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href='/nlab/show/looping'>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><a class='existingWikiWord' href='/nlab/show/homotopy+hypothesis'>homotopy hypothesis</a>-theorem</li> <li><a class='existingWikiWord' href='/nlab/show/delooping+hypothesis'>delooping hypothesis</a>-theorem</li> <li><a class='existingWikiWord' href='/nlab/show/periodic+table'>periodic table</a></li> <li><a class='existingWikiWord' href='/nlab/show/stabilization+hypothesis'>stabilization hypothesis</a>-theorem</li> <li><a class='existingWikiWord' href='/michaelshulman/show/exactness+hypothesis' title='michaelshulman'>exactness hypothesis</a></li> <li><a class='existingWikiWord' href='/nlab/show/holographic+principle+of+higher+category+theory'>holographic principle</a></li> </ul> <h2 id='applications'>Applications</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/applications+of+%28higher%29+category+theory'>applications of (higher) category theory</a></li> <li><a class='existingWikiWord' href='/nlab/show/higher+category+theory+and+physics'>higher category theory and physics</a></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/infinity-category'>∞-category</a>/<a class='existingWikiWord' href='/nlab/show/infinity-category'>∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/%28infinity%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/%28infinity%2C2%29-category'>(∞,2)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/%28infinity%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/%28infinity%2C0%29-category'>(∞,0)-category</a>/<a class='existingWikiWord' href='/nlab/show/infinity-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/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/partial+order'>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/2-poset'>(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/vertical+categorification'>categorification</a>/<a class='existingWikiWord' href='/nlab/show/decategorification'>decategorification</a></li> <li><a class='existingWikiWord' href='/nlab/show/geometric+definition+of+higher+categories'>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+omega-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+categories'>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+omega-category'>strict ∞-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/Batanin+omega-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+infinity-category'>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+%28infinity%2C1%29-category'>symmetric monoidal (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/stable+%28infinity%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-infinity-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/%28infinity%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+%28infinity%2C1%29-functor'>(∞,1)-adjunction</a></li> <li><a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-Kan+extension'>(∞,1)-Kan extension</a> <ul> <li><a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-limit'>(∞,1)-limit</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/%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/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+omega-category'>strict ∞-category</a>, <a class='existingWikiWord' href='/nlab/show/strict+omega-groupoid'>strict ∞-groupoid</a></li> </ul> </li> <li><a class='existingWikiWord' href='/nlab/show/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/%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/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> <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='#model_structure'>Model structure</a></li><li><a href='#examples'>Examples</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>Weak complicial sets are <a class='existingWikiWord' href='/nlab/show/simplicial+set'>simplicial sets</a> with <a class='existingWikiWord' href='/nlab/show/stuff%2C+structure%2C+property'>extra structure</a> that are closely related to the <a class='existingWikiWord' href='/nlab/show/omega-nerve'>∞-nerve</a>s of weak <a class='existingWikiWord' href='/nlab/show/infinity-category'>∞-categories</a>.</p> <p>The goal of characterizing such nerves, without an <em>a priori</em> definition of “weak <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>-category” to start from, is called <a class='existingWikiWord' href='/nlab/show/simplicial+weak+omega-category'>simplicial weak ∞-category</a> theory. It is expected that the (nerves of) weak <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>-categories will be weak complicial sets satisfying an extra “saturation” condition ensuring that “every <a class='existingWikiWord' href='/nlab/show/equivalence'>equivalence</a> is <a class='existingWikiWord' href='/nlab/show/thin+element'>thin</a>.” General weak complicial sets can be regarded as “presentations” of weak <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>-categories.</p> <p>Weak complicial sets are a joint generalization of</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/strict+omega-category'>strict ∞-categories</a>;</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Kan+complex'>Kan complexes</a>;</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/quasi-category'>quasi-categories</a>.</p> </li> </ul> <h2 id='definition'>Definition</h2> <div class='num_defn'> <h6 id='definition_2'>Definition</h6> <p>Let</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Delta^k[n]</annotation></semantics></math> be the <a class='existingWikiWord' href='/nlab/show/stratified+simplicial+set'>stratified simplicial set</a> whose underlying simplicial set is the <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/simplex'>simplex</a> <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Delta[n]</annotation></semantics></math>, and whose marked cells are precisely those simplices <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>→</mo><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[r] \to [n]</annotation></semantics></math> that contain <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>k</mi><mo>,</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy='false'>}</mo><mo>∩</mo><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\{k-1, k, k+1\} \cap [n]</annotation></semantics></math>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Λ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Lambda^k[n]</annotation></semantics></math> be the stratified simplicial set whose underlying simplicial set is the <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/horn'>horn</a> of <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Delta[n]</annotation></semantics></math>, with marked cells those that are marked in <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Delta^k[n]</annotation></semantics></math>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Λ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>′</mo></mrow><annotation encoding='application/x-tex'>\Lambda^k[n]'</annotation></semantics></math> be obtained from <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Delta^k[n]</annotation></semantics></math> by making the <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>k</mi><mo>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(k-1)</annotation></semantics></math>st <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n-1)</annotation></semantics></math>-face and the <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(k+1)</annotation></semantics></math>st <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n-1)</annotation></semantics></math> face thin;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>″</mo></mrow><annotation encoding='application/x-tex'>\Delta^k[n]''</annotation></semantics></math> be obtained from <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Delta^k[n]</annotation></semantics></math> by making all <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n-1)</annotation></semantics></math>-faces thin.</p> </li> </ul> <p>An <strong>elementary anodyne extension</strong> in <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Strat</mi></mrow><annotation encoding='application/x-tex'>Strat</annotation></semantics></math>, the category <a class='existingWikiWord' href='/nlab/show/stratified+simplicial+set'>stratified simplicial sets</a> is</p> <ul> <li>a <strong>complicial horn extension</strong> <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Λ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mover><mo>↪</mo><mrow><msub><mo>⊂</mo> <mi>r</mi></msub></mrow></mover><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Lambda^k[n] \stackrel{\subset_r}{\hookrightarrow} \Delta^k[n]</annotation></semantics></math></li> </ul> <p>or</p> <ul> <li>a <strong>complicial thinness extension</strong> <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Λ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>′</mo><mover><mo>↪</mo><mrow><msub><mo>⊂</mo> <mi>e</mi></msub></mrow></mover><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>″</mo></mrow><annotation encoding='application/x-tex'>\Lambda^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''</annotation></semantics></math></li> </ul> <p>for <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mi>⋯</mi></mrow><annotation encoding='application/x-tex'>n = 1,2, \cdots</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo>∈</mo><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>k \in [n]</annotation></semantics></math>.</p> </div> <div class='num_defn'> <h6 id='definition_3'>Definition</h6> <p>A <a class='existingWikiWord' href='/nlab/show/stratified+simplicial+set'>stratified simplicial set</a> is a <strong>weak complicial set</strong> if it has the <a class='existingWikiWord' href='/nlab/show/lift'>right lifting property</a> with respect to all</p> <p><math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Λ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mover><mo>↪</mo><mrow><msub><mo>⊂</mo> <mi>r</mi></msub></mrow></mover><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Lambda^k[n] \stackrel{\subset_r}{\hookrightarrow} \Delta^k[n]</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Λ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>′</mo><mover><mo>↪</mo><mrow><msub><mo>⊂</mo> <mi>e</mi></msub></mrow></mover><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>″</mo></mrow><annotation encoding='application/x-tex'>\Lambda^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''</annotation></semantics></math></p> <p>A <a class='existingWikiWord' href='/nlab/show/complicial+set'>complicial set</a> is a weak complicial set in which such liftings are unique.</p> </div> <h2 id='model_structure'>Model structure</h2> <p>There is a <a class='existingWikiWord' href='/nlab/show/model+category'>model category</a> structure that presents the <a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category'>(infinity,1)-category</a> of weak complicial sets, hence that of weak <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>-categories. See</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/model+structure+for+weak+complicial+sets'>model structure for weak complicial sets</a>.</li> </ul> <h2 id='examples'>Examples</h2> <ul> <li> <p>For <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/strict+omega-category'>strict ∞-category</a> and <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>N(C)</annotation></semantics></math> its <a class='existingWikiWord' href='/nlab/show/oriental'>∞-nerve</a>, the <em>Roberts stratification</em> which regards each identity morphism as a thin cell makes <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>N(C)</annotation></semantics></math> a strict <a class='existingWikiWord' href='/nlab/show/complicial+set'>complicial set</a>, hence a weak complicial set. This example is not “saturated.”</p> </li> <li> <p>There is also the <a class='existingWikiWord' href='/nlab/show/stratified+simplicial+set'>stratification</a> of <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>N(C)</annotation></semantics></math> which regards each <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>-equivalence morphism as a thin cell. <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>N(C)</annotation></semantics></math> with this stratification is a weak complicial set (example 17 of <a href='http://arxiv.org/abs/math/0604414'>Ver06</a>). This should be the “saturation” of the previous example, and exhibits the inclusion of strict <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>-categories into weak ones.</p> </li> <li> <p>A simplicial set is a weak complicial set when equipped with its maximal <a class='existingWikiWord' href='/nlab/show/stratified+simplicial+set'>stratification</a> (every simplex of dimension <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>></mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>\gt 0</annotation></semantics></math> is thin) if and only if it is a <a class='existingWikiWord' href='/nlab/show/Kan+complex'>Kan complex</a>. This example is, of course, saturated, and is viewed as embedding <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>-groupoids into <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>-categories.</p> </li> <li> <p>A simplicial set is a <a class='existingWikiWord' href='/nlab/show/quasi-category'>quasi-category</a> if and only if it is a weak complicial set when equipped with the stratification in which every simplex of dimension <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>></mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>\gt 1</annotation></semantics></math> is thin, and only degenerate 1-simplices are thin. This example is not saturated; in its saturation the thin 1-simplices are the internal equivalences in a quasi-category (equivalently, those that become isomorphisms in its <a class='existingWikiWord' href='/nlab/show/homotopy+category'>homotopy category</a>). It presents the embedding of <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_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>-<a class='existingWikiWord' href='/nlab/show/%28infinity%2C1%29-category'>categories</a> into weak <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>-categories.</p> <p>Note that 1-simplex equivalences in a quasi-category are automatically preserved by simplicial maps between quasi-categories; this is why <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>QCat</mi></mrow><annotation encoding='application/x-tex'>QCat</annotation></semantics></math> can “correctly” be regarded as a full subcategory of <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math>. This is not true at higher levels; for instance not every simplicial map between nerves of strict <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>-categories necessarily preserves <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi></mrow><annotation encoding='application/x-tex'>\omega</annotation></semantics></math>-equivalence morphisms.</p> </li> </ul> <h2 id='references'>References</h2> <p>The definition of weak complicial sets is definition 14, page 9 of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Dominic+Verity'>Dominic Verity</a>, <em>Weak complicial sets Part I: Basic homotopy theory</em> (<a href='http://arxiv.org/abs/math/0604414'>arXiv</a>)</li> </ul> <p>Further developments are in</p> <ul> <li>Dominic Verity, <em>Weak complicial sets Part II: Nerves of complicial Gray-categories</em> (<a href='http://arxiv.org/abs/math/0604416'>arXiv</a>)</li> </ul> <p>A <a class='existingWikiWord' href='/nlab/show/model+category'>model category</a> structure on <a class='existingWikiWord' href='/nlab/show/stratified+simplicial+set'>stratified simplicial sets</a> modelling <a class='existingWikiWord' href='/nlab/show/%28infinity%2Cn%29-category'>$(\infty,n)$-categories</a> in the guise of <a class='existingWikiWord' href='/nlab/show/weak+complicial+set'>$n$-complicial sets</a>:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Viktoriya+Ozornova'>Viktoriya Ozornova</a>, <a class='existingWikiWord' href='/nlab/show/Martina+Rovelli'>Martina Rovelli</a>, <em>Model structures for (∞,n)-categories on (pre)stratified simplicial sets and prestratified simplicial spaces</em>, Algebr. Geom. Topol. <strong>20</strong> (2020) 1543-1600 [[arxiv:1809.10621](https://arxiv.org/abs/1809.10621), <a href='https://doi.org/10.2140/agt.2020.20.1543'>doi:10.2140/agt.2020.20.1543</a>]</li> </ul> <p>A <a class='existingWikiWord' href='/nlab/show/Quillen+adjunction'>Quillen adjunction</a> relating <a class='existingWikiWord' href='/nlab/show/weak+complicial+set'>$n$-complicial sets</a> to <a class='existingWikiWord' href='/nlab/show/n-fold+complete+Segal+space'>$n$-fold complete Segal spaces</a>:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Viktoriya+Ozornova'>Viktoriya Ozornova</a>, <a class='existingWikiWord' href='/nlab/show/Martina+Rovelli'>Martina Rovelli</a>, <em>A Quillen adjunction between globular and complicial approaches to <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,n)</annotation></semantics></math>-categories</em>, Advances in Mathematics <strong>421</strong> (2023) 108980 [[doi:10.1016/j.aim.2023.108980](https://doi.org/10.1016/j.aim.2023.108980), <a href='https://arxiv.org/abs/2206.02689'>arXiv:2206.02689</a>]</li> </ul> <p>Review:</p> <ul> <li id='Rovelli2023'><a class='existingWikiWord' href='/nlab/show/Martina+Rovelli'>Martina Rovelli</a>, <em><math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-Complicial sets as a model for <math class='maruku-mathml' display='inline' id='mathml_b32ae67993ef865281c4b2372144e972a7015e3b_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,n)</annotation></semantics></math>-categories</em>, <a href='Center+for+Quantum+and+Topological+Systems#RovelliMay2023'>talk at</a> <em><a class='existingWikiWord' href='/nlab/show/Center+for+Quantum+and+Topological+Systems'>CQTS</a></em> (2023) [[web](Center+for+Quantum+and+Topological+Systems#RovelliMay2023), video: <a href='https://www.youtube.com/watch?v=T9Bg1AdaKv8'>YT</a>]</li> </ul> <p> </p> <p> </p> </div>  <div class="revisedby"> <p> Revision on September 13, 2023 at 18:57:26 by <a href="/nlab/author/Paolo+Sarti" style="color: #005c19">Paolo Sarti</a> See the <a href="/nlab/history/weak+complicial+set" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="https://nforum.ncatlab.org/discussion/16437/#Item_2">Discuss</a><span class="backintime"><a href="/nlab/show/weak+complicial+set" accesskey="F" class="navlinkbackintime" id="to_next_revision" rel="nofollow">Next revision</a> (to current)</span><span class="backintime"><a href="/nlab/revision/weak+complicial+set/20" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a> (20 more)</span><a href="/nlab/show/weak+complicial+set" class="navlink" id="to_current_revision">Current version of page</a><a href="/nlab/revision/diff/weak+complicial+set/21" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/weak+complicial+set" accesskey="S" class="navlink" id="history" rel="nofollow">History (21 revisions)</a><a href="/nlab/rollback/weak+complicial+set?rev=21" class="navlink" id="rollback" rel="nofollow">Rollback</a> <a href="/nlab/revision/weak+complicial+set/21/cite" style="color: black">Cite</a> <a href="/nlab/source/weak+complicial+set/21" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>