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href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalences in</a>/<a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">of</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>-categories</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sub-quasi-category">sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">reflective localization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/fibrations+of+quasi-categories">fibrations</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+fibration">left/right fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivator">derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation to simplicial categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure for Kan complexes</a></li> </ul> </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='#incarnations'>Incarnations</a></li> <ul> <li><a href='#in_quasicategories'>In quasi-categories</a></li> <ul> <li><a href='#QuasiCatPastingLaw'>Pasting law</a></li> </ul> <li><a href='#in_model_categories'>In model categories</a></li> <li><a href='#in_derivators'>In derivators</a></li> </ul> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#fiber_sequence'>Fiber sequence</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#in_homotopy_type_theory'>In homotopy type theory</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>An <strong><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>-pullback</strong> is a <a class="existingWikiWord" href="/nlab/show/limit+in+an+%28%E2%88%9E%2C1%29-category">limit in an (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> over a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of the shape</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>a</mi><mo>→</mo><mi>c</mi><mo>←</mo><mi>b</mi><mo stretchy="false">}</mo><mo>→</mo><mi>𝒞</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \{a \to c \leftarrow b\} \to \mathcal{C} \,. </annotation></semantics></math></div> <p>In other words it is a <a class="existingWikiWord" href="/nlab/show/cone">cone</a></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><mi>A</mi><msub><mo>×</mo> <mi>C</mi></msub><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>≅</mo><mo>⇙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A \times_C B &\to& B \\ \downarrow &\cong\swArrow& \downarrow \\ A &\to& C } </annotation></semantics></math></div> <p>which is <a class="existingWikiWord" href="/nlab/show/universal+property">universal</a> among all such cones in the <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>-categorical sense.</p> <p>This is the analog in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a> of the notion of <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>.</p> <h2 id="incarnations">Incarnations</h2> <h3 id="in_quasicategories">In quasi-categories</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a>. Recall the notion of <a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">limit in a quasi-category</a>.</p> <p>The non-degenerate cells of the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Delta[1] \times \Delta[1]</annotation></semantics></math> obtained as the <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> of the simplicial 1-<a class="existingWikiWord" href="/nlab/show/simplex">simplex</a> with itself look like</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><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (0,0) &\to& (1,0) \\ \downarrow &\searrow& \downarrow \\ (0,1) &\to& (1,1) } </annotation></semantics></math></div> <p>A <strong>square</strong> in a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is an image of this in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, i.e. a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>:</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>C</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> s : \Delta[1] \times \Delta[1] \to C \,. </annotation></semantics></math></div> <p>The simplicial square <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mrow><mo>×</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\Delta[1]^{\times 2}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a>, as a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>, to the <a class="existingWikiWord" href="/nlab/show/join+of+simplicial+sets">join of simplicial sets</a> of a 2-<a class="existingWikiWord" href="/nlab/show/horn">horn</a> with the point:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>≃</mo><mo stretchy="false">{</mo><mi>v</mi><mo stretchy="false">}</mo><mo>⋆</mo><mi>Λ</mi><mo stretchy="false">[</mo><mn>2</mn><msub><mo stretchy="false">]</mo> <mn>2</mn></msub><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>v</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>2</mn></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \Delta[1] \times \Delta[1] \simeq \{v\} \star \Lambda[2]_2 = \left( \array{ v &\to& 1 \\ \downarrow &\searrow& \downarrow \\ 0 &\to& 2 } \right) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>≃</mo><mi>Λ</mi><mo stretchy="false">[</mo><mn>2</mn><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo>⋆</mo><mo stretchy="false">{</mo><mi>v</mi><mo stretchy="false">}</mo><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>1</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mn>2</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>v</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Delta[1] \times \Delta[1] \simeq \Lambda[2]_0 \star \{v\} = \left( \array{ 0&\to& 1 \\ \downarrow &\searrow& \downarrow \\ 2 &\to& v } \right) \,. </annotation></semantics></math></div> <p>If a square <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>≃</mo><mi>Λ</mi><mo stretchy="false">[</mo><mn>2</mn><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo>⋆</mo><mo stretchy="false">{</mo><mi>v</mi><mo stretchy="false">}</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\Delta[1] \times \Delta[1] \simeq \Lambda[2]_0 \star \{v\} \to C</annotation></semantics></math> exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>v</mi><mo stretchy="false">}</mo><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\{v\} \to C</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">quasi-categorical limit</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>Λ</mi><mo stretchy="false">[</mo><mn>2</mn><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">F : \Lambda[2]_0 \to C</annotation></semantics></math>, we say the limit</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>:</mo><mo>=</mo><munder><mi>lim</mi> <mo>←</mo></munder><mi>F</mi><mo>:</mo><mo>=</mo><mi>F</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>F</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></munder><mi>F</mi><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> v := \lim_\leftarrow F := F(1) \prod_{F(0)} F(2) </annotation></semantics></math></div> <p>is <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <strong>quasi-categorical pullback</strong> of the diagram <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>.</p> <h4 id="QuasiCatPastingLaw">Pasting law</h4> <p>We have the following quasi-categorical analog of the familiar <a href="http://ncatlab.org/nlab/show/pullback#Pasting">pasting law of pullbacks</a> in ordinary <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>:</p> <p>A <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> diagram of two squares is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>𝒸</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \sigma : \Delta[2] \times \Delta[1] \to \mathcal{c} \,. </annotation></semantics></math></div> <p>Schematically this looks like</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><mi>a</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>b</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>f</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ a &\to& b &\to& c \\ \downarrow && \downarrow && \downarrow \\ d &\to& e &\to& f } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> <div class="un_prop"> <h6 id="proposition">Proposition</h6> <p><strong>(pasting law for quasi-categorical pullbacks)</strong></p> <p>If the right square is a pullback diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, then the left square is precisely if the outer square is.</p> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, lemma 4.4.2.1</a></p> <div class="proof"> <h6 id="proof">Proof</h6> <p>Consider the diagram inclusions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>f</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>→</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>f</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>←</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( \array{ && && c \\ && && \downarrow \\ d &\to& e &\to& f } \right) \;\;\to\;\; \left( \array{ && b &\to& c \\ && \downarrow && \downarrow \\ d &\to& e &\to& f } \right) \;\;\leftarrow\;\; \left( \array{ && b \\ && \downarrow \\ d &\to& e } \right) </annotation></semantics></math></div> <p>and the induced diagram of <a class="existingWikiWord" href="/nlab/show/over+quasi-categories">over quasi-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mover><mo>←</mo><mi>ϕ</mi></mover><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mover><mo>→</mo><mi>ψ</mi></mover><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C}_{/\sigma(c,d,f)} \stackrel{\phi}{\leftarrow} \mathcal{C}_{/\sigma(b,c,d,e,f)} \stackrel{\psi}{\to} \mathcal{C}_{/\sigma(b,d,e)} \,. </annotation></semantics></math></div> <p>Notice that by definition of <a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">limit in a quasi-category</a> the quasi-categorical pullback <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mi>σ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sigma(c) \times_{\sigma(f)} \sigma(d)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/\sigma(c,d,f)}</annotation></semantics></math>, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><mi>σ</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\sigma(d) \times_{\sigma(e)} \sigma(b)</annotation></semantics></math> is the terminal object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/\sigma(b,d,e)}</annotation></semantics></math>.</p> <p>The strategy now is to show that both these morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math> are acyclic <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>s. That will imply that these terminal objects coincide as objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> <p>First notice that the inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>→</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>f</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( \array{ && b \\ && \downarrow \\ d &\to& e } \right) \;\; \to \;\; \left( \array{ && b &\to& c \\ && \downarrow && \downarrow \\ d &\to& e &\to& f } \right) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/left+anodyne+morphism">left anodyne morphism</a>, being the composite of <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a>s of left <a class="existingWikiWord" href="/nlab/show/horn">horn</a> inclusions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd> <mtd><mo>→</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>→</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd> <mtd></mtd> <mtd><mi>f</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>→</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd> <mtd></mtd> <mtd><mi>f</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>→</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>b</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>f</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \left( \array{ && b \\ && \downarrow \\ d &\to& e } \right) & \to \left( \array{ && b &\to& c \\ && \downarrow && \\ d &\to& e } \right) \\ & \to \left( \array{ && b &\to& c \\ && \downarrow && \downarrow \\ d &\to& e && f } \right) \\ & \to \left( \array{ && b &\to& c \\ && \downarrow &\searrow& \downarrow \\ d &\to& e && f } \right) \\ & \to \left( \array{ && b &\to& c \\ && \downarrow &\searrow& \downarrow \\ d &\to& e &\to& f } \right) \end{aligned} \,. </annotation></semantics></math></div> <p>We could also prove this by showing that this functor is <a class="existingWikiWord" href="/nlab/show/homotopy+final+functor">homotopy initial</a> using the characterization in terms of slice categories, and then invoking the theorem of <a class="existingWikiWord" href="/nlab/show/HTT">HTT</a> 4.1.1.3(4) which says (in dual form) that an inclusion of simplicial sets is homotopy initial if and only if it is left anodyne.</p> <p>One of the <a href="http://ncatlab.org/nlab/show/right%2Fleft+Kan+fibration#PropRightAnodyne">properties of left anodyne morphisms</a> is that restriction of <a class="existingWikiWord" href="/nlab/show/over+quasi-categories">over quasi-categories</a> along left anodyne morphisms produces an acyclic <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>. This shows the desired statement for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi></mrow><annotation encoding="application/x-tex">\psi</annotation></semantics></math>.</p> <p>To see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is also an acyclic fibration observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> can be factored as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mo>←</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mo>←</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathcal{C}_{/\sigma(c,d,f)} \leftarrow \mathcal{C}_{/\sigma(c,d,e,f)} \leftarrow \mathcal{C}_{/\sigma(b,c,d,e,f)} </annotation></semantics></math></div> <p>Observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mo>←</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/\sigma(c,d,e,f)}\leftarrow\mathcal{C}_{/\sigma(b,c,d,e,f)}</annotation></semantics></math> fits into a pullback diagram</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>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub></mtd> <mtd><mo>←</mo></mtd> <mtd><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub></mtd> <mtd><mo>←</mo></mtd> <mtd><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C}_{/\sigma(c,d,e,f)} & \leftarrow & \mathcal{C}_{/\sigma(b,c,d,e,f)} \\ \downarrow & & \downarrow \\ \mathcal{C}_{/\sigma(c,e,f)} & \leftarrow & \mathcal{C}_{/\sigma(b,c,e,f)} } </annotation></semantics></math></div> <p>and hence is an acyclic Kan fibration since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mo>←</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/\sigma(c,e,f)} \leftarrow \mathcal{C}_{/\sigma(b,c,e,f)}</annotation></semantics></math> is one, on account of the fact that the square</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><mi>σ</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>σ</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>σ</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>σ</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \sigma(b) & \to & \sigma(c) \\ \downarrow & & \downarrow \\ \sigma(e) & \to & \sigma(f) } </annotation></semantics></math></div> <p>is a pullback in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. Finally, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub><mo>←</mo><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>σ</mi><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_{/\sigma(c,d,f)} \leftarrow \mathcal{C}_{/\sigma(c,d,e,f)}</annotation></semantics></math> is a trivial fibration since</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><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>f</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd> <mtd><mo>→</mo></mtd> <mtd><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>c</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>d</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>e</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>f</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \left( \array{ & & c \\ & & \downarrow \\ d & \to & f } \right) & \to & \left( \array{ & & & & c \\ & & & & \downarrow \\ d & \to & e & \to & f } \right) } </annotation></semantics></math></div> <p>is left anodyne; clearly this is a pushout of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>d</mi><mo>→</mo><mi>e</mi><mo>→</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(d\to f)\to (d\to e\to f)</annotation></semantics></math> and so it suffices to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow></msup><mo>→</mo><msup><mi>Δ</mi> <mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\Delta^{\{0,2\}}\to \Delta^{\{0,1,2\}}</annotation></semantics></math> is left anodyne. But this map factors as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow></msup><mo>→</mo><msubsup><mi>Λ</mi> <mn>0</mn> <mn>2</mn></msubsup><mo>→</mo><msup><mi>Δ</mi> <mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\Delta^{\{0,2\}}\to \Lambda^2_0 \to \Delta^{\{0,1,2\}}</annotation></semantics></math> and clearly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow></msup><mo>→</mo><msubsup><mi>Λ</mi> <mn>0</mn> <mn>2</mn></msubsup></mrow><annotation encoding="application/x-tex">\Delta^{\{0,2\}}\to \Lambda^2_0</annotation></semantics></math> is left anodyne since it is a pushout of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Δ</mi> <mrow><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mrow></msup><mo>→</mo><msup><mi>Δ</mi> <mrow><mo stretchy="false">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">}</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\Delta^{\{0\}}\to \Delta^{\{0,1\}}</annotation></semantics></math>.</p> </div> <h3 id="in_model_categories">In model categories</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></li> </ul> <h3 id="in_derivators">In derivators</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/pullback+in+a+derivator">pullback in a derivator</a></li> </ul> <h2 id="examples">Examples</h2> <h3 id="fiber_sequence">Fiber sequence</h3> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>C</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">* \to C \in \mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed object</a>, then the <strong>fiber</strong> or <strong><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>-kernel</strong> of a morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">f : B \to C</annotation></semantics></math> is the <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>-pullback</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><mi>ker</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ ker(f) &\to& * \\ \downarrow && \downarrow \\ B &\stackrel{f}{\to}& C } \,. </annotation></semantics></math></div> <p>For more on this see <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>.</p> <h2 id="related_concepts">Related concepts</h2> <div> <p><strong>Notions of pullback:</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback">pullback</a>, <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> (<a class="existingWikiWord" href="/nlab/show/limit">limit</a> over a <a class="existingWikiWord" href="/nlab/show/cospan">cospan</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/wide+pullback">wide pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+pullback">lax pullback</a>, <a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a> (<a class="existingWikiWord" href="/nlab/show/lax+limit">lax limit</a> over a cospan)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a>, (<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-limit</a> over a cospan)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a>, <a class="existingWikiWord" href="/nlab/show/context+extension">context extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+bundle">pullback bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/contravariant+functor">contravariant functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+in+cohomology">pullback in cohomology</a>, <a class="existingWikiWord" href="/nlab/show/d-invariant">d-invariant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+of+a+distribution">pullback of a distribution</a></p> </li> </ul> </li> </ul> </div> <h2 id="references">References</h2> <h3 id="in_homotopy_type_theory">In homotopy type theory</h3> <p>A formalization of homotopy pullbacks in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> is <a class="existingWikiWord" href="/nlab/show/Coq">Coq</a>-coded in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Guillaume+Brunerie">Guillaume Brunerie</a>, <em><a href="https://github.com/guillaumebrunerie/HoTT/blob/master/Coq/Limits/Pullbacks.v">Hott/Coq/Limits/Pullbacks.v</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 2, 2020 at 17:34:02. 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