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encoding="application/x-tex">(n+1,r+1)</annotation></semantics></math>-categories of <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-categories">(n,r)-categories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pos">Pos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Set">Set</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Rel">Rel</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ho%28Cat%29">Ho(Cat)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AccCat">AccCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PrCat">PrCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/LexCat">LexCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/MonCat">MonCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/VCat">VCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CatAdj">CatAdj</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Prof">Prof</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Operad">Operad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2Cat">2Cat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ModCat">ModCat</a>, <a class="existingWikiWord" href="/nlab/show/CombModCat">CombModCat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Cat">(∞,1)Cat</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Pr%28%E2%88%9E%2C1%29Cat">Pr(∞,1)Cat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Operad">(∞,1)Operad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29Cat">(∞,n)Cat</a></p> </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='#properties'>Properties</a></li> <ul> <li><a href='#as_an_topos'>As an <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>-topos</a></li> <li><a href='#limits_and_colimits_in_'>Limits and colimits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math></a></li> </ul> <li><a href='#subcategories'>Subcategories</a></li> <li><a href='#related_categories'>Related categories</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math></strong> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a>, i.e. of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-categories">(∞,0)-categories</a>. This is the archetypical <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>, the home of classical <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">Equivalently</a> this means all of the following:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> of the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">{}_k</annotation></semantics></math> of (<a class="existingWikiWord" href="/nlab/show/weakly+Hausdorff+topological+space">weakly Hausdorff</a>) <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+spaces">locally compact topological spaces</a> at the <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a>. As such it is the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a>-enhancement of the <a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">\tau_0(\infty Grpd) \simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a>, itself <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presented</a> by the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi><mo>≃</mo><msub><mi>L</mi> <mi>whe</mi></msub><msub><mi>Top</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\infty Grpd \simeq L_{whe} Top_k</annotation></semantics></math>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> of the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> at the simplicial <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a>. As such it is the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a>-enhancement of the <a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">)</mo><mo>≃</mo></mrow><annotation encoding="application/x-tex">\tau_0(\infty Grpd) \simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Ho%28sSet%29">Ho(sSet)</a>, itself <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presented</a> by the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi><mo>≃</mo><msub><mi>L</mi> <mi>whe</mi></msub><mi>sSet</mi></mrow><annotation encoding="application/x-tex">\infty Grpd \simeq L_{whe} sSet</annotation></semantics></math>.</p> <p>Hence, as a <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">Kan-complex enriched category</a> (a <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant object</a> in the <a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">model structure on sSet-categories</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full</a> <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">sSet enriched</a>-<a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> of <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> on the <a class="existingWikiWord" href="/nlab/show/Kan+complexes">Kan complexes</a>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/full+sub-%28%E2%88%9E%2C1%29-category">full sub-(∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Cat">(∞,1)Cat</a> on those <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a> that are <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a>.</p> </li> </ol> <h2 id="properties">Properties</h2> <h3 id="as_an_topos">As an <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>-topos</h3> <p>As an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> <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>-topos: for every other <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-sheaves">(∞,1)-sheaf</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> there is up to a <a class="existingWikiWord" href="/nlab/show/contractible">contractible</a> space of choices a unique <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>LConst</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><mo>→</mo><mo>←</mo></mover><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">(LConst \dashv \Gamma) : \mathbf{H}\stackrel{\leftarrow}{\to} \infty Grpd</annotation></semantics></math> – the <a class="existingWikiWord" href="/nlab/show/global+section">global section</a> geometric morphism. See there for more details.</p> <h3 id="limits_and_colimits_in_">Limits and colimits in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math></h3> <p><a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">Limits and colimits</a> over a <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math> may be reformulation in terms of the <a class="existingWikiWord" href="/nlab/show/universal+fibration+of+%28infinity%2C1%29-categories">universal fibration of (infinity,1)-categories</a>.</p> <p>Let the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><msub><mo stretchy="false">|</mo> <mi>Grpd</mi></msub><mo>→</mo><mn>∞</mn><msup><mi>Grpd</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Z|_{Grpd} \to \infty Grpd^{op}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/universal+fibration+of+%28infinity%2C1%29-categories">universal ∞-groupoid fibration</a> whose fiber over the object denoting some <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>-groupoid is that very <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>-groupoid.</p> <p>Then let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>X</mi><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> F : X \to \infty Grpd </annotation></semantics></math></div> <p>an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>. Recall that the <a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">coCartesian fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>F</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E_F \to X</annotation></semantics></math> classified by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is the pullback of the <a class="existingWikiWord" href="/nlab/show/universal+fibration+of+%28%E2%88%9E%2C1%29-categories">universal fibration of (∞,1)-categories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> along F:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>E</mi> <mi>F</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Z</mi><msub><mo stretchy="false">|</mo> <mi>Grpd</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>F</mi></mover></mtd> <mtd><mn>∞</mn><mi>Grpd</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ E_F &\to& Z|_{Grpd} \\ \downarrow && \downarrow \\ X &\stackrel{F}{\to}& \infty Grpd } </annotation></semantics></math></div> <div class="un_prop"> <h6 id="proposition">Proposition</h6> <p>Let the assumptions be as above. Then:</p> <ul> <li> <p>The colimit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">E_F</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>F</mi></msub><mo>≃</mo><mi>colim</mi><mi>F</mi></mrow><annotation encoding="application/x-tex"> E_F \simeq colim F </annotation></semantics></math></div></li> <li> <p>The limit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is equivalent to the <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+of+cartesian+section">(∞,1)-groupoid of sections</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>F</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">E_F \to X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>F</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>lim</mi><mi>F</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma_X(E_F) \simeq lim F \,. </annotation></semantics></math></div></li> </ul> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The statement for the colimit is corollary 3.3.4.6 in <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT</a>. The statement for the limit is corollary 3.3.3.4.</p> </div> <h2 id="subcategories">Subcategories</h2> <p>The <a class="existingWikiWord" href="/nlab/show/n-truncated+object+of+an+%28infinity%2C1%29-category">n-truncated objects</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/n-groupoids">n-groupoids</a> (including <a class="existingWikiWord" href="/nlab/show/%28-1%29-groupoid">(-1)-groupoid</a>s and the <a class="existingWikiWord" href="/nlab/show/%28-2%29-groupoid">(-2)-groupoid</a>).</p> <h2 id="related_categories">Related categories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Set">Set</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a>,</p> <p><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd</annotation></semantics></math></strong></p> <p><a class="existingWikiWord" href="/nlab/show/Top">Top</a>, <a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Cat">(∞,1)Cat</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29Cat">(∞,n)Cat</a></p> </li> </ul> <div class="property">category: <a class="category_link" href="/nlab/all_pages/category">category</a></div></body></html> </div> <div class="revisedby"> <p> Last revised on August 3, 2020 at 06:56:58. 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