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href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28n%2Cr%29-category'>(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hom-object+in+a+quasi-category'>hom-objects</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivalence+in+an+%28infinity%2C1%29-category'>equivalences in</a>/<a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+%28infinity%2C1%29-categories'>of</a> <math class='maruku-mathml' display='inline' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_2' 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</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sub-%28infinity%2C1%29-category'>sub-(∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-%28infinity%2C1%29-category'>reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+of+an+%28infinity%2C1%29-category'>reflective localization</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/opposite+quasi-category'>opposite (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-category'>over (∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/join+of+quasi-categories'>join of quasi-categories</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/flat+%28infinity%2C1%29-functor'>exact (∞,1)-functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-functors'>(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves'>(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/fibration+of+quasi-categories'>fibrations</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inner+fibration'>inner fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/right%2Fleft+Kan+fibration'>left/right fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cartesian+fibration'>Cartesian fibration</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Cartesian+morphism'>Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>limit</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/terminal+object+in+a+quasi-category'>terminal object</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>locally presentable</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/essentially+small+%28infinity%2C1%29-category'>essentially small</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+small+%28infinity%2C1%29-category'>locally small</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/accessible+%28infinity%2C1%29-category'>accessible</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/idempotent+complete+%28infinity%2C1%29-category'>idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories'>(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-Grothendieck+construction'>(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor+theorem'>adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+with+weak+equivalences'>category with weak equivalences</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derivator'>derivator</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+quasi-categories'>model structure for quasi-categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+Cartesian+fibrations'>model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relation+between+quasi-categories+and+simplicial+categories'>relation to simplicial categories</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+nerve'>homotopy coherent nerve</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+category'>simplicial model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+sets'>model structure for Kan complexes</a></li> </ul> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#relation_to_simplicially_enriched_categories'>Relation to simplicially enriched categories</a></li><li><a href='#higher_associahedra_in_quasicategories'>Higher associahedra in quasi-categories</a></li></ul></li><li><a href='#examples'>Examples</a></li><li><a href='#constructions_in_quasicategories'>Constructions in quasi-categories</a></li><li><a href='#RelatedConcepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>The notion of <em>quasi-category</em> is a <a class='existingWikiWord' href='/nlab/show/diff/geometric+definition+of+higher+categories'>geometric model</a> for <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a>.</p> <p>In analogy to how a <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a> is a model in terms of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a>s of an <a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>∞-groupoid</a> – also called an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C0%29-category'>(∞,0)-category</a> – a <a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-category</a> is a model in terms of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a>s of an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a>.</p> <div class='un_remark'> <h6 id='warning'>Warning</h6> <p>In older literature, such as <a class='existingWikiWord' href='/nlab/show/diff/Abstract+and+Concrete+Categories'>The Joy of Cats</a>, the term “quasicategory” was sometimes used for a “very large” category whose objects are <a class='existingWikiWord' href='/nlab/show/diff/large+category'>large categories</a> or otherwise built out of <a class='existingWikiWord' href='/nlab/show/diff/class'>proper classes</a>, but nowadays this usage is fairly archaic. See also <a class='existingWikiWord' href='/nlab/show/diff/metacategory'>metacategory</a>.</p> </div> <h2 id='definition'>Definition</h2> <div class='num_defn'> <h6 id='definition_2'>Definition</h6> <p>A <strong>quasi-category</strong> or <strong><a class='existingWikiWord' href='/nlab/show/diff/weak+Kan+complex'>weak Kan complex</a></strong> is a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a> <math class='maruku-mathml' display='inline' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> satisfying the following equivalent conditions</p> <ul> <li> <p>all <em>inner</em> <a class='existingWikiWord' href='/nlab/show/diff/horn'>horn</a>s in <math class='maruku-mathml' display='inline' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> have fillers. This means that the lifting condition given at <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a> is imposed only for horns <math class='maruku-mathml' display='inline' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Λ</mi> <mi>i</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Lambda^i[n]</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn><mo><</mo><mi>i</mi><mo><</mo><mi>n</mi></mrow><annotation encoding='application/x-tex'>0 \lt i \lt n</annotation></semantics></math>.</p> </li> <li> <p>the morphism of simplicial sets</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>2</mn><mo stretchy='false'>]</mo><mo>,</mo><mi>C</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>sSet</mi><mo stretchy='false'>(</mo><msup><mi>Λ</mi> <mn>1</mn></msup><mo stretchy='false'>[</mo><mn>2</mn><mo stretchy='false'>]</mo><mo>,</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> sSet(\Delta[2],C) \to sSet(\Lambda^1[2],C) </annotation></semantics></math></div> <p>(induced from the inner <a class='existingWikiWord' href='/nlab/show/diff/horn'>horn</a> inclusion <math class='maruku-mathml' display='inline' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Λ</mi> <mn>1</mn></msup><mo stretchy='false'>[</mo><mn>2</mn><mo stretchy='false'>]</mo><mo>→</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>2</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Lambda^1[2] \to \Delta[2]</annotation></semantics></math>) is an acyclic <a class='existingWikiWord' href='/nlab/show/diff/Kan+fibration'>Kan fibration</a>.</p> </li> </ul> </div> <p>The equivalence of these two definitions is due to <a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>Andre Joyal</a> and recalled as <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>HTT, corollary 2.3.2.2</a>. Quasi-categories are the <a class='existingWikiWord' href='/nlab/show/diff/fibrant+object'>fibrant objects</a> in the <a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+quasi-categories'>model structure for quasi-categories</a>.</p> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p>The second condition says manifestly that a quasi-category is a simplicial set in which composition of any two composable edges is defined up to a contractible space of choices. This is the <a class='existingWikiWord' href='/nlab/show/diff/coherence+law'>coherence law</a> on composition.</p> </div> <div class='un_defn'> <h6 id='definition_3'>Definition</h6> <p>An <strong><a class='existingWikiWord' href='/nlab/show/diff/algebraic+quasi-category'>algebraic quasi-category</a></strong> is a quasi-category equipped with a <em>choice</em> of inner horn fillers.</p> </div> <p>While quasi-categories provide a <a class='existingWikiWord' href='/nlab/show/diff/geometric+definition+of+higher+categories'>geometric definition of higher categories</a>, algebraic quasi-categories provide an <a class='existingWikiWord' href='/nlab/show/diff/algebraic+definition+of+higher+categories'>algebraic definition of higher categories</a>. For more details on this see <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+algebraic+fibrant+objects'>model structure on algebraic fibrant objects</a>.</p> <h2 id='properties'>Properties</h2> <h3 id='relation_to_simplicially_enriched_categories'>Relation to simplicially enriched categories</h3> <p>The <a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+nerve'>homotopy coherent nerve</a> relates quasi-categories with another model for <math class='maruku-mathml' display='inline' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_9' 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: <a class='existingWikiWord' href='/nlab/show/diff/simplicially+enriched+category'>simplicially enriched categories</a>.</p> <p>See <a class='existingWikiWord' href='/nlab/show/diff/relation+between+quasi-categories+and+simplicial+categories'>relation between quasi-categories and simplicial categories</a> for more.</p> <h3 id='higher_associahedra_in_quasicategories'>Higher associahedra in quasi-categories</h3> <p>While the geometric definition of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a> in terms of quasi-categories elegantly captures all the higher categorical data automatically, it may be of interest in applications to explicitly extract the associators and higher associators encoded by this structure, that would show up in any <a class='existingWikiWord' href='/nlab/show/diff/algebraic+definition+of+higher+categories'>algebraic definition of the same categorical structure</a>, such as <a class='existingWikiWord' href='/nlab/show/diff/algebraic+quasi-category'>algebraic quasi-categories</a>.</p> <p>For a discussion of this see</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Emily+Riehl'>Emily Riehl</a>, <em>Associativity data in an <math class='maruku-mathml' display='inline' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_10' 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</em> (<a href='http://www.math.jhu.edu/~eriehl/associativity.pdf'>pdf</a> <a href='http://golem.ph.utexas.edu/category/2009/10/associativity_data_in_an_1cate.html'>blog</a>)</li> </ul> <h2 id='examples'>Examples</h2> <p>The two basic examples for quasi-categories are</p> <ul> <li> <p>Every <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a> is, in particular, a quasi-category.</p> </li> <li> <p>The <a class='existingWikiWord' href='/nlab/show/diff/nerve'>nerve</a> of a <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> is a quasi-category.</p> </li> </ul> <p>Since the nerve of a category is a <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a> iff the category is a <a class='existingWikiWord' href='/nlab/show/diff/groupoid'>groupoid</a>, quasi-categories are a minimal common generalization of Kan complexes and nerves of categories.</p> <p>By the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+hypothesis'>homotopy hypothesis</a>-theorem every Kan complex arises, up to equivalence, as the <a class='existingWikiWord' href='/nlab/show/diff/fundamental+infinity-groupoid'>fundamental ∞-groupoid</a> of a <a class='existingWikiWord' href='/nlab/show/diff/topological+space'>topological space</a>.</p> <p>Analogously, every <a class='existingWikiWord' href='/nlab/show/diff/directed+topological+space'>directed topological space</a> <math class='maruku-mathml' display='inline' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> has naturally a <a class='existingWikiWord' href='/nlab/show/diff/fundamental+%28infinity%2C1%29-category'>fundamental (∞,1)-category</a> given by a quasi-category whose <math class='maruku-mathml' display='inline' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-cells are maps <math class='maruku-mathml' display='inline' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Δ</mi> <mi>Top</mi> <mi>k</mi></msubsup><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>\Delta^k_{Top} \to X</annotation></semantics></math> that map the 1-<a class='existingWikiWord' href='/nlab/show/diff/simplicial+skeleton'>skeleton</a> of the topological simplex in an order-preserving way to directed paths in <math class='maruku-mathml' display='inline' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/directed+homotopy+theory'>directed homotopy theory</a><span> <del class='diffmod'> that</del><ins class='diffmod'> which</ins> would state that<del class='diffmod'> this</del><ins class='diffmod'> this,</ins> or a similar<del class='diffmod'> construction</del><ins class='diffmod'> construction,</ins> exhausts all quasicategories up to<del class='diffmod'> equivalence,</del><ins class='diffmod'> equivalence</ins><del class='diffmod'> does</del><ins class='diffmod'> is</ins> not<del class='diffmod'> quite</del><ins class='diffmod'> yet</ins><del class='diffmod'> exist</del><ins class='diffmod'> known.</ins><del class='diffdel'> yet.</del></span></p> <h2 id='constructions_in_quasicategories'>Constructions in quasi-categories</h2> <p>The point of quasi-categories is that they are supposed to provide a fully <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy-theoretic</a><span> refinement of the ordinary notion of<ins class='diffins'> a</ins></span><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a>. In particular, all the familiar constructions of <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a><span> have natural<del class='diffmod'> analogs</del><ins class='diffmod'> analogues</ins> in the context of quasi-categories. See for instance</span></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hom-object+in+a+quasi-category'>hom-object in a quasi-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivalence+in+an+%28infinity%2C1%29-category'>equivalence in a quasi-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+%28infinity%2C1%29-categories'>equivalence of quasi-categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/join+of+quasi-categories'>join of quasi-categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-category'>over quasi-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/terminal+object+in+a+quasi-category'>terminal object in a quasi-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monomorphism+in+an+%28infinity%2C1%29-category'>monomorphism in an (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>limit in quasi-categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sub-%28infinity%2C1%29-category'>sub-quasi-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/opposite+quasi-category'>opposite quasi-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fibration+of+quasi-categories'>fibrations of quasi-categories</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inner+fibration'>inner Kan fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cartesian+fibration'>Cartesian fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/right%2Fleft+Kan+fibration'>left Kan fibration</a>/<a class='existingWikiWord' href='/nlab/show/diff/right%2Fleft+Kan+fibration'>right Kan fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/minimal+inner+fibration'>minimal Joyal fibration</a></p> </li> </ul> </li> </ul> <h2 id='RelatedConcepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relation+between+quasi-categories+and+simplicial+categories'>relation between quasi-categories and sSet-enriched categories</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+nerve'>homotopy coherent nerve</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/rigidification+of+quasi-categories'>rigidification of quasi-categories</a></p> </li> <li> <p>One may try to further weaken the filler conditions in order to describe <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29-category'>(∞,n)-categories</a> for <math class='maruku-mathml' display='inline' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n \gt 1</annotation></semantics></math>. One approach along these lines is the theory of <em><a class='existingWikiWord' href='/nlab/show/diff/weak+complicial+set'>weak complicial sets</a></em>.</p> </li> <li> <p>Or one may change the shape category to pass from <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial sets</a> to <a class='existingWikiWord' href='/nlab/show/diff/cellular+set'>cellular sets</a>. A quasi-category-like definition of <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29-category'>(∞,n)-categories</a> on these – <em><a class='existingWikiWord' href='/nlab/show/diff/n-quasicategory'>n-quasicategories</a></em> – is discussed at <em><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+cellular+sets'>model structure on cellular sets</a></em>.</p> </li> </ul> <h2 id='references'>References</h2> <p><span> The notion of quasi-categories<del class='diffmod'> were</del><ins class='diffmod'> was</ins> originally defined, under the name</span><em><a class='existingWikiWord' href='/nlab/show/diff/weak+Kan+complex'>weak Kan complexes</a></em> in:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael+Boardman'>Michael Boardman</a>, <a class='existingWikiWord' href='/nlab/show/diff/Rainer+Vogt'>Rainer Vogt</a>, <em>Homotopy invariant algebraic structures on topological spaces</em>, Lecture Notes in Mathematics <strong>347</strong> Springer (1973) [[doi:10.1007/BFb0068547](https://doi.org/10.1007/BFb0068547)]</p> </li> <li id='Vogt73'> <p><a class='existingWikiWord' href='/nlab/show/diff/Rainer+Vogt'>Rainer Vogt</a>, <em>Homotopy limits and colimits</em>, Math. Z., <strong>134</strong> (1973) 11-52 [[doi:10.1007/BF01219090](https://doi.org/10.1007/BF01219090), <a href='https://eudml.org/doc/171965'>eudml:171965</a>]</p> </li> </ul> <p>The main theorem of <a href='#Vogt73'>Vogt (1973)</a> involved a category of <a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+diagram'>homotopy coherent diagrams</a><span> defined on a topologically enriched category and showed<ins class='diffins'> that</ins> it was equivalent to a quotient category of the category of (commutative) diagrams on the same category.</span></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jean-Marc+Cordier'>J.-M. Cordier</a>, <em>Sur la notion de diagramme homotopiquement cohérent</em>, Cahiers de Top. Géom. Diff., 23, (1982), 93 –112,</li> </ul> <p>defined the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+nerve'>homotopy coherent nerve</a> of any <a class='existingWikiWord' href='/nlab/show/diff/simplicially+enriched+category'>simplicially enriched category</a>. This generalised the <a class='existingWikiWord' href='/nlab/show/diff/nerve'>nerve</a> of an ordinary category. In</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jean-Marc+Cordier'>J.-M. Cordier</a> and <a class='existingWikiWord' href='/nlab/show/diff/Tim+Porter'>Tim Porter</a>, <em>Vogt’s theorem on categories of homotopy coherent diagrams</em>, Math. Proc. Cambridge Philos. Soc., 100, (1986), 65–90,</li> </ul> <p>it was shown that this homotopy coherent nerve was a quasi-category if the simplicial enrichment was by Kan complexes.</p> <p>A systematic study of SSet-enriched categories in this context is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jean-Marc+Cordier'>J.-M. Cordier</a>, <a class='existingWikiWord' href='/nlab/show/diff/Tim+Porter'>Tim Porter</a> <em>Homotopy coherent category theory</em> Trans. Amer. Math. Soc. 349 (1997), no. 1, 1-54. (<a href='http://www.ams.org/tran/1997-349-01/S0002-9947-97-01752-2/S0002-9947-97-01752-2.pdf'>pdf</a>)</li> </ul> <p>The importance of quasi-categories as a basis for <a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a> has been particularly emphasized in work by <a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>André Joyal</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>André Joyal</a>, <em>Quasi-categories and Kan complexes</em>, J. Pure Appl. Algebra <strong>175</strong> (2002) 207-222 []</li> </ul> <p><span> For several<del class='diffmod'> years</del><ins class='diffmod'> years,</ins> Joyal has been preparing a textbook on the subject which<ins class='diffins'> has</ins> never<del class='diffmod'> became</del><ins class='diffmod'> become</ins> publically available, but an extensive writeup of lecture notes<del class='diffmod'> is:</del><ins class='diffmod'> is</ins></span></p> <ul> <li id='Joyal08'> <p><a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>André Joyal</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/The+Theory+of+Quasi-Categories+and+its+Applications'>The Theory of Quasi-Categories and its Applications</a></em>, lectures at <em><a href='https://lists.lehigh.edu/pipermail/algtop-l/2007q4/000017.html'>Advanced Course on Simplicial Methods in Higher Categories</a></em>, CRM (2008) [[pdf](http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf), <a class='existingWikiWord' href='/nlab/files/JoyalTheoryOfQuasiCategories.pdf' title='pdf'>pdf</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Andr%C3%A9+Joyal'>André Joyal</a>, <em>Notes on quasi-categories</em> (2008) [[pdf](http://www.math.uchicago.edu/~may/IMA/Joyal.pdf), <a class='existingWikiWord' href='/nlab/files/JoyalNotesOnQuasiCategories.pdf' title='pdf'>pdf</a>]</p> </li> </ul> <p><span><del class='diffmod'> Meanwhile</del><ins class='diffmod'> Meanwhile,</ins></span><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a><span> , building on Joyal’s work, has<del class='diffdel'> considerably</del> pushed the theory<ins class='diffins'> considerably</ins> further. A comprehensive discussion of the theory of</span><math class='maruku-mathml' display='inline' id='mathml_8f09542c9a48e5023e8621329a8974d1649b2750_16' 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 in terms of the models <a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-category</a> and <a class='existingWikiWord' href='/nlab/show/diff/simplicially+enriched+category'>simplicially enriched category</a> is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>Higher Topos Theory</a></em> .</li> </ul> <p>An overview of the material there is contained in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Denis-Charles+Cisinski'>Denis-Charles Cisinski</a>, <em>Catégories supérieures et théorie des topos</em>, Séminaire Bourbaki, 21.3.2015 [[pdf](http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf)]</li> </ul> <p>Textbook accounts:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Emily+Riehl'>Emily Riehl</a>, Part IV in: <em><a class='existingWikiWord' href='/nlab/show/diff/Categorical+Homotopy+Theory'>Categorical Homotopy Theory</a></em>, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), <a href='http://www.math.jhu.edu/~eriehl/cathtpy.pdf'>pdf</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Denis-Charles+Cisinski'>Denis-Charles Cisinski</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Higher+categories+and+homotopical+algebra'>Higher Categories and Homotopical Algebra</a></em>, Cambridge University Press (2019) [[doi:10.1017/9781108588737](https://doi.org/10.1017/9781108588737), <a href='http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf'>pdf</a>]</p> </li> </ul> <p>The relation between <a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-categories</a> and <a class='existingWikiWord' href='/nlab/show/diff/simplicially+enriched+category'>simplicially enriched categories</a> was discussed in detail in</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Dugger'>Dan Dugger</a>, <a class='existingWikiWord' href='/nlab/show/diff/David+Spivak'>David Spivak</a>, <em>Rigidification of quasi-categories</em> (<a href='http://arxiv.org/abs/0910.0814'>arXiv:0910.0814</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Dugger'>Dan Dugger</a>, <a class='existingWikiWord' href='/nlab/show/diff/David+Spivak'>David Spivak</a>, <em>Mapping spaces in quasi-categories</em> (<a href='http://arxiv.org/abs/0911.0469'>arXiv:0911.0469</a>)</p> </li> </ul> <p>Further survey:</p> <ul> <li id='Rezk16'> <p><a class='existingWikiWord' href='/nlab/show/diff/Charles+Rezk'>Charles Rezk</a>, <em>Stuff about quasicategories</em>, Lecture Notes for course at University of Illinois at Urbana-Champaign, 2016, version May 2017 (<a href='http://math.uiuc.edu/~rezk/595-fal16/quasicats.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/RezkQuasicategories.pdf' title='pdf'>pdf</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Charles+Rezk'>Charles Rezk</a>, <em>Introduction to quasicategories</em> (2022) [[pdf](https://faculty.math.illinois.edu/~rezk/quasicats.pdf), <a class='existingWikiWord' href='/nlab/files/Rezk-IntroToQuasicategories.pdf' title='pdf'>pdf</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Moritz+Groth'>Moritz Groth</a>, <em>A short course on ∞-categories</em> (<a href='https://arxiv.org/abs/1007.2925'>arXiv:1007.2925</a>)</p> </li> </ul> <p>An in-depth study of adjunctions between quasi-categories and the monadicity theorem is given in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Emily+Riehl'>Emily Riehl</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dominic+Verity'>Dominic Verity</a> <em>The 2-category theory of quasi-categories</em> (<a href='http://arxiv.org/abs/1306.5144'>arXiv</a>), <em>Homotopy coherent adjunctions and the formal theory of monads</em> (<a href='http://arxiv.org/abs/1310.8279'>arXiv</a>)</li> </ul> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on January 26, 2025 at 09:48:13. 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