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concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>, <a class="existingWikiWord" href="/nlab/show/coherence">coherence</a></li> <li><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></li> <li><a class="existingWikiWord" href="/nlab/show/stabilization">looping and suspension</a></li> </ul> <h2 id="basic_theorems">Basic theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/periodic+table">periodic table</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization+hypothesis">stabilization hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/michaelshulman/show/exactness+hypothesis">exactness hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a></p> </li> </ul> <h2 id="models">Models</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Theta-space">Theta-space</a></li> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-fold+complete+Segal+space">n-fold complete Segal space</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+quasi-category">algebraic quasi-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a></li> <li><a class="existingWikiWord" href="/nlab/show/complete+Segal+space">complete Segal space</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-category">(∞,0)-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complex">algebraic Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+T-complex">simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2CZ%29-category">(∞,Z)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> = (n,n)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/1-category">1-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/0-category">0-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-1%29-category">(-1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-2%29-category">(-2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-poset">n-poset</a> = <a class="existingWikiWord" href="/nlab/show/n-poset">(n-1,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/poset">poset</a> = <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a> = <a class="existingWikiWord" href="/nlab/show/%281%2C2%29-category">(1,2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a> = (n,0)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>/<a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a></li> <li><a class="existingWikiWord" href="/nlab/show/geometric+definition+of+higher+category">geometric definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+model+for+weak+%E2%88%9E-categories">simplicial model for weak ∞-categories</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/complicial+set">complicial set</a></li> <li><a class="existingWikiWord" href="/nlab/show/weak+complicial+set">weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraic definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></li> <li><a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/tetracategory">tetracategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Batanin+%E2%88%9E-category">Batanin ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck-Maltsiniotis+%E2%88%9E-categories">Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></li> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+category">A-∞ category</a></li> <li><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id="morphisms">Morphisms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/transfor">transfor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></li> <li><a class="existingWikiWord" href="/nlab/show/modification">modification</a></li> </ul> </li> </ul> <h2 id="functors">Functors</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/functor">functor</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></li> <li><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></li> <li><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">(∞,1)-adjunction</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Kan+extension">(∞,1)-Kan extension</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">(∞,1)-limit</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></li> </ul> <h2 id="extra_properties_and_structure">Extra properties and structure</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/cosmic+cube">cosmic cube</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a>, <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></li> </ul> <h2 id="1categorical_presentations">1-categorical presentations</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></li> </ul> </div></div> </div> </div> <h1 id="categories"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The term <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-category</em> broadly refers to <a class="existingWikiWord" href="/nlab/show/higher+categories">higher categories</a> with no bound on the dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> of their <a class="existingWikiWord" href="/nlab/show/n-morphisms"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>n</mi> </mrow> <annotation encoding="application/x-tex">n</annotation> </semantics> </math>-morphisms</a>.</p> <p>There are two different ways this is understood more in detail:</p> <p>Historically, a tower of notions of <a class="existingWikiWord" href="/nlab/show/n-categories">n-categories</a> (<a class="existingWikiWord" href="/nlab/show/0-category">0-</a>, <a class="existingWikiWord" href="/nlab/show/1-category">1-</a>, <a class="existingWikiWord" href="/nlab/show/2-category">2-</a>, <a class="existingWikiWord" href="/nlab/show/3-category">3-</a>, <a class="existingWikiWord" href="/nlab/show/4-categories">4-categories</a>, …) was continued to a limiting case of “<a class="existingWikiWord" href="/nlab/show/omega-categories">omega-categories</a>”, and sometimes “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories” is used synonymously in this sense, broadly referring to the most general notion of <a class="existingWikiWord" href="/nlab/show/higher+categories">higher categories</a>. Beware that this general notion is not well developed nor is the equivalence or not of the few existing models, but see for instance the entries <em><a class="existingWikiWord" href="/nlab/show/weak+omega-category">weak omega-category</a></em>, <em><a class="existingWikiWord" href="/nlab/show/weak+complicial+set">weak complicial set</a></em> and <em><a class="existingWikiWord" href="/nlab/show/opetopic+omega-categories">opetopic omega-categories</a></em> .</p> <p>The more fine-grained organization of types of higher categories by bidegree is better understood: <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-categories">(n,r)-categories</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">n = \infty</annotation></semantics></math>, hence <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-categories"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>, have by now a very well developed theory with ubiquitous applications, and many authors refer to these <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-categories"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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> (aka: <em><a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a></em>) as just “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories”, for short (some authors even just say “categories” now, for this case, with a <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy-theoretic</a> background foundation tacitly understood).</p> <p>In terms of the terminology of <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-categories"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(n,r)</annotation> </semantics> </math>-categories</a>, the previous general notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-categories may be referred to as <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,\infty)</annotation></semantics></math>-categories</em></p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/0-category">0-category</a>, <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-category">3-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-category">(∞,0)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-category">(n,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a>, <a class="existingWikiWord" href="/nlab/show/internal+%28%E2%88%9E%2C1%29-category">internal (∞,1)-category</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <p><a class="existingWikiWord" href="/nlab/show/table+-+models+for+%28%E2%88%9E%2C1%29-categories">table - models for (∞,1)-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+omega-category">weak omega-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/opetopic+omega-category">opetopic omega-category</a></p> </li> </ul> <h2 id="references">References</h2> <blockquote> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></em>, <em><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a></em>, <em><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></em>.</p> </blockquote> <p>A versatile approach via <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> and <a class="existingWikiWord" href="/nlab/show/infinity-cosmoi">infinity-cosmoi</a>:</p> <ul> <li id="RiehlVerity16"><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <a class="existingWikiWord" href="/nlab/show/Dominic+Verity">Dominic Verity</a>, <em>Infinity category theory from scratch</em>, Higher Structures <strong>4</strong> 1 (2020) [<a href="https://arxiv.org/abs/1608.05314">arXiv:1608.05314</a>, <a href="http://www.math.jhu.edu/~eriehl/scratch.pdf">pdf</a>, <a href="https://www.epfl.ch/labs/hessbellwald-lab/seminar/ytm2015/">lectures</a>]</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C%E2%88%9E%29-categories">(∞,∞)-categories</a></p> <p>in terms of <a class="existingWikiWord" href="/nlab/show/induction">inductively</a> and <a class="existingWikiWord" href="/nlab/show/coinduction">coinductively</a> defined <a class="existingWikiWord" href="/nlab/show/equivalences">equivalences</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Zach+Goldthorpe">Zach Goldthorpe</a>, <em>Homotopy theories of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,\infty)</annotation></semantics></math>-categories as universal fixed points with respect to enrichment</em>, International Mathematics Research Notices <strong>2023</strong> 22 (2023) 19592–19640 [<a href="https://arxiv.org/abs/2307.00442">arXiv:2307.00442</a>, <a href="https://doi.org/10.1093/imrn/rnad196">doi:10.1093/imrn/rnad196</a>]</li> </ul> <p>and generalization to higher sheaves/stacks of these:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Zach+Goldthorpe">Zach Goldthorpe</a>, <em>Sheaves of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,\infty)</annotation></semantics></math>-categories</em> [<a href="https://arxiv.org/abs/2403.06926">arXiv:2403.069262</a>]</li> </ul> <p>see also</p> <ul> <li>Félix Loubaton, <em>Theory and models of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,\omega)</annotation></semantics></math>-categories</em> [<a href="https://arxiv.org/abs/2307.11931">arXiv:2307.11931</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 12, 2024 at 09:35:02. 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