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<ul> <li><a class='existingWikiWord' href='/nlab/show/diff/k-morphism'>k-morphism</a>, <a class='existingWikiWord' href='/nlab/show/diff/coherence+law'>coherence</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/looping'>looping and delooping</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/stabilization'>looping and suspension</a></li> </ul> <h2 id='basic_theorems'>Basic theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/delooping+hypothesis'>delooping hypothesis</a>-theorem</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/periodic+table'>periodic table</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stabilization+hypothesis'>stabilization hypothesis</a>-theorem</p> </li> <li> <p><a class='existingWikiWord' href='/michaelshulman/show/diff/exactness+hypothesis' title='michaelshulman'>exactness hypothesis</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/applications+of+%28higher%29+category+theory'>applications of (higher) category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/%28n%2Cr%29-category'>(n,r)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Theta-space'>Theta-space</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/infinity-category'>∞-category</a>/<a class='existingWikiWord' href='/nlab/show/diff/infinity-category'>∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2Cn%29-category'>(∞,n)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/n-fold+complete+Segal+space'>n-fold complete Segal space</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C2%29-category'>(∞,2)-category</a></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/quasi-category'>quasi-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/algebraic+quasi-category'>algebraic quasi-category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/simplicially+enriched+category'>simplicially enriched category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/complete+Segal+space'>complete Segal space</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C0%29-category'>(∞,0)-category</a>/<a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>∞-groupoid</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/algebraic+Kan+complex'>algebraic Kan complex</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/simplicial+T-complex'>simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2CZ%29-category'>(∞,Z)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/n-category'>n-category</a> = (n,n)-category <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/2-category'>2-category</a>, <a class='existingWikiWord' href='/nlab/show/diff/%282%2C1%29-category'>(2,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/1-category'>1-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/0-category'>0-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28-1%29-category'>(-1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28-2%29-category'>(-2)-category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/n-poset'>n-poset</a> = <a class='existingWikiWord' href='/nlab/show/diff/n-poset'>(n-1,n)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/partial+order'>poset</a> = <a class='existingWikiWord' href='/nlab/show/diff/%280%2C1%29-category'>(0,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/2-poset'>2-poset</a> = <a class='existingWikiWord' href='/nlab/show/diff/2-poset'>(1,2)-category</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/n-groupoid'>n-groupoid</a> = (n,0)-category <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/2-groupoid'>2-groupoid</a>, <a class='existingWikiWord' href='/nlab/show/diff/3-groupoid'>3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/vertical+categorification'>categorification</a>/<a class='existingWikiWord' href='/nlab/show/diff/decategorification'>decategorification</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/geometric+definition+of+higher+categories'>geometric definition of higher category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+for+weak+omega-categories'>simplicial model for weak ∞-categories</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/complicial+set'>complicial set</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/weak+complicial+set'>weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/algebraic+definition+of+higher+categories'>algebraic definition of higher category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/bicategory'>bicategory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/bigroupoid'>bigroupoid</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/tricategory'>tricategory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/tetracategory'>tetracategory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/strict+omega-category'>strict ∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Batanin+omega-category'>Batanin ∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Trimble+n-category'>Trimble ∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck-Maltsiniotis+infinity-category'>Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/stable+homotopy+theory'>stable homotopy theory</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric monoidal category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+%28infinity%2C1%29-category'>symmetric monoidal (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/dg-category'>dg-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-category'>A-∞ category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/triangulated+category'>triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id='morphisms'>Morphisms</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/k-morphism'>k-morphism</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/2-morphism'>2-morphism</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/transfor'>transfor</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/modification'>modification</a></li> </ul> </li> </ul> <h2 id='functors'>Functors</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/2-functor'>2-functor</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/pseudofunctor'>pseudofunctor</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/lax+functor'>lax functor</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a></li> </ul> <h2 id='universal_constructions'>Universal constructions</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/2-limit'>2-limit</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>(∞,1)-adjunction</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-Kan+extension'>(∞,1)-Kan extension</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-limit</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%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/diff/cosmic+cube'>cosmic cube</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/k-tuply+monoidal+n-category'>k-tuply monoidal n-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/strict+omega-category'>strict ∞-category</a>, <a class='existingWikiWord' href='/nlab/show/diff/strict+omega-groupoid'>strict ∞-groupoid</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a></li> </ul> <h2 id='1categorical_presentations'>1-categorical presentations</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a></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='#properties'>Properties</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><ins class='diffins'><ul><li><a href='#weak_4categories'>Weak 4-categories</a></li><li><a href='#strict_4categories'>Strict 4-categories</a></li></ul></ins></li></ul></div> <h2 id='idea'>Idea</h2> <p>Tetracategories are the <a class='existingWikiWord' href='/nlab/show/diff/algebraic+definition+of+higher+categories'>algebraic definition of higher category</a> for the most general (i.e. weak) <a class='existingWikiWord' href='/nlab/show/diff/n-category'>4-categories</a>.</p> <h2 id='properties'>Properties</h2> <p>By the discussion at <a class='existingWikiWord' href='/nlab/show/diff/k-tuply+monoidal+%28n%2Cr%29-category'>k-tuply monoidal (n,r)-category</a> a tetracategory with a single object may be regarded as the <a class='existingWikiWord' href='/nlab/show/diff/delooping'>delooping</a> of a monoidal <a class='existingWikiWord' href='/nlab/show/diff/tricategory'>tricategory</a>. This is discussed in (<a href='#Hoffnung'>Hoffnung</a>).</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category'>category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/bicategory'>bicategory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tricategory'>tricategory</a></p> </li> <li> <p><strong>tetracategory</strong></p> </li> </ul> <h2 id='references'>References</h2> <ins class='diffins'><h3 id='weak_4categories'>Weak 4-categories</h3></ins><ins class='diffins'> </ins><p>The structure of a tetracategories was given by <a class='existingWikiWord' href='/nlab/show/diff/Todd+Trimble'>Todd Trimble</a>.</p> <ul> <li> <p>Todd Trimble <a href='http://math.ucr.edu/home/baez/trimble/tetracategories.html'>Notes on Tetracategories</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/files/TetracategoricalData.pdf' title='Tetracategorical data'>Tetracategorical data</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/files/K6associativity.pdf' title='K6 associativity'>K6 associativity</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/files/U52unitcondition.pdf' title='U52 unit condition'>U52 unit condition</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/files/U53unitconditionA.pdf' title='U53 unit condition'>U53 unit condition</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/files/U54unitcondition.pdf' title='U54 unit condition'>U54 unit condition</a></p> </li> </ul> </li> </ul> <p>A polished writeup of the definition appears in section 3.2 of</p> <ul id='Hoffnung'> <li><a class='existingWikiWord' href='/nlab/show/diff/Alex+Hoffnung'>Alex Hoffnung</a>, <em>Spans in 2-Categories: A monoidal tricategory</em> (<a href='http://arxiv.org/abs/1112.0560'>arXiv:1112.0560</a>)</li> </ul><ins class='diffins'> </ins><ins class='diffins'><h3 id='strict_4categories'>Strict 4-categories</h3></ins><ins class='diffins'> </ins><ins class='diffins'><p>On <a class='existingWikiWord' href='/nlab/show/diff/strict+n-category'>strict</a> 4-categories</p></ins><ins class='diffins'> </ins><ins class='diffins'><p>via a <a class='existingWikiWord' href='/nlab/show/diff/string+diagram'>string diagram</a>-calculus:</p></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li>Manuel Araújo: <em>String diagrams for 4-categories and fibrations of mapping 4-groupoids</em>, Theory and Applications of Categories <strong>41</strong> 38 (2024) 1352-1398 [[arXiv:2012.03797](https://arxiv.org/abs/2012.03797), <a href='http://www.tac.mta.ca/tac/volumes/41/38/41-38abs.html'>tac:41-38</a>]</li> </ul></ins> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on September 26, 2024 at 11:19:50. 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