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this.value = 'Search' : true" /></fieldset> </form> </div> <div id="revision"> Showing changes from revision #91 to #92: <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> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#Idea'>Idea</a><ul><li><a href='#GeneralIdea'>General idea</a></li><li><a href='#TopologicalHomotopyTheoryInIdeaSection'>Topological homotopy theory</a></li><li><a href='#abstract_homotopy_theory'>Abstract homotopy theory</a></li><li><a href='#RelationBetweenTopologicalAndAbstractHomotopyTheoryInIdeaSection'>Relation between topological and abstract homotopy theory: Cohesion.</a></li><li><a href='#simplicial_homotopy_theory'>Simplicial homotopy theory</a></li><li><a href='#flavors_of_homotopy_theory'>Flavors of homotopy theory</a></li></ul></li><li><a href='#presentations'>Presentations</a><ul><li><a href='#model_categories'>Model Categories</a></li><li><a href='#examples'>Examples</a></li><li><a href='#generalized_morphisms'>Generalized Morphisms</a></li><li><a href='#quillen_equivalences'>Quillen Equivalences</a></li></ul></li><li><a href='#RelatedConcepts'>Related entries</a><ul><li><a href='#general'>General</a></li><li><a href='#flavors_of_homotopy_theory_2'>Flavors of homotopy theory</a></li><li><a href='#basic_concepts_in_homotopy_theory'>Basic concepts in homotopy theory</a></li><li><a href='#categorical_homotopy_theory'>Categorical homotopy theory</a></li></ul></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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_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> <h3 id='GeneralIdea'>General idea</h3> In generality, homotopy theory is the study of mathematical contexts in which <a class='existingWikiWord' href='/nlab/show/diff/function'>functions</a> or rather (<a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homo</a>-)<a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a> are equipped with a concept of <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a> between them, hence with a concept of “equivalent <a class='existingWikiWord' href='/nlab/show/diff/deformation+theory'>deformations</a>” of morphisms, \begin{imagefromfile} “file_name”: “2Cell.jpg”, “width”: 200, “unit”: “px”, “margin”: { “top”: -30, “bottom”: 10, “right”: 0, “left”: 10 } \end{imagefromfile} and then iteratively with <a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy'>homotopies of homotopies</a> between those: \begin{imagefromfile} “file_name”: “3Cell.jpg”, “width”: 215, “unit”: “px”, “margin”: { “top”: -30, “bottom”: 0, “right”: 0, “left”: 10 } \end{imagefromfile} and so forth. A key aspect here is that in such homotopy theoretic contexts the concept of <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> is relaxed to that of <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>: Where a <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> is regarded as <a class='existingWikiWord' href='/nlab/show/diff/inverse'>invertible</a> if there is a reverse function such that both <a class='existingWikiWord' href='/nlab/show/diff/composition'>composites</a> are <a class='existingWikiWord' href='/nlab/show/diff/equality'>equal</a> to the <a class='existingWikiWord' href='/nlab/show/diff/identity+morphism'>identity morphism</a>, for a <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a> one only requires the composites to be <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopic</a> to the identity. Regarding objects in a homotopical context up to <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a> this way is to regard them as <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy types</a>. It is not wrong to summarize this by saying that: <blockquote> Homotopy theory embodies the <a class='existingWikiWord' href='/nlab/show/diff/gauge+theory'>gauge principle</a> in mathematics. </blockquote> Namely, in <a class='existingWikiWord' href='/nlab/show/diff/physics'>physics</a>, the <a class='existingWikiWord' href='/nlab/show/diff/gauge+theory'>gauge principle</a> says that it is wrong to ask for any two <a class='existingWikiWord' href='/nlab/show/diff/field+history'>things</a> to be <a class='existingWikiWord' href='/nlab/show/diff/equality'>equal</a>, instead one always has to ask whether there is a <a class='existingWikiWord' href='/nlab/show/diff/gauge+transformation'>gauge transformation</a> relating them, and then a <a class='existingWikiWord' href='/nlab/show/diff/higher+gauge+transformation'>gauge-of-gauge transformation</a> relating these, etc. Similarly in homotopy theory it is wrong to ask whether any two objects are equal, instead one has to ask whether there is a <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a> between them and then <a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy'>higher homotopies</a> between these, etc. \linebreak For more exposition on the general idea of homotopy theory see: <ul> <li><a class='existingWikiWord' href='/schreiber/show/diff/Higher+Structures' title='schreiber'>Higher Structures</a></li> </ul> \linebreak <h3 id='TopologicalHomotopyTheoryInIdeaSection'>Topological homotopy theory</h3> \begin{imagefromfile} “file_name”: “AHomotopy.jpg”, “float”: “right”, “width”: 380, “unit”: “px”, “margin”: { “top”: -40, “bottom”: 20, “right”: 0, “left”: 10 } \end{imagefromfile} The archetypical example of a homotopy theory is the <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical homotopy theory of topological spaces</a>, where one considers <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a> with <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> between them, and with the original concept of <a href='Introduction+to+Topology+--+2#LeftHomotopy'>topological homotopies</a> between these continuous functions. The <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> whose <a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a> are <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological spaces</a> and whose <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a> are <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>-<a class='existingWikiWord' href='/nlab/show/diff/equivalence+class'>classes</a> of <a class='existingWikiWord' href='/nlab/show/diff/continuous+map'>continuous functions</a> is also called the <a class='existingWikiWord' href='/nlab/show/diff/Ho%28Top%29'>classical homotopy category</a>. Classical constructions in topological homotopy theory are <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy groups</a>, <a class='existingWikiWord' href='/nlab/show/diff/Toda+bracket'>Toda brackets</a>, <a class='existingWikiWord' href='/nlab/show/diff/long+exact+sequence+of+homotopy+groups'>long exact sequences of homotopy groups</a>, … This <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical homotopy theory of topological spaces</a> has many applications, for example to <a class='existingWikiWord' href='/nlab/show/diff/covering+space'>covering space</a> theory, to <a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a> theory, to <a class='existingWikiWord' href='/nlab/show/diff/generalized+%28Eilenberg-Steenrod%29+cohomology'>Whitehead-generalized cohomology theory</a> and many more. (See also at <a class='existingWikiWord' href='/nlab/show/diff/shape+theory'>shape theory</a>.) Accordingly, homotopy theory has a large overlap with <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a>. \linebreak For more exposition of topological homotopy theory see: <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Topology+--+2'>Introduction to Basic Homotopy Theory</a></li> </ul> \linebreak <h3 id='abstract_homotopy_theory'>Abstract homotopy theory</h3> The basic structures seen in <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical homotopy theory of topological spaces</a> may be abstracted to yield an “abstract homotopy theory” that applies to a large variety of contexts. There are several more or less equivalent formalizations of the concept of “abstract homotopy theory”, including <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model categories</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-categories</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theories</a>. </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/higher+observational+type+theory'>higher observational type theories</a>. </li> </ul> The terminology <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> is short for “category of models of homotopy types”. The idea here is to consider <a class='existingWikiWord' href='/nlab/show/diff/category'>categories</a> equipped with suitable <a class='existingWikiWord' href='/nlab/show/diff/stuff%2C+structure%2C+property'>extra structure and properties</a> that encodes the existence of <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopies</a> between all <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a> and convenient means to handle and control these, in particular a means to construct the corresponding <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a>. \linebreak For a detailed introduction to abstract homotopy theory via model cateories see: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Homotopy+Theory'>Introduction to 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> \linebreak The approach of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-categories</a> to homotopy theory is meant to be more truthful to the intrinsic nature of homotopy theory. Instead of equipping an ordinary category with a extra concept of homotopy between its morphisms, here one regards the resulting structure as a <a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category</a> where the <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopies</a> themselves appear as a kind of higher order morphisms, called <a class='existingWikiWord' href='/nlab/show/diff/2-morphism'>2-morphisms</a> and where higher <a class='existingWikiWord' href='/nlab/show/diff/higher+homotopy'>homotopies of homotopies</a> are regarded as <a class='existingWikiWord' href='/nlab/show/diff/k-morphism'>k-morphisms</a> for all <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>. The terminology “<a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a>” signifies that homotopy theory is but one special case of general <a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a>, namely <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-categories</a> (hence homotopy theories) are those <a class='existingWikiWord' href='/nlab/show/diff/infinity-category'>infinity-categories</a> in which all <a class='existingWikiWord' href='/nlab/show/diff/k-morphism'>k-morphisms</a> for <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>></mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>k \gt 1</annotation></semantics></math> are <a class='existingWikiWord' href='/nlab/show/diff/inverse'>invertible</a> up to <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a>. If one drops this constraint, so that homotopies become “directed” then one might still speak of “<a class='existingWikiWord' href='/nlab/show/diff/directed+homotopy+theory'>directed homotopy theory</a>”. The archetypical example of an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a> is the <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_4' 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 <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a> of <a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>∞-groupoids</a>, just as <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> is the archetypical 1-<a class='existingWikiWord' href='/nlab/show/diff/category'>category</a>. This turns out to be equivalent, as homotopy theories, to the <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical homotopy theory of topological spaces</a> if restricted to those that admit the structure of <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complexes</a>. \linebreak <h3 id='RelationBetweenTopologicalAndAbstractHomotopyTheoryInIdeaSection'>Relation between topological and abstract homotopy theory: Cohesion.</h3> The case of the homotopy theory of topological spaces (<a href='#TopologicalHomotopyTheoryInIdeaSection'>above</a>) is so classical that often the distinction between <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a> and homotopy theory has been (certainly so in the early literature) and still is often blurred (for instance in textbook titles or lecture notes). A transparent conceptual way to understand how <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a> and homotopy theory are closely related and yet different is provided by the notion of <a class='existingWikiWord' href='/nlab/show/diff/cohesive'>cohesion</a>: A <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> has (or presents) a <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a> through the data that is retained in its <a class='existingWikiWord' href='/nlab/show/diff/path+infinity-groupoid'>path ∞-groupoid</a>, conveniently modeled by its <a class='existingWikiWord' href='/nlab/show/diff/singular+simplicial+complex'>singular simplicial set</a> <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Sing</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Sing(X)</annotation></semantics></math> (which is a <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a> and as such an <a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>∞-groupoid</a>). But the topological space contains more information than that retained in its <a class='existingWikiWord' href='/nlab/show/diff/path+infinity-groupoid'>path ∞-groupoid</a> – the latter is just a shadow of it, called the <a class='existingWikiWord' href='/nlab/show/diff/shape+modality'>shape</a> of the topological space. The extra information in a topological space is the <a class='existingWikiWord' href='/nlab/show/diff/cohesive'>cohesion</a> on its <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>underlying</a> <a class='existingWikiWord' href='/nlab/show/diff/set'>set</a> of points: The <a class='existingWikiWord' href='/nlab/show/diff/geometry'>geometric</a> information of how these <a class='existingWikiWord' href='/nlab/show/diff/point'>points</a> stick together within <a class='existingWikiWord' href='/nlab/show/diff/open+subspace'>open subsets</a>. This information is lost when one regards topological spaces as stand-ins for their <a class='existingWikiWord' href='/nlab/show/diff/shape+modality'>shapes</a> (<a class='existingWikiWord' href='/nlab/show/diff/path+infinity-groupoid'>path ∞-groupoids</a>) in plain homotopy theory. More formally: The <a class='existingWikiWord' href='/nlab/show/diff/Top'>category of</a> <a class='existingWikiWord' href='/nlab/show/diff/Delta-generated+topological+space'>D-topological spaces</a> is a <a class='existingWikiWord' href='/nlab/show/diff/sub-%28infinity%2C1%29-category'>full subcategory</a> of the <a class='existingWikiWord' href='/nlab/show/diff/cohesive+%28infinity%2C1%29-topos'>cohesive $(\infty,1)$-topos</a> of <a class='existingWikiWord' href='/nlab/show/diff/D-topological+infinity-groupoid'>D-topological $\infty$-groupoids</a> (also of <a class='existingWikiWord' href='/nlab/show/diff/smooth+infinity-groupoid'>smooth $\infty$-groupoids</a>). These carry a <a class='existingWikiWord' href='/nlab/show/diff/modal+homotopy+type+theory'>qualitative aspect</a> called their <a class='existingWikiWord' href='/nlab/show/diff/shape+modality'>shape</a>, which is a <a class='existingWikiWord' href='/nlab/show/diff/shape+via+cohesive+path+%E2%88%9E-groupoid'>generalization</a> of the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a>-construction. Now, regarding a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a> as an object in (plain) <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>, as traditionally done, really means to first regard it as a <a class='existingWikiWord' href='/nlab/show/diff/topological+groupoid'>topological groupoid</a> in <a class='existingWikiWord' href='/nlab/show/diff/cohesive+%28infinity%2C1%29-topos'>cohesive homotopy theory</a> and then, as such, to retain only its <a class='existingWikiWord' href='/nlab/show/diff/shape+modality'>shape</a>. The result is the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type'>homotopy type</a> which is “presented” by the topological space: \begin{imagefromfile} “file_name”: “RelationTopologyHomotopyTheory_20210921.jpg”, “width”: 700, “unit”: “px”, “margin”: { “top”: -20, “bottom”: 40, “right”: 0, “left”: 10 } \end{imagefromfile} \linebreak <h3 id='simplicial_homotopy_theory'>Simplicial homotopy theory</h3> The most immediate way<ins class='diffins'> to</ins> model an<a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>∞-groupoid</a> is as a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a> which is a <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a>. Accordingly, another homotopy theory equivalent to archetypical homotopy theory of <a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>∞-groupoids</a> is the <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+simplicial+sets'>classical homotopy theory of simplicial sets</a>, typically referred to as <a class='existingWikiWord' href='/nlab/show/diff/simplicial+homotopy+theory'>simplicial homotopy theory</a>. \linebreak <h3 id='flavors_of_homotopy_theory'>Flavors of homotopy theory</h3> But there are many other homotopy theories besides (the various incarnations of) this classical one. Important sub-classes of homotopy theories include: <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable homotopy theory</a> modeled by <a class='existingWikiWord' href='/nlab/show/diff/stable+model+category'>stable model categories</a> and <a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-categories</a>, where the operations of <a class='existingWikiWord' href='/nlab/show/diff/looping'>looping and delooping</a> are an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+%28infinity%2C1%29-categories'>equivalence</a>. This includes the <a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category+of+spectra'>stable (∞,1)-category of spectra</a> as well as those of <a class='existingWikiWord' href='/nlab/show/diff/sheaf+of+spectra'>sheaves of spectra</a>. </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+type+theory'>geometric homotopy theory</a> modeled by <a class='existingWikiWord' href='/nlab/show/diff/model+topos'>model toposes</a> and <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-toposes</a>, which are the homotopy theoretic analogs of ordinary <a class='existingWikiWord' href='/nlab/show/diff/topos'>toposes</a>. This includes the homotopy theoretic analog of <a class='existingWikiWord' href='/nlab/show/diff/category+of+sheaves'>categories of sheaves</a>, called <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves'>(∞,1)-categories of (∞,1)-sheaves</a> or of <a class='existingWikiWord' href='/nlab/show/diff/infinity-stack'>∞-stacks</a>, but it potentially also contains “<a class='existingWikiWord' href='/nlab/show/diff/elementary+%28infinity%2C1%29-topos'>elementary (∞,1)-toposes</a>”. </li> </ul> The <a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+type+theory'>geometric homotopy theory</a> of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-toposes</a> in particular serves as the foundation for <a class='existingWikiWord' href='/nlab/show/diff/higher+geometry'>higher geometry</a>/<a class='existingWikiWord' href='/nlab/show/diff/derived+geometry'>derived geometry</a>. This is relevant notably in the <a class='existingWikiWord' href='/nlab/show/diff/physics'>physics</a> of <a class='existingWikiWord' href='/nlab/show/diff/gauge+theory'>gauge theory</a>, where <a class='existingWikiWord' href='/nlab/show/diff/gauge+transformation'>gauge transformations</a> are identified with <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopies</a> in <a class='existingWikiWord' href='/nlab/show/diff/geometric+homotopy+type+theory'>geometric homotopy theory</a>. For more on this see at <a class='existingWikiWord' href='/nlab/show/diff/geometry+of+physics+--+homotopy+types'>geometry of physics -- homotopy types</a>. On the other hand, the incarnation of homotopy theory as <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a> exhibits the remarkably <a class='existingWikiWord' href='/nlab/show/diff/foundation+of+mathematics'>foundational</a> nature of homotopy theory. Contrary to its original appearance as a fairly complicated-looking theory built on top of classical <a class='existingWikiWord' href='/nlab/show/diff/set+theory'>set theory</a> and classical <a class='existingWikiWord' href='/nlab/show/diff/topology'>topology</a>, homotopy theory turns out to be intrinsically simple: it arises from plain <a class='existingWikiWord' href='/nlab/show/diff/dependent+type+theory'>dependent type theory</a> just by adopting a fully <a class='existingWikiWord' href='/nlab/show/diff/constructive+mathematics'>constructive</a> attitude towards the concept of <a class='existingWikiWord' href='/nlab/show/diff/identity'>identity</a>/<a class='existingWikiWord' href='/nlab/show/diff/equality'>equality</a>, see at <a class='existingWikiWord' href='/nlab/show/diff/identity+type'>identity type</a> for more on this. For exposition of this perspective see (<a href='#Shulman17'>Shulman 17</a>). \linebreak <h2 id='presentations'>Presentations</h2> A convenient, powerful, and traditional way to deal with <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-categories</a> is to “present” them by 1-categories with specified classes of morphisms called <a class='existingWikiWord' href='/nlab/show/diff/weak+equivalence'>weak equivalences</a> : a <a class='existingWikiWord' href='/nlab/show/diff/category+with+weak+equivalences'>category with weak equivalences</a> or <a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a>. The idea is as follows. Given a category <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> with a class <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math> of weak equivalences, we can form its <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a> or <a class='existingWikiWord' href='/nlab/show/diff/calculus+of+fractions'>category of fractions</a> <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>[</mo><msup><mi>W</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>C[W^{-1}]</annotation></semantics></math> by adjoining formal inverses to all the morphisms in <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math>. The <a class='existingWikiWord' href='/nlab/show/diff/simplicial+localization'>(∞,1)-category presented by</a> <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(C,W)</annotation></semantics></math>“ can be thought of as the result of regarding <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> as an <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-category with only identity <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-cells for <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>></mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>k\gt 1</annotation></semantics></math>, then adjoining formal inverses to morphisms in <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi></mrow><annotation encoding='application/x-tex'>W</annotation></semantics></math> in the <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-categorical sense; that is, making them into <a class='existingWikiWord' href='/nlab/show/diff/equivalence'>equivalences</a> rather than isomorphisms. It is remarkable that most <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_18' 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>-categories that arise in mathematics can be presented in this way. As with presentations of <a class='existingWikiWord' href='/nlab/show/diff/group'>group</a>s and other algebraic structures, very different presentations can give rise to equivalent <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_19' 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>-categories. For example, several different presentations of the <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_20' 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 of <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids are: <ul> <li><math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>C=</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/CW+complex'>CW-complex</a>es, <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>W=</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+equivalence'>homotopy equivalence</a>s - see <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>Quillen model structure on topological spaces</a></li> <li><math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>C=</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>s, <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>W=</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalences</a></li> <li><math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>C=</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a>es, <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>W=</annotation></semantics></math> simplicial homotopy equivalences</li> <li><math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>C=</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a>s, <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>W=</annotation></semantics></math> weak homotopy equivalences - see <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+simplicial+sets'>Quillen model structure on simplicial sets</a></li> <li><math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>C=</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/small+category'>small categories</a>, <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>W=</annotation></semantics></math> functors whose nerves are weak homotopy equivalences – see <a class='existingWikiWord' href='/nlab/show/diff/Thomason+model+structure'>Thomason model structure</a>.</li> </ul> The latter three can hence be regarded as providing “combinatorial models” for the homotopy theory of topological spaces. <h3 id='model_categories'>Model Categories</h3> The value of working with presentations of <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_32' 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>-categories rather than the <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_33' 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>-categories themselves is that the presentations are ordinary 1-categories, and thus much simpler to work with. For instance, ordinary limits and colimits are easy to construct in the category of topological spaces, or of simplicial sets, and we can then use these to get a handle on <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_34' 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>-categorical limits and colimits in the <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_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 of <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoids. However, we always have to make sure that we use only 1-categorical constructions that are homotopically meaningful, which essentially means that they induce <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_37' 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>-categorical meaningful constructions in the presented <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_38' 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. In particular, they must be invariant under weak equivalence. Most presentations of <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_39' 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>-categories come with additional classes of morphisms, called fibrations and cofibrations, that are very useful in performing constructions in a homotopically meaningful way. Quillen defined a <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> to be a 1-category together with classes of morphisms called weak equivalences, cofibrations, and fibrations that fit together in a very precise way (the term is meant to suggest “a category of models for a homotopy theory”). Many, perhaps most, presentations of <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_40' 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>-categories are model categories. Moreover, even when we do not have a model category, we often have classes of cofibrations and fibrations with many of the properties possessed by cofibrations and fibrations in a model category, and even when we do have a model category, there may be classes of cofibrations and fibrations, different from those in the model structure, that are useful for some purposes. Unlike the weak equivalences, which determine the “homotopy theory” and the <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_41' 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 that it presents, fibrations and cofibrations should be regarded as technical tools which make working directly with the presentation easier (or possible). Whether a morphism is a fibration or cofibration has no meaning after we pass to the presented <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_42' 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. In fact, every morphism is weakly equivalent to a fibration and to a cofibration. In particular, despite the common use of double-headed arrows for fibrations and hooked arrows for cofibrations, they do not correspond to <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_43' 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>-categorical epimorphisms and monomorphisms. In a model category, a morphism which is both a fibration and a weak equivalence is called an acyclic fibration or a trivial fibration. Dually we have acyclic or trivial cofibrations. An object <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is called cofibrant if the map <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>0\to X</annotation></semantics></math> from the initial object to <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a cofibration, and fibrant if the map <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>X\to 1</annotation></semantics></math> to the terminal# object is a fibration. The axioms of a model category ensure that for every object <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> there is an acyclic fibration <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>Q X \to X</annotation></semantics></math> where <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>Q X</annotation></semantics></math> is cofibrant and an acyclic cofibration <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>R</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>X\to R X</annotation></semantics></math> where <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>R X</annotation></semantics></math> is fibrant. <h3 id='examples'>Examples</h3> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/topological+homotopy+theory'>topological homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure on topological spaces</a>, </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/simplicial+homotopy+theory'>simplicial homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+simplicial+sets'>classical model structure on simplicial sets</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/localic+homotopy+theory'>localic homotopy theory</a> </li> </ul> For a (higher) category theorist, the following examples of model categories are perhaps the most useful to keep in mind: <ul> <li><math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>C=</annotation></semantics></math> sets, <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>W=</annotation></semantics></math> isomorphisms. All morphisms are both fibrations and cofibrations. The <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_55' 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 presented is again the 1-category <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math>.</li> <li><math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>C=</annotation></semantics></math> categories, <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>W=</annotation></semantics></math> equivalences of categories. The cofibrations are the functors which are injective on objects, and the fibrations are the <a class='existingWikiWord' href='/nlab/show/diff/isofibration'>isofibrations</a>. The acyclic fibrations are the equivalences of categories which are literally surjective on objects. Every object is both fibrant and cofibrant. The <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_59' 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 presented is the 2-category <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math>. This is often called the folk model structure.</li> <li><math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>C=</annotation></semantics></math> (strict) 2-categories and (strict) 2-functors, <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>W</mi><mo>=</mo></mrow><annotation encoding='application/x-tex'>W=</annotation></semantics></math> 2-functors which are <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalences of bicategories</a>. The fibrations are the 2-functors which are isofibrations on hom-categories and have an equivalence-lifting property. Every object is fibrant; the cofibrant 2-categories are those whose underlying 1-category is freely generated by some directed graph. The <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_63' 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 presented is the (weak) 3-category <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>2Cat</annotation></semantics></math>. This model structure is due to Steve Lack.</li> </ul> <h3 id='generalized_morphisms'>Generalized Morphisms</h3> The morphisms from <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> in the <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_67' 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 presented by <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mi>W</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(C,W)</annotation></semantics></math> are zigzags <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mo>←</mo><mo>≃</mo></mover><mo>→</mo><mover><mo>←</mo><mo>≃</mo></mover><mo>→</mo><mi>⋯</mi></mrow><annotation encoding='application/x-tex'> \stackrel{\simeq}{\leftarrow} \to \stackrel{\simeq}{\leftarrow} \to \cdots </annotation></semantics></math>; these are sometimes called generalized morphisms. Many presentations (including every model category) have the property that any such morphism is equivalent to one with a single zag, as in <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mo>←</mo><mo>≃</mo></mover><mo>→</mo><mover><mo>←</mo><mo>≃</mo></mover></mrow><annotation encoding='application/x-tex'>\stackrel{\simeq}{\leftarrow} \to \stackrel{\simeq}{\leftarrow}</annotation></semantics></math>. In a model category, a canonical form for such a zigzag is <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mi>Q</mi><mi>X</mi><mo>→</mo><mi>R</mi><mi>Y</mi><mover><mo>←</mo><mo>≃</mo></mover><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X \stackrel{\simeq}{\leftarrow} Q X \to R Y \stackrel{\simeq}{\leftarrow} Y</annotation></semantics></math> where <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>Q X</annotation></semantics></math> is cofibrant and <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mi>Y</mi></mrow><annotation encoding='application/x-tex'>R Y</annotation></semantics></math> is fibrant. In this case we can moreover take <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>Q X\to X</annotation></semantics></math> to be an acyclic fibration and <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi><mo>→</mo><mi>R</mi><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y\to R Y</annotation></semantics></math> to be an acyclic cofibration. Often it suffices to consider even shorter zigzags of the form <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mo>←</mo><mo>≃</mo></mover><mo>→</mo></mrow><annotation encoding='application/x-tex'>\stackrel{\simeq}{\leftarrow} \to</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>→</mo><mover><mo>←</mo><mo>≃</mo></mover></mrow><annotation encoding='application/x-tex'>\to \stackrel{\simeq}{\leftarrow}</annotation></semantics></math>. In particular, this is the case if every object is fibrant or every object is cofibrant. For example: <ul> <li>If <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> are strict 2-categories, then pseudofunctors <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X\to Y</annotation></semantics></math> are equivalent to strict 2-functors <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Q X \to Y</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Q</mi><mi>X</mi></mrow><annotation encoding='application/x-tex'>Q X</annotation></semantics></math> is a cofibrant replacement for <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</li> <li><a class='existingWikiWord' href='/nlab/show/diff/anafunctor'>anafunctors</a> are zigzags <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mo>←</mo><mo>≃</mo></mover><mo>→</mo></mrow><annotation encoding='application/x-tex'>\stackrel{\simeq}{\leftarrow} \to</annotation></semantics></math> in the <a class='existingWikiWord' href='/nlab/show/diff/canonical+model+structure'>folk model structure</a> on 1-categories whose first factor is an acyclic (i.e. surjective) fibration.</li> <li><a class='existingWikiWord' href='/nlab/show/diff/Morita+equivalence'>Morita morphisms</a> in the theory of <a class='existingWikiWord' href='/nlab/show/diff/Lie+groupoid'>Lie groupoids</a> are generalized morphisms of length one where both maps are acyclic fibrations.</li> </ul> If <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is cofibrant and <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> is fibrant, then every generalized morphism from <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Y</mi></mrow><annotation encoding='application/x-tex'>Y</annotation></semantics></math> is equivalent to an ordinary morphism. For example, if <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> is a cofibrant 2-category, then every pseudofunctor <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X\to Y</annotation></semantics></math> is equivalent to a strict 2-functor <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>X\to Y</annotation></semantics></math> <h3 id='quillen_equivalences'>Quillen Equivalences</h3> Quillen also introduced a highly structured notion of equivalence between model categories, now called a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+equivalence'>Quillen equivalence</a>, which among other things ensures that they present the same <math class='maruku-mathml' display='inline' id='mathml_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_92' 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. Quillen equivalences are now being used to compare different definitions of higher categories. <h2 id='RelatedConcepts'>Related entries</h2> <h3 id='general'>General</h3> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory+FAQ'>homotopy theory FAQ</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/higher+structure'>higher structures</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/algebraic+homotopy'>algebraic homotopy</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a> </li> </ul> <h3 id='flavors_of_homotopy_theory_2'>Flavors of homotopy theory</h3> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/synthetic+homotopy+theory'>synthetic homotopy theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/simplicial+homotopy+theory'>simplicial homotopy theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable homotopy theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/proper+homotopy+theory'>proper homotopy theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/parameterized+homotopy+theory'>parameterized homotopy theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/equivariant+homotopy+theory'>equivariant homotopy theory</a>, <a class='existingWikiWord' href='/nlab/show/diff/global+equivariant+homotopy+theory'>global equivariant homotopy theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/equivariant+stable+homotopy+theory'>equivariant stable homotopy theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational homotopy theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/rational+parameterized+stable+homotopy+theory'>rational parameterized stable homotopy theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/p-adic+homotopy+theory'>p-adic homotopy theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/chromatic+homotopy+theory'>chromatic homotopy theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/persistent+homotopy'>persistent homotopy theory</a> </li> </ul> <h3 id='basic_concepts_in_homotopy_theory'>Basic concepts in homotopy theory</h3> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/interval+object'>interval object</a>, <a class='existingWikiWord' href='/nlab/show/diff/cylinder+object'>cylinder object</a>, <a class='existingWikiWord' href='/nlab/show/diff/path+space+object'>path space object</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy'>homotopy</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+group'>homotopy group</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/deformation+retract'>deformation retract</a>, <a class='existingWikiWord' href='/nlab/show/diff/neighborhood+retract'>neighborhood retract</a> </li> </ul> <h3 id='categorical_homotopy_theory'>Categorical homotopy theory</h3> <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/category+with+weak+equivalences'>category with weak equivalences</a> <a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a>, <a class='existingWikiWord' href='/nlab/show/diff/relative+category'>relative 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/model+category'>model category</a> </li> </ul> (…) <ul> <li> <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+deformation+retract'>closed monoidal deformation retract</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+homotopical+category'>closed monoidal homotopical category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/cylinder+functor'>cylinder functor</a>, <a class='existingWikiWord' href='/nlab/show/diff/cocylinder'>cocylinder</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/derived+functor'>derived functor</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/deformation+retract+of+a+homotopical+category'>deformation retract of a homotopical category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/Dold+fibration'>Dold fibration</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/equivalence'>equivalence</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/canonical+model+structure'>folk model structure</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/fundamental+groupoid'>fundamental groupoid</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homological+resolution'>homological resolution</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopical+cohomology+theory'>homotopical cohomology theory</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimit</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+category+theory'>homotopy coherent category theory</a> </li> <li> <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/Hurewicz+cofibration'>Hurewicz cofibration</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+crossed+complexes'>model structure on crossed complexes</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+sets'>model structure on simplicial sets</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dendroidal+sets'>model structure on dendroidal sets</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/path+space+object'>path object</a>, <a class='existingWikiWord' href='/nlab/show/diff/path+groupoid'>path groupoid</a>, <a class='existingWikiWord' href='/nlab/show/diff/path+n-groupoid'>path n-groupoid</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/test+category'>test category</a> </li> <li> <a class='existingWikiWord' href='/nlab/show/diff/weak+equivalence'>weak equivalence</a> </li> </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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_93' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_94' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_95' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_96' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_97' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_98' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_99' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_100' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_101' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_102' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_103' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_104' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_105' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_106' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_107' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_108' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_109' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_110' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_111' 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_e3dbaffcefc8a6b7d764d87ba08e236cf9ced9b5_112' 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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