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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="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></li> <ul> <li><a href='#coherence_theorem'>Coherence theorem</a></li> <li><a href='#canonical_model_structure'>Canonical model structure</a></li> </ul> <li><a href='#groupoids'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gray</mi></mrow><annotation encoding="application/x-tex">Gray</annotation></semantics></math>-groupoids</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_pages'>Related pages</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gray</mi></mrow><annotation encoding="application/x-tex">Gray</annotation></semantics></math>-category</em> (or <em>Gray category</em>) is a certain type of <a class="existingWikiWord" href="/nlab/show/semi-strict+infinity-category">semi-strict</a> <a class="existingWikiWord" href="/nlab/show/3-category">3-category</a>, in which <a class="existingWikiWord" href="/nlab/show/composition">composition</a> is strictly <a class="existingWikiWord" href="/nlab/show/associativity">associative</a> and <a class="existingWikiWord" href="/nlab/show/unitality">unital</a>, but the <a class="existingWikiWord" href="/nlab/show/interchange+law">interchange law</a> holds only up to <a class="existingWikiWord" href="/nlab/show/coherence">coherent</a> <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <h2 id="definition">Definition</h2> <p>A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gray</mi></mrow><annotation encoding="application/x-tex">Gray</annotation></semantics></math>-category</strong> is a category <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> over the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gray</mi></mrow><annotation encoding="application/x-tex">Gray</annotation></semantics></math>, which is the category of <a class="existingWikiWord" href="/nlab/show/strict+2-category">2-categories</a> and <a class="existingWikiWord" href="/nlab/show/strict+2-functors">strict 2-functors</a> with the <a class="existingWikiWord" href="/nlab/show/Gray+tensor+product">Gray tensor product</a>.</p> <h2 id="properties">Properties</h2> <h3 id="coherence_theorem">Coherence theorem</h3> <p>Gordon, Power, and Street proved that every <a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a> (that is, weak 3-category) is equivalent to a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gray</mi></mrow><annotation encoding="application/x-tex">Gray</annotation></semantics></math>-category. Not every tricategory is equivalent to a fully <a class="existingWikiWord" href="/nlab/show/strict+3-category">strict 3-category</a>; any doubly-degenerate <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> which is not <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a> is an example. So this is “the best one can do” in one sense, although there are other incomparable paths one can take, such as <a class="existingWikiWord" href="/nlab/show/Simpson%27s+conjecture">weakening units</a> but keeping interchange strict.</p> <p>The inclusion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gray</mi></mrow><annotation encoding="application/x-tex">Gray</annotation></semantics></math>-categories into tricategories is not uniquely determined – there is a left and right-hand version (from a remark in Example 9.3.9 of Leinster’s book cited below). However, the two possible ways are canonically equivalent as tricategories.</p> <h3 id="canonical_model_structure">Canonical model structure</h3> <p>Gray-categories support a <a class="existingWikiWord" href="/nlab/show/canonical+model+structure">canonical model structure</a> (<a href="#Lack">Lack</a>)</p> <h2 id="groupoids"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gray</mi></mrow><annotation encoding="application/x-tex">Gray</annotation></semantics></math>-groupoids</h2> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gray</mi></mrow><annotation encoding="application/x-tex">Gray</annotation></semantics></math>-category that is a <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a> is a <a class="existingWikiWord" href="/nlab/show/Gray-groupoid">Gray-groupoid</a>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>The prototypical Gray-category is <a class="existingWikiWord" href="/nlab/show/Gray">Gray</a>, which consists of <a class="existingWikiWord" href="/nlab/show/strict+2-categories">strict 2-categories</a>, strict 2-functors, pseudonatural transformations, and modifications.</p> </li> <li> <p>A Gray-category with one object is called a Gray-monoid, and is a semi-strict version of a <a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a>.</p> </li> <li> <p>A doubly-degenerate Gray-category is the same as a category with two <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal structures</a> satisfying an <a class="existingWikiWord" href="/nlab/show/exchange+law">exchange law</a>. This is essentially the same as a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a>. (<a href="#GurskiCheng">Gurski & Cheng</a>)</p> </li> <li> <p>Since any tricategory is equivalent to a Gray-category, one can obtain examples of Gray-categories in this way. For example, the tricategory <a class="existingWikiWord" href="/nlab/show/Bicat">Bicat</a> of bicategories, pseudofunctors, pseudonatural transformations, and modifications is equivalent to some Gray-category.</p> <p>It is important to note that Bicat is <em>not</em> equivalent to Gray, due to the absence of pseudofunctors in the latter. It is equivalent to the sub-Gray-category of Gray determined by the “flexible” or “cofibrant” 2-categories, however, since between such 2-categories any pseudofunctor is equivalent to a strict one.</p> </li> <li> <p>Since pseudofunctors between strict 2-categories compose strictly associatively, and between any 2-categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> there is a strict 2-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ps</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ps(A,B)</annotation></semantics></math> of pseudofunctors, pseudonatural transformations, and modifications, one might hope that there is a Gray-category consisting of strict 2-categories, <em>pseudofunctors</em>, pseudonatural transformations, and modifications, despite the fact that the prototypical example <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Gray</mi></mrow><annotation encoding="application/x-tex">Gray</annotation></semantics></math> contains only strict 2-functors. However, this is false, because in a Gray-category the <a class="existingWikiWord" href="/nlab/show/whiskering">whiskering</a> of 2-cells by a 1-cell is strictly functorial relative to composition of 2-cells along 1-cells, but this fails for whiskering of pseudonaturals by a pseudofunctor.</p> </li> </ul> <h2 id="related_pages">Related pages</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/intercategory">intercategory</a></li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Robert+Gordon">Robert Gordon</a>, <a class="existingWikiWord" href="/nlab/show/John+Power">John Power</a>, <a class="existingWikiWord" href="/nlab/show/Ross+Street">Ross Street</a>, <em>Coherence for tricategories</em>, Memoirs of the American Mathematical Society, <strong>117</strong>, 1995. (<a href="http://dx.doi.org/10.1090/memo/0558">doi:10.1090/memo/0558</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Nick+Gurski">Nick Gurski</a>, <em>An algebraic theory of tricategories</em>, PhD thesis (2007) [<a href="https://dokumen.tips/documents/an-algebraic-theory-of-tricategories.html">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Gurski-AlgebraicTricategories.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tom+Leinster">Tom Leinster</a>, <em>Higher operads, higher categories</em>, Cambridge University Press, 2004. (<a href="https://arxiv.org/abs/math/0305049">arXiv:math/0305049</a>,</p> <p><a href="https://doi.org/10.1017/CBO9780511525896">doi:10.1017/CBO9780511525896</a>)</p> </li> <li id="Lack"> <p><a class="existingWikiWord" href="/nlab/show/Steve+Lack">Steve Lack</a>, <em>A Quillen model structure for Gray-categories</em>, Journal of K-Theory, <strong>8</strong>, 2011. (<a href="http://arxiv.org/abs/1001.2366">arxiv:1001.2366</a>, <a href="https://doi.org/10.1017/is010008014jkt127">doi:10.1017/is010008014jkt127</a>)</p> </li> <li id="GurskiCheng"> <p><a class="existingWikiWord" href="/nlab/show/Nick+Gurski">Nick Gurski</a>, <a class="existingWikiWord" href="/nlab/show/Eugenia+Cheng">Eugenia Cheng</a>, <em>The periodic table of n-categories II: degenerate tricategories</em>, Cahiers de Topologie et Géométrie Différentielle Catégoriques, <strong>52</strong>, 2011. (<a href="http://www.numdam.org/item/?id=CTGDC_2011__52_2_82_0">link</a>, <a href="https://arxiv.org/abs/0706.2307">arxiv:0706.2307</a>)</p> </li> <li> <p>Peter Guthmann, <em>The tricategory of formal composites and its strictification</em>, <a href="https://arxiv.org/abs/1903.05777">arXiv:1903.05777</a></p> </li> <li id="DiVittorio2023"> <p><a class="existingWikiWord" href="/nlab/show/Nicola+Di+Vittorio">Nicola Di Vittorio</a>, <em>A Gray-categorical pasting theorem</em>, Theory and Applications of Categories <strong>39</strong> 5 (2023) 150-171 [<a href="http://www.tac.mta.ca/tac/volumes/39/5/39-05abs.html">tac:39-05</a>]</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 16, 2023 at 23:58:38. 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