, but typically loses some information (say getting from a topological spaces with a continuum or more points to rather discrete algebraic structures). <h3 id='the_idea_of_functorial_invariants'>The idea of functorial invariants</h3> The basic idea of the functorial method for the problem of existence of morphisms is the following: If <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>F:A\to B</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> (we present here a general statement, but in the above context <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is a category of topological objects and <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> some category of algebraic objects) and <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>d:D\to A</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> in <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> then <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>∘</mo><mi>d</mi></mrow><annotation encoding='application/x-tex'>F\circ d</annotation></semantics></math> is a diagram in <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>. If one can fill certain additional arrow <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> in the diagram <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math> making the extended diagram commutative, then <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>F(f)</annotation></semantics></math> is a morphism between the corresponding vertices in <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> extending <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>∘</mo><mi>d</mi></mrow><annotation encoding='application/x-tex'>F\circ d</annotation></semantics></math> to a commutative diagram. Thus if we prove that there is no morphism extending <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><mo>∘</mo><mi>d</mi></mrow><annotation encoding='application/x-tex'>F\circ d</annotation></semantics></math> then there was no morphism extending <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi></mrow><annotation encoding='application/x-tex'>d</annotation></semantics></math> in the first place. Therefore, the functorial method is very suitable to prove negative existence for morphisms. Sometimes, however, there is a theorem showing that some set of invariants completely characterizes a problem hence being able to show positive existence or uniqueness for maps or spaces. For the uniqueness for morphisms, it is enough to show that <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> is faithful and that there is at most one solution for the existence problem in the target category. Faithful functors in this context are rare, but it is sufficient for <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi></mrow><annotation encoding='application/x-tex'>F</annotation></semantics></math> to be faithful on some subcategory <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>A</mi> <mi>p</mi></msub></mrow><annotation encoding='application/x-tex'>A_p</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> containing at least all morphisms which are the possible candidates for the solution of the particular existence problem for morphisms. <h2 id='overview_of_methods'>Overview of methods</h2> The archetypical example is the classification of <a class='existingWikiWord' href='/nlab/show/diff/surface'>surfaces</a> via their <a class='existingWikiWord' href='/nlab/show/diff/Euler+characteristic'>Euler characteristic</a>. But as this example already shows, algebraic topology tends to be less about <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a> themselves as rather about the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy types</a> which they <a class='existingWikiWord' href='/nlab/show/diff/homotopy+hypothesis'>present</a>. Therefore the topological invariants in question are typically homotopy invariants of spaces with some exceptions, like the <a class='existingWikiWord' href='/nlab/show/diff/shape+theory'>shape invariants</a> for spaces with bad local behaviour. Hence modern algebraic topology is to a large extent the application of algebraic methods to <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>. A general and powerful such method is the assignment of <a class='existingWikiWord' href='/nlab/show/diff/homology'>homology</a> and <a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a> <a class='existingWikiWord' href='/nlab/show/diff/group'>groups</a> to topological spaces, such that these <a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian groups</a> depend only on the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a>. The simplest such are <a class='existingWikiWord' href='/nlab/show/diff/ordinary+homology'>ordinary homology</a> and <a class='existingWikiWord' href='/nlab/show/diff/ordinary+cohomology'>ordinary cohomology</a> groups, given by <a class='existingWikiWord' href='/nlab/show/diff/singular+simplicial+complex'>singular simplicial complexes</a>. This way algebraic topology makes use of tools of <a class='existingWikiWord' href='/nlab/show/diff/homological+algebra'>homological algebra</a>. The <a class='existingWikiWord' href='/nlab/show/diff/axiom'>axiomatization</a> of the properties of such <a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a> group assignments is what led to the formulation of the trinity of concepts of <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a>, <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> and <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformations</a>, and algebraic topology has come to make intensive use of <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a>. In particular this leads to the formulation of <a class='existingWikiWord' href='/nlab/show/diff/generalized+%28Eilenberg-Steenrod%29+cohomology'>generalized (Eilenberg-Steenrod) cohomology</a> theories which detect more information about classes of homotopy types. By the <a class='existingWikiWord' href='/nlab/show/diff/Brown+representability+theorem'>Brown representability theorem</a> such are represented by <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectra</a> (generalizing <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complexes</a>), hence <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+type'>stable homotopy types</a>, and this way algebraic topology comes to use and be about <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable homotopy theory</a>. Still finer invariants of <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy types</a> are detected by further refinements of these “algebraic” structures, for instance to <a class='existingWikiWord' href='/nlab/show/diff/multiplicative+cohomology+theory'>multiplicative cohomology theories</a>, to <a class='existingWikiWord' href='/nlab/show/diff/equivariant+homotopy+theory'>equivariant homotopy theory</a>/<a class='existingWikiWord' href='/nlab/show/diff/equivariant+stable+homotopy+theory'>equivariant stable homotopy theory</a> and so forth. The construction and analysis of these requires the intimate combination of algebra and homotopy theory to <a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a> and <a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebra</a>, notably embodied in the <a class='existingWikiWord' href='/nlab/show/diff/universal+algebra'>universal</a> higher algebra of <a class='existingWikiWord' href='/nlab/show/diff/operad'>operads</a>. The central tool for breaking down all this <a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebraic</a> data into computable pieces are <a class='existingWikiWord' href='/nlab/show/diff/spectral+sequence'>spectral sequences</a>, which are maybe the main heavy-lifting workhorses of algebraic topology. <h2 id='related_entries'>Related entries</h2> <ul> <li><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a>, <a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a></del><ins class='diffmod'> </ins></li> <li><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/homology'>homology</a>/<a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/shape+theory'>shape theory</a></del><ins class='diffmod'> </ins><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/Nonabelian+Algebraic+Topology'>nonabelian algebraic topology</a></del><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational homotopy theory</a></del></li> <li><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+lifting+property'>homotopy lifting property</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/shape+theory'>shape theory</a></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+fibration'>Hurewicz fibration</a></del><ins class='diffmod'> </ins><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+connection'>Hurewicz connection</a></del><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/Serre+fibration'>Serre fibration</a></del></li> <li><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+extension+property'>homotopy extension property</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational homotopy theory</a></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a></del><ins class='diffmod'> </ins><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/deformation+retract'>deformation retract</a></del></li> <li><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/suspension'>suspension</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Nonabelian+Algebraic+Topology'>nonabelian algebraic topology</a></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a></del><ins class='diffmod'> </ins><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a></del><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a></del><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a></del></li> <li><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/topological+data+analysis'>topological data analysis</a></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></del><ins class='diffmod'> </ins><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/Brown+representability+theorem'>Brown representability theorem</a></del></li> <li><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+lifting+property'>homotopy lifting property</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+fibration'>Hurewicz fibration</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+connection'>Hurewicz connection</a>, <a class='existingWikiWord' href='/nlab/show/diff/Serre+fibration'>Serre fibration</a></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/fundamental+groupoid'>fundamental groupoid</a></del><ins class='diffmod'> </ins></li> <li><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+extension+property'>homotopy extension property</a>, <a class='existingWikiWord' href='/nlab/show/diff/Hurewicz+cofibration'>Hurewicz cofibration</a>, <a class='existingWikiWord' href='/nlab/show/diff/deformation+retract'>deformation retract</a></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Eckmann-Hilton+duality'>Eckmann-Hilton duality</a></del><ins class='diffmod'> </ins><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/H-space'>H-space</a></del><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+product'>Whitehead product</a></del></li> <li><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/suspension'>suspension</a>, <a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cylinder'>mapping cylinder</a>, <a class='existingWikiWord' href='/nlab/show/diff/mapping+cone'>mapping cone</a>, <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>mapping cocylinder</a></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/MU'>complex cobordism</a></del><ins class='diffmod'> </ins><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/elliptic+cohomology'>elliptic cohomology</a></del><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/tmf'>tmf</a></del></li> <li><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW complex</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/cohomology'>cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a>, <a class='existingWikiWord' href='/nlab/show/diff/Brown+representability+theorem'>Brown representability theorem</a></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/CW+approximation'>CW approximation</a></del><ins class='diffmod'> </ins><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/simplicial+complex'>simplicial complex</a></del><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a></del></li> <li><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/fundamental+group'>fundamental group</a>, <a class='existingWikiWord' href='/nlab/show/diff/fundamental+groupoid'>fundamental groupoid</a></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+topological+spaces'>model structure on topological spaces</a></del><ins class='diffmod'> </ins><del class='diffdel'>, </del><del class='diffdel'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a></del></li> <li><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>fibration sequence</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a>, <a class='existingWikiWord' href='/nlab/show/diff/Eckmann-Hilton+duality'>Eckmann-Hilton duality</a>, <a class='existingWikiWord' href='/nlab/show/diff/H-space'>H-space</a>, <a class='existingWikiWord' href='/nlab/show/diff/Whitehead+product'>Whitehead product</a></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/cofiber+sequence'>cofibration sequence</a></del><ins class='diffmod'> </ins></li> <li><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a></del><ins class='diffmod'> </ins><del class='diffmod'>, </del><ins class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/topological+K-theory'>topological K-theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/MU'>complex cobordism</a>, <a class='existingWikiWord' href='/nlab/show/diff/elliptic+cohomology'>elliptic cohomology</a>, <a class='existingWikiWord' href='/nlab/show/diff/tmf'>tmf</a></ins><del class='diffmod'><a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead theorem</a></del><ins class='diffmod'> </ins></li><ins class='diffins'> </ins><ins class='diffins'><li> <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/CW+approximation'>CW approximation</a>, <a class='existingWikiWord' href='/nlab/show/diff/simplicial+complex'>simplicial complex</a>, <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a> </li></ins><ins class='diffins'> </ins><ins class='diffins'><li> <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+topological+spaces'>model structure on topological spaces</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a> </li></ins><ins class='diffins'> </ins><ins class='diffins'><li> <a class='existingWikiWord' href='/nlab/show/diff/fiber+sequence'>fibration sequence</a>, <a class='existingWikiWord' href='/nlab/show/diff/cofiber+sequence'>cofibration sequence</a> </li></ins><ins class='diffins'> </ins><ins class='diffins'><li> <a class='existingWikiWord' href='/nlab/show/diff/Freudenthal+suspension+theorem'>Freudenthal suspension theorem</a>, <a class='existingWikiWord' href='/nlab/show/diff/Whitehead+theorem'>Whitehead theorem</a> </li></ins> </ul> <h2 id='References'>References</h2> The following lists basic references on <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a> and some <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>$(\infty,1)$-category theory</a> and <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a>, but see these entries for more pointers. <h3 id='prehistory'>Pre-history</h3> Historical article at the origin of all these subjects: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Henri+Poincar%C3%A9'>Henri Poincaré</a>, <a class='existingWikiWord' href='/nlab/show/diff/Analysis+Situs'>Analysis Situs</a>, Journal de l’École Polytechnique. (2). 1: 1–123 (1895) (<a href='https://gallica.bnf.fr/ark:/12148/bpt6k4337198/f7'>gallica:12148/bpt6k4337198/f7</a>, Engl: <a href='https://www.maths.ed.ac.uk/~v1ranick/papers/poincare2009.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Stillwell_AnalysisSitus.pdf' title='pdf'>pdf</a>)</li> </ul> On early developments from there, such as the eventual understanding of the notion of higher <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy groups</a>: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Peter+Hilton'>Peter Hilton</a>, Subjective History of Homology and Homotopy Theory, Mathematics Magazine 61 5 (1988) 282-291 <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://doi.org/10.2307/2689545'>doi:10.2307/2689545</a><math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math></li> </ul> <h3 id='ReferencesTopologicalHomotopyTheory'>Topological homotopy theory</h3> Textbook accounts of <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a> of <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a> (i.e. via “<a class='existingWikiWord' href='/nlab/show/diff/general+topology'>point-set topology</a>”): <ul> <li id='Hilton53'> <a class='existingWikiWord' href='/nlab/show/diff/Peter+Hilton'>Peter J. Hilton</a>, An introduction to homotopy theory, Cambridge University Press 1953 (<a href='https://doi.org/10.1017/CBO9780511666278'>doi:10.1017/CBO9780511666278</a>) </li> <li id='SzeTsen59'> <a class='existingWikiWord' href='/nlab/show/diff/Sze-Tsen+Hu'>Sze-Tsen Hu</a>, Homotopy Theory, Academic Press 1959 (<a href='https://www.maths.ed.ac.uk/~v1ranick/papers/hu2.pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Robert+Mosher'>Robert E. Mosher</a>, <a class='existingWikiWord' href='/nlab/show/diff/Martin+Tangora'>Martin C. Tangora</a>, Cohomology operations and applications in homotopy theory, Harper & Row, 1968, reprinted by <a href='https://store.doverpublications.com/0486466647.html'>Dover 2008</a> <a href='https://www.google.com/books/edition/Cohomology_Operations_and_Applications_i/wu79f-7V_6AC'>GoogleBooks</a> </li> <li id='Homotopietheorie'> <a class='existingWikiWord' href='/nlab/show/diff/Tammo+tom+Dieck'>Tammo tom Dieck</a>, <a class='existingWikiWord' href='/nlab/show/diff/Klaus+Heiner+Kamps'>Klaus Heiner Kamps</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dieter+Puppe'>Dieter Puppe</a>, Homotopietheorie, Lecture Notes in Mathematics 157 Springer 1970 (<a href='https://link.springer.com/book/10.1007/BFb0059721'>doi:10.1007/BFb0059721</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Brayton+Gray'>Brayton Gray</a>, Homotopy Theory: An Introduction to Algebraic Topology, Academic Press (1975) <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/64/suppl/C'>978-0-12-296050-5</a>, <a href='https://www.maths.ed.ac.uk/~v1ranick/papers/gray.pdf'>pdf</a><math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/George+Whitehead'>George W. Whitehead</a>, Elements of Homotopy Theory, Springer 1978 (<a href='https://link.springer.com/book/10.1007/978-1-4612-6318-0'>doi:10.1007/978-1-4612-6318-0</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Ioan+James'>Ioan Mackenzie James</a>, General Topology and Homotopy Theory, Springer 1984 (<a href='https://doi.org/10.1007/978-1-4613-8283-6'>doi:10.1007/978-1-4613-8283-6</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Renzo+A.+Piccinini'>Renzo A. Piccinini</a>, Lectures on Homotopy Theory, Mathematics Studies 171, North Holland 1992 (<a href='https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/171/suppl/C'>ISBN:978-0-444-89238-6</a>) </li> <li id='Bredon93'> <a class='existingWikiWord' href='/nlab/show/diff/Glen+Bredon'>Glen Bredon</a>, Chapter VII of: Topology and Geometry, Graduate texts in mathematics 139, Springer 1993 (<a href='https://link.springer.com/book/10.1007/978-1-4757-6848-0'>doi:10.1007/978-1-4757-6848-0</a>, <a href='http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Hans-Joachim+Baues'>Hans-Joachim Baues</a>, Homotopy types, in <a class='existingWikiWord' href='/nlab/show/diff/Ioan+James'>Ioan Mackenzie James</a> (ed.) <a class='existingWikiWord' href='/nlab/show/diff/Handbook+of+Algebraic+Topology'>Handbook of Algebraic Topology</a>, North Holland, 1995 (<a href='https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2'>ISBN:9780080532981</a>, <a href='https://doi.org/10.1016/B978-0-444-81779-2.X5000-7'>doi:10.1016/B978-0-444-81779-2.X5000-7</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Bourbaki'>Nicolas Bourbaki</a>, Topologie Algébrique, Chapitres 1 à 4, Springer (1998, 2016) [ISBN 978-3-662-49361-8, <a href='https://doi.org/10.1007/978-3-662-49361-8'>doi:10.1007/978-3-662-49361-8</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Marcelo+Aguilar'>Marcelo Aguilar</a>, <a class='existingWikiWord' href='/nlab/show/diff/Samuel+Gitler'>Samuel Gitler</a>, <a class='existingWikiWord' href='/nlab/show/diff/Carlos+Prieto'>Carlos Prieto</a>, Algebraic topology from a homotopical viewpoint, Springer (2008) (<a href='https://link.springer.com/book/10.1007/b97586'>doi:10.1007/b97586</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Jeffrey+Strom'>Jeffrey Strom</a>, Modern classical homotopy theory, Graduate Studies in Mathematics 127, American Mathematical Society (2011) [<a href='http://www.ams.org/books/gsm/127'>ams:gsm/127</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Martin+Arkowitz'>Martin Arkowitz</a>, Introduction to Homotopy Theory, Springer (2011) [<a href='https://doi.org/10.1007/978-1-4419-7329-0'>doi:10.1007/978-1-4419-7329-0</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Anatoly+Fomenko'>Anatoly Fomenko</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dmitry+Fuchs'>Dmitry Fuchs</a>: Homotopical Topology, Graduate Texts in Mathematics 273, Springer (2016) [<a href='https://doi.org/10.1007/978-3-319-23488-5'>doi:10.1007/978-3-319-23488-5</a>, <a href='https://www.cimat.mx/~gil/docencia/2020/topologia_diferencial/[Fomenko,Fuchs]Homotopical_Topology(2016).pdf'>pdf</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Dai+Tamaki'>Dai Tamaki</a>, Fiber Bundles and Homotopy, World Scientific (2021) [<a href='https://doi.org/10.1142/12308'>doi:10.1142/12308</a>] <blockquote> (motivated from <a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying spaces</a> for <a class='existingWikiWord' href='/nlab/show/diff/principal+bundle'>principal bundles</a>/<a class='existingWikiWord' href='/nlab/show/diff/fiber+bundle'>fiber bundles</a>) </blockquote> </li> </ul> <h3 id='ReferencesAlegbraicTopology'>Algebraic topology</h3> On <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a>: Monographs: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Samuel+Eilenberg'>Samuel Eilenberg</a>, <a class='existingWikiWord' href='/nlab/show/diff/Norman+Steenrod'>Norman Steenrod</a>, Foundations of Algebraic Topology, Princeton University Press 1952 (<a href='https://www.maths.ed.ac.uk/~v1ranick/papers/eilestee.pdf'>pdf</a>, <a href='https://press.princeton.edu/books/hardcover/9780691653297/foundations-of-algebraic-topology'>ISBN:9780691653297</a>) </li> <li id='Godement58'> <a class='existingWikiWord' href='/nlab/show/diff/Roger+Godement'>Roger Godement</a>, Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://www.editions-hermann.fr/livre/topologie-algebrique-et-theorie-des-faisceaux-roger-godement'>webpage</a>, <a class='existingWikiWord' href='/nlab/files/Godement-TopologieAlgebrique.pdf' title='pdf'>pdf</a><math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math> </li> <li id='Spanier66'> <a class='existingWikiWord' href='/nlab/show/diff/Edwin+Spanier'>Edwin Spanier</a>, Algebraic topology, McGraw Hill (1966), Springer (1982) (<a href='https://link.springer.com/book/10.1007/978-1-4684-9322-1'>doi:10.1007/978-1-4684-9322-1</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/William+S.+Massey'>William S. Massey</a>, Algebraic Topology: An Introduction, Harcourt Brace & World 1967, reprinted in: Graduate Texts in Mathematics, Springer 1977 (<a href='https://link.springer.com/book/9780387902715'>ISBN:978-0-387-90271-5</a>) </li> <li id='Maunder70'> <a class='existingWikiWord' href='/nlab/show/diff/C.+R.+F.+Maunder'>C. R. F. Maunder</a>, Algebraic Topology, Cambridge University Press, Cambridge (1970, 1980) <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://www.maths.ed.ac.uk/~v1ranick/papers/maunder.pdf'>pdf</a><math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math> </li> <li id='Switzer75'> <a class='existingWikiWord' href='/nlab/show/diff/Robert+Switzer'>Robert Switzer</a>, Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975 (<a href='https://link.springer.com/book/10.1007/978-3-642-61923-6'>doi:10.1007/978-3-642-61923-6</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Raoul+Bott'>Raoul Bott</a>, <a class='existingWikiWord' href='/nlab/show/diff/Loring+Tu'>Loring Tu</a>, <a class='existingWikiWord' href='/nlab/show/diff/Differential+Forms+in+Algebraic+Topology'>Differential Forms in Algebraic Topology</a>, Graduate Texts in Mathematics 82, Springer (1982) <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://link.springer.com/book/10.1007/978-1-4757-3951-0'>doi:10.1007/978-1-4757-3951-0</a>] <blockquote> (with focus on <a class='existingWikiWord' href='/nlab/show/diff/differential+form'>differential forms</a>, <a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a>) </blockquote> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/James+Munkres'>James Munkres</a>, Elements of Algebraic Topology, Addison-Wesley (1984) <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://people.dm.unipi.it/benedett/MUNKRES-ETA.pdf'>pdf</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Joseph+J.+Rotman'>Joseph J. Rotman</a>, An Introduction to Algebraic Topology, Graduate Texts in Mathematics 119 (1988) <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://doi.org/10.1007/978-1-4612-4576-6'>doi:10.1007/978-1-4612-4576-6</a>] </li> <li id='Bredon93'> <a class='existingWikiWord' href='/nlab/show/diff/Glen+Bredon'>Glen Bredon</a>, Topology and Geometry, Graduate texts in mathematics 139, Springer 1993 (<a href='https://link.springer.com/book/10.1007/978-1-4757-6848-0'>doi:10.1007/978-1-4757-6848-0</a>, <a href='http://virtualmath1.stanford.edu/~ralph/math215b/Bredon.pdf'>pdf</a>) </li> <li id='Dold95'> <a class='existingWikiWord' href='/nlab/show/diff/Albrecht+Dold'>Albrecht Dold</a>, Lectures on Algebraic Topology, Springer 1995 (<a href='https://www.springer.com/gp/book/9783540586609'>doi:10.1007/978-3-642-67821-9</a>, <a href='https://link.springer.com/content/pdf/bfm%3A978-3-642-67821-9%2F1.pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/William+Fulton'>William Fulton</a>, Algebraic Topology – A First Course, Graduate Texts in Mathematics 153, Springer (1995) [<a href='https://doi.org/10.1007/978-1-4612-4180-5'>doi:10.1007/978-1-4612-4180-5</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>Peter May</a>, <a class='existingWikiWord' href='/nlab/show/diff/A+Concise+Course+in+Algebraic+Topology'>A concise course in algebraic topology</a>, University of Chicago Press 1999 (<a href='https://www.press.uchicago.edu/ucp/books/book/chicago/C/bo3777031.html'>ISBN: 9780226511832</a>, <a href='http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Tammo+tom+Dieck'>Tammo tom Dieck</a>, Topologie, De Gruyter 2000 (<a href='https://doi.org/10.1515/9783110802542'>doi:10.1515/9783110802542</a>) </li> <li id='Hatcher02'> <a class='existingWikiWord' href='/nlab/show/diff/Allen+Hatcher'>Allen Hatcher</a>, Algebraic Topology, Cambridge University Press (2002) [<a href='https://www.cambridge.org/gb/academic/subjects/mathematics/geometry-and-topology/algebraic-topology-1?format=PB=9780521795401'>ISBN:9780521795401</a>, <a href='https://pi.math.cornell.edu/~hatcher/AT/ATpage.html'>webpage</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Dai+Tamaki'>Dai Tamaki</a>, <a class='existingWikiWord' href='/nlab/show/diff/Akira+Kono'>Akira Kono</a>, Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (<a href='https://bookstore.ams.org/mmono-230'>ISBN: 978-0-8218-3514-2</a>) </li> <li id='tomDieck2008'> <a class='existingWikiWord' href='/nlab/show/diff/Tammo+tom+Dieck'>Tammo tom Dieck</a>, Algebraic topology, European Mathematical Society, Zürich (2008) (<a href='https://www.ems-ph.org/books/book.php?proj_nr=86'>doi:10.4171/048</a>, <a href='https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf'>pdf</a>) </li> <li id='Warner05'> <a class='existingWikiWord' href='/nlab/show/diff/Garth+Warner'>Garth Warner</a>: Topics in Topology and Homotopy Theory, EPrint Collection, University of Washington (2005) [<a href='http://hdl.handle.net/1773/2641'>hdl:1773/2641</a>, <a href='https://sites.math.washington.edu//~warner/TTHT_Warner.pdf'>pdf</a>, <a href='https://arxiv.org/abs/2007.02467'>arXiv:2007.02467</a>] </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>Peter May</a>, <a class='existingWikiWord' href='/nlab/show/diff/Kate+Ponto'>Kate Ponto</a>, <a class='existingWikiWord' href='/nlab/show/diff/A+Concise+Course+in+Algebraic+Topology'>More concise algebraic topology</a>, University of Chicago Press (2012) (<a href='https://press.uchicago.edu/ucp/books/book/chicago/M/bo12322308.html'>ISBN:9780226511795</a>, <a href='https://www.math.uchicago.edu/~may/TEAK/KateBookFinal.pdf'>pdf</a>) </li> <li> Clark Bray, Adrian Butcher, Simon Rubinstein-Salzedo: Algebraic Topology, Springer (2021) [<a href='https://doi.org/10.1007/978-3-030-70608-1'>doi:10.1007/978-3-030-70608-1</a>, <a href='https://link.springer.com/content/pdf/10.1007/978-3-030-70608-1.pdf'>pdf</a>] </li> </ul> On <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive</a> methods (<a class='existingWikiWord' href='/nlab/show/diff/computational+topology'>constructive algebraic topology</a>): <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Julio+Rubio'>Julio Rubio</a>, <a class='existingWikiWord' href='/nlab/show/diff/Francis+Sergeraert'>Francis Sergeraert</a>, Constructive Algebraic Topology, Bulletin des Sciences Mathématiques 126 5 (2002) 389-412 [<a href='https://doi.org/10.1016/S0007-4497(02)01119-3'>doi:10.1016/S0007-4497(02)01119-3</a>, <a href='https://arxiv.org/abs/math/0111243'>arXiv:math/0111243</a>]</li> </ul> Lecture notes: <ul> <li id='HopkinsMathew'> <a class='existingWikiWord' href='/nlab/show/diff/Michael+Hopkins'>Michael Hopkins</a> (notes by <a class='existingWikiWord' href='/nlab/show/diff/Akhil+Mathew'>Akhil Mathew</a>), algebraic topology – Lectures (<a href='http://people.fas.harvard.edu/~amathew/ATnotes.pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Friedhelm+Waldhausen'>Friedhelm Waldhausen</a>, Algebraische Topologie I (<a href='https://www.math.uni-bielefeld.de/~fw/at.pdf'>pdf</a>) , II (<a href='https://www.math.uni-bielefeld.de/~fw/at_II.pdf'>pdf</a>), III (<a href='https://www.math.uni-bielefeld.de/~fw/at_III.pdf'>pdf</a>) (<a href='https://www.math.uni-bielefeld.de/~fw/'>web</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/James+Davis'>James F. Davis</a> and <a class='existingWikiWord' href='/nlab/show/diff/Paul+Kirk'>Paul Kirk</a>, Lecture notes in algebraic topology (<a href='http://www.indiana.edu/~jfdavis/teaching/m623/book.pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Gereon+Quick'>Gereon Quick</a>, <a href='https://folk.ntnu.no/gereonq/Math231br.html'>Advanced algebraic topology</a>, 2014 </li> </ul> Survey of various subjects in algebraic topology: <ul> <li id='James95'><a class='existingWikiWord' href='/nlab/show/diff/Ioan+James'>Ioan Mackenzie James</a>, <a class='existingWikiWord' href='/nlab/show/diff/Handbook+of+Algebraic+Topology'>Handbook of Algebraic Topology</a> 1995</li> </ul> Survey with relation to <a class='existingWikiWord' href='/nlab/show/diff/differential+topology'>differential topology</a>: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Sergei+Novikov'>Sergei Novikov</a>, Topology I – General survey, in: Encyclopedia of Mathematical Sciences Vol. 12, Springer 1986 (<a href='https://link.springer.com/book/10.1007/978-3-662-10579-5'>doi:10.1007/978-3-662-10579-5</a>, <a href='https://web.math.rochester.edu/people/faculty/doug/otherpapers/novikovsurv.pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Jean+Dieudonn%C3%A9'>Jean Dieudonné</a>, A History of Algebraic and Differential Topology, 1900 - 1960, Modern Birkhäuser Classics 2009 (<a href='https://www.springer.com/de/book/9780817649067'>ISBN:978-0-8176-4907-4</a>) </li> </ul> With focus on <a class='existingWikiWord' href='/nlab/show/diff/ordinary+homology'>ordinary homology</a>, <a class='existingWikiWord' href='/nlab/show/diff/ordinary+cohomology'>ordinary cohomology</a> and <a class='existingWikiWord' href='/nlab/show/diff/abelian+sheaf+cohomology'>abelian sheaf cohomology</a>: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jean+Gallier'>Jean Gallier</a>, <a class='existingWikiWord' href='/nlab/show/diff/Jocelyn+Quaintance'>Jocelyn Quaintance</a>, Homology, Cohomology, and Sheaf Cohomology for Algebraic Topology, Algebraic Geometry, and Differential Geometry, World Scientific (2022) [<a href='https://doi.org/10.1142/12495'>doi:10.1142/12495</a>, <a href='https://www.cis.upenn.edu/~jean/gbooks/sheaf-coho.html'>webpage</a>]</li> </ul> Some interactive 3D demos: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Neil+Strickland'>Neil Strickland</a>, Interactive pages for Algebraic Topology, <a href='http://neil-strickland.staff.shef.ac.uk/courses/MAS435/demos/'>web site</a></li> </ul> Further pointers: <ul> <li><a href='http://mathoverflow.net/questions/18041/algebraic-topology-beyond-the-basicsany-texts-bridging-the-gap'>a thread on AlgTop literature at MathOverflow</a></li> </ul> <h3 id='ReferencesAbstractHomotopyTheory'>Abstract homotopy theory</h3> On <a class='existingWikiWord' href='/nlab/show/diff/localization'>localization</a> at <a class='existingWikiWord' href='/nlab/show/diff/weak+equivalence'>weak equivalences</a> to <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy categories</a>: <ul> <li id='Brown65'><a class='existingWikiWord' href='/nlab/show/diff/Edgar+Brown'>Edgar Brown</a>, Abstract homotopy theory, Trans. AMS 119 no. 1 (1965) (<a href='https://doi.org/10.1090/S0002-9947-1965-0182970-6'>doi:10.1090/S0002-9947-1965-0182970-6</a>)</li> </ul> On <a class='existingWikiWord' href='/nlab/show/diff/localization'>localization</a> via <a class='existingWikiWord' href='/nlab/show/diff/calculus+of+fractions'>calculus of fractions</a>: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Pierre+Gabriel'>Pierre Gabriel</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michel+Zisman'>Michel Zisman</a>, <a class='existingWikiWord' href='/nlab/show/diff/Calculus+of+fractions+and+homotopy+theory'>Calculus of fractions and homotopy theory</a>, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35. Springer, New York (1967)</li> </ul> On <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category+of+a+model+category'>localization via</a> <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a>-theory: <ul> <li id='Quillen67'> <a class='existingWikiWord' href='/nlab/show/diff/Daniel+Quillen'>Daniel Quillen</a>, Homotopical algebra, Lecture Notes in Mathematics 43, Berlin, New York, 1967 </li> <li id='Hovey99'> <a class='existingWikiWord' href='/nlab/show/diff/Mark+Hovey'>Mark Hovey</a>, <a class='existingWikiWord' href='/nlab/show/diff/Model+Categories'>Model Categories</a>, Mathematical Surveys and Monographs, Volume 63, AMS (1999) (<a href='https://bookstore.ams.org/surv-63-s'>ISBN:978-0-8218-4361-1</a>, <a href='https://doi.org/http://dx.doi.org/10.1090/surv/063'>doi:10.1090/surv/063</a>, <a href='https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf'>pdf</a>, <a href='http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover'>Google books</a>) </li> <li id='Hirschhorn02'> <a class='existingWikiWord' href='/nlab/show/diff/Philip+Hirschhorn'>Philip Hirschhorn</a>, Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (<a href='https://bookstore.ams.org/surv-99-s/'>ISBN:978-0-8218-4917-0</a>, <a href='http://www.gbv.de/dms/goettingen/360115845.pdf'>pdf toc</a>, <a href='http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/William+Dwyer'>William G. Dwyer</a>, <a class='existingWikiWord' href='/nlab/show/diff/Philip+S.+Hirschhorn'>Philip S. Hirschhorn</a>, <a class='existingWikiWord' href='/nlab/show/diff/Daniel+Kan'>Daniel M. Kan</a>, <a class='existingWikiWord' href='/nlab/show/diff/Jeff+Smith'>Jeffrey H. Smith</a>, <a class='existingWikiWord' href='/nlab/show/diff/Homotopy+Limit+Functors+on+Model+Categories+and+Homotopical+Categories'>Homotopy Limit Functors on Model Categories and Homotopical Categories</a>, Mathematical Surveys and Monographs 113 (2004) <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://bookstore.ams.org/surv-113-s'>ISBN: 978-1-4704-1340-8</a>, <a href='http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf'>pdf</a><math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math> </li> </ul> On <a class='existingWikiWord' href='/nlab/show/diff/localization'>localization</a> (especially of categories of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+sheaf'>simplicial sheaves</a>/<a class='existingWikiWord' href='/nlab/show/diff/simplicial+presheaf'>simplicial presheaves</a>) via <a class='existingWikiWord' href='/nlab/show/diff/category+of+fibrant+objects'>categories of fibrant objects</a>: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Kenneth+Brown'>Kenneth S. Brown</a>, <a class='existingWikiWord' href='/nlab/files/BrownAbstractHomotopyTheory.pdf' title='Abstract Homotopy Theory and Generalized Sheaf Cohomology'>Abstract Homotopy Theory and Generalized Sheaf Cohomology</a>, Transactions of the American Mathematical Society 186 (1973) 419-458 <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='http://www.jstor.org/stable/1996573'>jstor:1996573</a><math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math>.</li> </ul> See also: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Klaus+Heiner+Kamps'>Klaus Heiner Kamps</a>, <a class='existingWikiWord' href='/nlab/show/diff/Tim+Porter'>Tim Porter</a>, Abstract Homotopy and Simple Homotopy Theory, World Scientific 1997 (<a href='https://doi.org/10.1142/2215'>doi:10.1142/2215</a>, <a href='http://books.google.de/books?id=7JYKxInRMdAC&dq=Porter+Kamps&printsec=frontcover&source=bl&ots=uuyl_tIjs4&sig=Lt8I92xQBZ4DNKVXD0x76WkcxCE&hl=de&sa=X&oi=book_result&resnum=3&ct=result#PPP1,M1'>GoogleBooks</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Haynes+Miller'>Haynes Miller</a> (ed.), <a class='existingWikiWord' href='/nlab/show/diff/Handbook+of+Homotopy+Theory'>Handbook of Homotopy Theory</a>, 2019 </li> </ul> Lecture notes: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/William+Dwyer'>William Dwyer</a>, Homotopy theory and classifying spaces, Copenhagen, June 2008 (<a href='http://www.math.ku.dk/~jg/homotopical2008/Dwyer.CopenhagenNotes.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Dwyer_HomotopyTheoryOfClassifyingSpaces.pdf' title='pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Jesper+Michael+M%C3%B8ller'>Jesper Michael Møller</a>, Homotopy theory for beginners, 2015 (<a href='http://www.math.ku.dk/~moller/e01/algtopI/comments.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Moller_HomotopyTheory.pdf' title='pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Urs+Schreiber'>Urs Schreiber</a>, <a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Homotopy+Theory'>Introduction to Homotopy Theory</a> (2016) </li> <li id='Martins20'> <a class='existingWikiWord' href='/nlab/show/diff/Yuri+Ximenes+Martins'>Yuri Ximenes Martins</a>, Introduction to Abstract Homotopy Theory (<a href='https://arxiv.org/abs/2008.05302'>arXiv:2008.05302</a>) </li> </ul> Introduction, from <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a> to (mostly <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>abstract</a>, <a class='existingWikiWord' href='/nlab/show/diff/simplicial+homotopy+theory'>simplicial</a>) homotopy theory: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Emily+Riehl'>Emily Riehl</a>, <a class='existingWikiWord' href='/nlab/show/diff/Categorical+Homotopy+Theory'>Categorical Homotopy Theory</a>, Cambridge University Press, 2014 (<a href='http://www.math.jhu.edu/~eriehl/cathtpy.pdf'>pdf</a>, <a href='https://doi.org/10.1017/CBO9781107261457'>doi:10.1017/CBO9781107261457</a>) </li> <li id='Richter19'> <a class='existingWikiWord' href='/nlab/show/diff/Birgit+Richter'>Birgit Richter</a>, From categories to homotopy theory, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press 2020 (<a href='https://doi.org/10.1017/9781108855891'>doi:10.1017/9781108855891</a>, <a href='https://www.math.uni-hamburg.de/home/richter/catbook.html'>book webpage</a>, <a href='https://www.math.uni-hamburg.de/home/richter/bookdraft.pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Urs+Schreiber'>Urs Schreiber</a>, <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+categories+and+toposes'>geometry of physics -- categories and toposes</a> </li> </ul> See also: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/William+Dwyer'>William Dwyer</a>, <a class='existingWikiWord' href='/nlab/show/diff/Philip+Hirschhorn'>Philip Hirschhorn</a>, <a class='existingWikiWord' href='/nlab/show/diff/Daniel+Kan'>Daniel Kan</a>, <a class='existingWikiWord' href='/nlab/show/diff/Jeff+Smith'>Jeff Smith</a>, Homotopy Limit Functors on Model Categories and Homotopical Categories, volume 113 of Mathematical Surveys and Monographs, American Mathematical Society (2004) (there exists <a href='http://dodo.pdmi.ras.ru/~topology/books/dhks.pdf'>this</a> pdf copy of what seems to be a preliminary version of this book) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Zhen+Lin+Low'>Zhen Lin Low</a>, <a class='existingWikiWord' href='/nlab/show/diff/Notes+on+homotopical+algebra'>Notes on homotopical algebra</a>, 2015 </li> <li id='MunsonVolic15'> <a class='existingWikiWord' href='/nlab/show/diff/Brian+Munson'>Brian Munson</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ismar+Voli%C4%87'>Ismar Volic</a>, Cubical homotopy theory, Cambridge University Press, 2015 (<a href='http://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf'>pdf</a>, <a href='https://doi.org/10.1017/CBO9781139343329'>doi:10.1017/CBO9781139343329</a>) <blockquote> (with emphasis on <a class='existingWikiWord' href='/nlab/show/diff/cubical+object'>cubical objects</a> such as in <a class='existingWikiWord' href='/nlab/show/diff/n-excisive+%28%E2%88%9E%2C1%29-functor'>n-excisive functors</a> and <a class='existingWikiWord' href='/nlab/show/diff/Goodwillie+calculus'>Goodwillie calculus</a>) </blockquote> </li> </ul> <h3 id='ReferencesSimplicialHomotopyTheory'>Simplicial homotopy theory</h3> On <a class='existingWikiWord' href='/nlab/show/diff/simplicial+homotopy+theory'>simplicial homotopy theory</a>: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>Peter May</a>, Simplicial objects in algebraic topology, University of Chicago Press 1967 (<a href='https://press.uchicago.edu/ucp/books/book/chicago/S/bo5956688.html'>ISBN:9780226511818</a>, <a href='http://www.math.uchicago.edu/~may/BOOKS/Simp.djvu'>djvu</a>, <a class='existingWikiWord' href='/nlab/files/May_SimplicialObjectsInAlgebraicTopology.pdf' title='pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Edward+Curtis'>Edward B. Curtis</a>, Simplicial homotopy theory, Advances in Mathematics 6 (1971) 107–209 (<a href='https://doi.org/10.1016/0001-8708(71)90015-6'>doi:10.1016/0001-8708(71)90015-6</a>, <a href='http://www.ams.org/mathscinet-getitem?mr=279808'>MR279808</a>) </li> <li id='JoyalTierney05'> <a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>André Joyal</a>, <a class='existingWikiWord' href='/nlab/show/diff/Myles+Tierney'>Myles Tierney</a> Notes on simplicial homotopy theory, Lecture at <a href='https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html'>Advanced Course on Simplicial Methods in Higher Categories</a>, CRM 2008 (<a class='existingWikiWord' href='/nlab/files/JoyalTierneyNotesOnSimplicialHomotopyTheory.pdf' title='pdf'>pdf</a>) </li> <li id='JoyalTierney05'> <a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>André Joyal</a>, <a class='existingWikiWord' href='/nlab/show/diff/Myles+Tierney'>Myles Tierney</a>, An introduction to simplicial homotopy theory, 2009 (<a href='http://hopf.math.purdue.edu/cgi-bin/generate?/Joyal-Tierney/JT-chap-01'>web</a>, <a class='existingWikiWord' href='/nlab/files/JoyalTierneySimplicialHomotopyTheory.pdf' title='pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Paul+Goerss'>Paul Goerss</a>, <a class='existingWikiWord' href='/nlab/show/diff/Kristen+Schemmerhorn'>Kirsten Schemmerhorn</a>, Model categories and simplicial methods, Notes from lectures given at the University of Chicago, August 2004, in: Interactions between Homotopy Theory and Algebra, Contemporary Mathematics 436, AMS 2007 (<a href='http://arxiv.org/abs/math.AT/0609537'>arXiv:math.AT/0609537</a>, <a href='http://dx.doi.org/10.1090/conm/436'>doi:10.1090/conm/436</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Francis+Sergeraert'>Francis Sergeraert</a>, Introduction to Combinatorial Homotopy Theory, 2008 (<a href='https://www-fourier.ujf-grenoble.fr/~%20sergerar/Papers/Trieste-Lecture-Notes.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/SergeraertCombinatorialHomotopyTheory.pdf' title='pdf'>pdf</a>) </li> <li id='GoerssJardine09'> <a class='existingWikiWord' href='/nlab/show/diff/Paul+Goerss'>Paul Goerss</a>, <a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>J. F. Jardine</a>, Section V.4 of: <a class='existingWikiWord' href='/nlab/show/diff/Simplicial+homotopy+theory'>Simplicial homotopy theory</a>, Progress in Mathematics, Birkhäuser (1999) Modern Birkhäuser Classics (2009) (<a href='https://link.springer.com/book/10.1007/978-3-0346-0189-4'>doi:10.1007/978-3-0346-0189-4</a>, <a href='http://web.archive.org/web/19990208220238/http://www.math.uwo.ca/~jardine/papers/simp-sets/'>webpage</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Garth+Warner'>Garth Warner</a>: Categorical Homotopy Theory, EPrint Collection, University of Washington (2012) [<a href='http://hdl.handle.net/1773/19589'>hdl:1773/19589</a>, <a href='https://digital.lib.washington.edu/researchworks/bitstreams/0082c74f-f4e0-4578-a44e-d57a0ea29112/download'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Waner-CategoricalHomotopy.pdf' title='pdf'>pdf</a>] </li> </ul> <h3 id='ReferencesBasicInfinityCategoryTheory'>Basic <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-category theory</h3> On <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a> and <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos+theory'>(∞,1)-topos theory</a>: <ul> <li id='Joyal08'> <a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>André Joyal</a>, The theory of quasicategories and its applications lectures at <a href='https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html'>Advanced Course on Simplicial Methods in Higher Categories</a>, CRM 2008 (<a href='http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/JoyalTheoryOfQuasiCategories.pdf' title='pdf'>pdf</a>) </li> <li id='Joyal08'> <a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>André Joyal</a>, Notes on Logoi, 2008 (<a href='http://www.math.uchicago.edu/~may/IMA/JOYAL/Joyal.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/JoyalOnLogoi2008.pdf' title='pdf'>pdf</a>) </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>Higher Topos Theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Denis-Charles+Cisinski'>Denis-Charles Cisinski</a>, Higher category theory and homotopical algebra (<a href='http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf'>pdf</a>) </li> </ul> <h3 id='ReferencesBasicHomotopyTypeTheory'>Basic homotopy type theory</h3> On <a class='existingWikiWord' href='/nlab/show/diff/synthetic+homotopy+theory'>synthetic homotopy theory</a> in <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a>: Exposition: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Daniel+Licata'>Dan Licata</a>: Homotopy theory in type theory (2013) <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='http://dlicata.web.wesleyan.edu/pubs/bll13homotopy/bll13homotopy.pdf'>pdf slides</a>, <a class='existingWikiWord' href='/nlab/files/Licata-HomotopyInTypeTheory.pdf' title='pdf'>pdf</a>, <a href='https://homotopytypetheory.org/2013/03/08/homotopy-theory-in-homotopy-type-theory-introduction'>blog entry 1</a>, <a href='https://homotopytypetheory.org/2013/05/20/homotopy-theory-in-type-theory-progress-report/'>blog entry 2</a><math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math> </li> <li id='Shulman17'> <a class='existingWikiWord' href='/nlab/show/diff/Mike+Shulman'>Mike Shulman</a>, The logic of space, in: <a class='existingWikiWord' href='/nlab/show/diff/Gabriel+Catren'>Gabriel Catren</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mathieu+Anel'>Mathieu Anel</a> (eds.), <a class='existingWikiWord' href='/nlab/show/diff/New+Spaces+for+Mathematics+and+Physics'>New Spaces for Mathematics and Physics</a>, Cambridge University Press (2021) 322-404 <math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo></mrow><annotation encoding='application/x-tex'>[</annotation></semantics></math><a href='https://arxiv.org/abs/1703.03007'>arXiv:1703.03007</a>, <a href='https://doi.org/10.1017/9781108854429.009'>doi:10.1017/9781108854429.009</a><math class='maruku-mathml' display='inline' id='mathml_ea19e6dcc42faa94c6b2cc5dda09ec4ead245b97_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>]</annotation></semantics></math> </li> </ul> Textbook accounts: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/UF-IAS-2012'>Univalent Foundations Project</a>: <a class='existingWikiWord' href='/nlab/show/diff/Homotopy+Type+Theory+--+Univalent+Foundations+of+Mathematics'>Homotopy Type Theory -- Univalent Foundations of Mathematics</a> (2013) (<a href='http://homotopytypetheory.org/book/'>webpage</a>, <a href='http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf'>pdf</a>) </li> <li id='Rijke19'> <a class='existingWikiWord' href='/nlab/show/diff/Egbert+Rijke'>Egbert Rijke</a>, <a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Homotopy+Type+Theory'>Introduction to Homotopy Type Theory</a> (2019) (<a href='http://www.andrew.cmu.edu/user/erijke/hott/'>web</a>, <a href='http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf'>pdf</a>, <a href='https://github.com/EgbertRijke/HoTT-Intro'>GitHub</a>) </li> </ul> For more see also at <a href='homotopy+type+theory#HomotopyTheoryInHomotopyTyepTheoryReferences'>homotopy theory formalized in homotopy type theory</a>. <h3 id='ReferencesOutlook'>Outlook</h3> Indications of open questions and possible future directions in <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a> and (<a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable</a>) <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Mark+Hovey'>Mark Hovey</a>, <a href='https://www-users.cse.umn.edu/~tlawson/hovey/'>Algebraic Topology Problem List</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Tyler+Lawson'>Tyler Lawson</a>, The future, Talbot lectures 2013 (<a href='http://math.mit.edu/conferences/talbot/2013/19-Lawson-thefuture.pdf'>pdf</a>) </li> <li id='ProblemsInHomotopyTheoryWiki'> Problems in homotopy theory (<a href='http://topology-octopus.herokuapp.com/problemsinhomotopytheory/show/HomePage'>wiki</a>) </li> </ul> More regarding the sociology of the field (such as its <a class='existingWikiWord' href='/nlab/show/diff/folklore'>folklore</a> results): <ul> <li id='Barwick17'><a class='existingWikiWord' href='/nlab/show/diff/Clark+Barwick'>Clark Barwick</a>, The future of homotopy theory, 2017 (<a href='http://www.maths.ed.ac.uk/~cbarwick/papers/future.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/BarwickFutureOfHomotopyTheory.pdf' title='pdf'>pdf</a>)</li> </ul> </div> <div class="revisedby"> Last revised 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