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href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="category_theory_2"><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>-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" 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='#definition'>Definition</a></li> <ul> <li><a href='#in_categories'>In categories</a></li> <ul> <li><a href='#traditional_definition'>Traditional definition</a></li> <li><a href='#CartInOrdCatReformulation'>Reformulations</a></li> </ul> <li><a href='#in_categories_2'>In <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</a></li> <ul> <li><a href='#in_quasicategories'>In quasi-categories</a></li> <li><a href='#in_categories_3'>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math>-categories</a></li> </ul> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#pullback_along_cartesian_morphisms'>Pullback along Cartesian morphisms</a></li> <li><a href='#cartesian_morphisms_and_equivalences'>Cartesian morphisms and equivalences</a></li> <ul> <li><a href='#observation'>Observation</a></li> </ul> </ul> <li><a href='#related_pages'>Related pages</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Given a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p\colon X \to Y</annotation></semantics></math> between categories one may ask for each <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msub><mi>y</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>y</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f\colon y_1 \to y_2</annotation></semantics></math> that, if given a lift of its target</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>X</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mover><mi>y</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mi>p</mi></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>y</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><msub><mi>y</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &&& && \hat y_2 \\ \downarrow^p \\ Y &&& y_1 &\stackrel{f}{\to}& y_2 } </annotation></semantics></math></div> <p>there be a <em>universal</em> lift 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></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>X</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mover><mi>y</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mover><mi>f</mi><mo stretchy="false">^</mo></mover></mover></mtd> <mtd><msub><mover><mi>y</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mi>p</mi></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>y</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><msub><mi>y</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &&& \hat y_1 &\stackrel{\hat f}{\to}& \hat y_2 \\ \downarrow^p \\ Y &&& y_1 &\stackrel{f}{\to}& y_2 } \,. </annotation></semantics></math></div> <p>There may also be other lifts 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>, but the universal one is essentially unique, as usual for anything having a <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a>. Specifically, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat f</annotation></semantics></math> in <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> is essentially uniquely determined by its target <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>y</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\hat y_2</annotation></semantics></math> and its image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mover><mi>f</mi><mo stretchy="false">^</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f = p(\hat f)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, and is called a <strong>cartesian morphism</strong>. A morphism which is cartesian relative to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mi>op</mi></msup><mo lspace="verythinmathspace">:</mo><msup><mi>X</mi> <mi>op</mi></msup><mo>→</mo><msup><mi>Y</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">p^{op}\colon X^{op}\to Y^{op}</annotation></semantics></math> is called <strong>opcartesian</strong> or <strong>cocartesian</strong>.</p> <p>If there are enough cartesian morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, they may be used to define <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><msub><mi>X</mi> <mrow><msub><mi>y</mi> <mn>2</mn></msub></mrow></msub><mo>→</mo><msub><mi>X</mi> <mrow><msub><mi>y</mi> <mn>1</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> f^* : X_{y_2} \to X_{y_1} </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>y</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">y_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>y</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">y_2</annotation></semantics></math>.</p> <p>This way a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> with enough Cartesian morphisms – called a <a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a> or <a class="existingWikiWord" href="/nlab/show/Grothendieck+fibration">Grothendieck fibration</a> – determines and is determined by a fiber-assigning functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><msup><mi>Cat</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Y \to Cat^{op}</annotation></semantics></math>.</p> <p>This has its analog in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher categories</a>.</p> <h2 id="definition">Definition</h2> <h3 id="in_categories">In categories</h3> <h4 id="traditional_definition">Traditional definition</h4> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p><strong>(cartesian morphism)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a>. A <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f : x_1 \to x_2</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/category">category</a> <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> is <strong>strongly cartesian</strong> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> (nowadays often just <em>cartesian</em>), or (strongly) <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-cartesian</strong> if for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>′</mo><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x'\in X</annotation></semantics></math>, for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>x</mi><mo>′</mo><mo>→</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">h:x'\to x_2</annotation></semantics></math> and every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>:</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo>′</mo><mo stretchy="false">)</mo><mo>→</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u:p(x')\to p(x_1)</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">p(h) = p(f) u</annotation></semantics></math>, there exists a unique <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>:</mo><mi>x</mi><mo>′</mo><mo>→</mo><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">v:x'\to x_1</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>=</mo><mi>f</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">h = f v</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">u = p(v)</annotation></semantics></math>:</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>∀</mo><mi>x</mi><mo>′</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∃</mo><mo>!</mo><mi>v</mi></mrow></mpadded></msup></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mo>∀</mo><mi>h</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>x</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><msub><mi>x</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mover><mo>↦</mo><mi>p</mi></mover><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo>′</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∀</mo><mi>u</mi></mrow></mpadded></msup></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \forall x' \\ \downarrow^{\mathrlap{\exists! v}} & \searrow^{\mathrlap{\forall h}} \\ x_1 &\stackrel{f}{\to}& x_2 } \;\;\; \;\;\; \stackrel{p}{\mapsto} \;\;\; \;\;\; \array{ p(x') \\ \downarrow^{\mathrlap{\forall u}} & \searrow^{\mathrlap{p(h)}} \\ p(x_1) &\stackrel{p(f)}{\to}& p(x_2) } </annotation></semantics></math></div> <p>In imprecise words: for all commuting triangles in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> (involving <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(f)</annotation></semantics></math> as above) and all lifts through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> of its 2-<a class="existingWikiWord" href="/nlab/show/horn">horn</a> to <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> (involving <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> as above), there is a unique refinement to a lift of the entire commuting triangle.</p> <p>We can make this definition slightly more explicit by working with the fibres of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>: let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">X_x</annotation></semantics></math> denote <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p^{-1}(y)</annotation></semantics></math>, the set of objects living over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>; and, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>y</mi><mo>→</mo><mi>y</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f : y \to y'</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><msub><mi>X</mi> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">x : X_y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>′</mo><mo>:</mo><msub><mi>X</mi> <mrow><mi>y</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex">x' : X_{y'}</annotation></semantics></math>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>f</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X_f(x,x')</annotation></semantics></math> denote <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>X</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p^{-1}(f) \cap X(x,x')</annotation></semantics></math>, the set of morphisms from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">x'</annotation></semantics></math> that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> maps to <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> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>:</mo><msub><mi>X</mi> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">x : X_y</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>′</mo><mo>:</mo><msub><mi>X</mi> <mrow><mi>y</mi><mo>′</mo></mrow></msub></mrow><annotation encoding="application/x-tex">x' : X_{y'}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>y</mi><mo>′</mo><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f : y' \to y</annotation></semantics></math>, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">¯</mo></mover><mo>:</mo><msub><mi>X</mi> <mi>f</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>′</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\bar f : X_f(x',x)</annotation></semantics></math> is <em>cartesian</em> iff for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>:</mo><mi>y</mi><mo>″</mo><mo>→</mo><mi>y</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">u : y'' \to y'</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>″</mo><mo>:</mo><msub><mi>X</mi> <mrow><mi>y</mi><mo>″</mo></mrow></msub></mrow><annotation encoding="application/x-tex">x'' : X_{y''}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><msub><mi>X</mi> <mrow><mi>f</mi><mo>∘</mo><mi>u</mi></mrow></msub><mo stretchy="false">(</mo><mi>x</mi><mo>″</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h : X_{f \circ u}(x'', x)</annotation></semantics></math>, there is a unique <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>u</mi><mo stretchy="false">¯</mo></mover><mo>:</mo><msub><mi>X</mi> <mi>u</mi></msub><mo stretchy="false">(</mo><mi>x</mi><mo>″</mo><mo>,</mo><mi>x</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\bar u : X_u(x'',x')</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>=</mo><mover><mi>f</mi><mo stretchy="false">¯</mo></mover><mo>∘</mo><mover><mi>u</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">h = \bar f \circ \bar u</annotation></semantics></math>. 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style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-dasharray:0.47818,1.99255;stroke-miterlimit:10;" d="M 82.304401 -8.650257 L 82.304401 -56.885322 " transform="matrix(0.998609,0,0,-0.998609,92.786649,77.78365)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 0.00155493 1.962138 L 0.00155493 -1.961289 " transform="matrix(0,-0.998609,-0.998609,0,174.975033,86.302334)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485878 2.868941 C -2.032122 1.147797 -1.018994 0.334165 -0.00195437 0.00167119 C -1.018994 -0.334734 -2.032122 -1.148366 -2.485878 -2.86951 " transform="matrix(0,0.998609,0.998609,0,174.974894,134.830077)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 15.449686 -65.780508 L 71.304725 -65.780508 " transform="matrix(0.998609,0,0,-0.998609,92.786649,77.78365)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485435 2.868211 C -2.031679 1.147067 -1.018551 0.333435 -0.00151147 0.000941607 C -1.018551 -0.335464 -2.031679 -1.149096 -2.485435 -2.87024 " transform="matrix(0.998609,0,0,-0.998609,164.231978,143.473597)"></path> <g style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#mZQpTsjfz-tS41yc5wOU5IqVT2I=-glyph3-3" x="133.254279" y="138.026734"></use> </g> </g> </svg> <p>If we pass to the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(Y)</annotation></semantics></math> of the categories, then in terms of diagrams in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> this means that the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f : x \to y</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-cartesian precisely if for all <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><mtr><mtd><msup><mi>Δ</mi> <mrow><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><msub><mi>Λ</mi> <mn>2</mn></msub><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>N</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>N</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Delta^{\{1,2\}} \\ \downarrow & \searrow^f \\ \Lambda_2[2] &\to& N(X) \\ \downarrow && \downarrow^{\mathrlap{p}} \\ \Delta[2] &\to& N(Y) } </annotation></semantics></math></div> <p>such that the last edge of the 2-<a class="existingWikiWord" href="/nlab/show/horn">horn</a> is the given edge <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>, a unique lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></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><msup><mi>Δ</mi> <mrow><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><msub><mi>Λ</mi> <mn>2</mn></msub><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>N</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mi>σ</mi></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>N</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Delta^{\{1,2\}} \\ \downarrow & \searrow^f \\ \Lambda_2[2] &\to& N(X) \\ \downarrow &{}^\sigma\nearrow& \downarrow^{\mathrlap{p}} \\ \Delta[2] &\to& N(Y) } </annotation></semantics></math></div> <p>exists.</p> </div> <p> <div class='num_remark' id='WeakCartesianMorphisms'> <h6>Remark</h6> <p><strong>(weak/local Cartesian morphisms)</strong> <br /> There is a weaker universal property, originally devised by Grothendieck and Gabriel, where one requires the above lifting property only for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>=</mo><msub><mi>id</mi> <mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">u = id_{p(x_1)}</annotation></semantics></math>. Morphisms satisfying this universal property have in recent years been called <em><a class="existingWikiWord" href="/nlab/show/locally+Cartesian+morphisms">locally Cartesian morphisms</a></em>, although historically they have been called simply <em>Cartesian</em>, or sometimes <em>weak Cartesian</em>. For the case of Grothendieck <a class="existingWikiWord" href="/nlab/show/fibered+categories">fibered categories</a> the notion of weak Cartesian morphisms already coincides with that of actual Cartesian morphisms.</p> <p>Equivalently, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><msup><mi>x</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup></mrow><annotation encoding="application/x-tex">f:x\to x^\prime</annotation></semantics></math> is called <em>locally <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-Cartesian</em> (relative to a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X\to Y</annotation></semantics></math>) if it is Cartesian with respect to the projection functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>pf</mi></msub><mo>:</mo><msub><mi>X</mi> <mi>pf</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mn>2</mn></mstyle></mrow><annotation encoding="application/x-tex">p_{pf}:X_{pf}\to \mathbf{2}</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>pf</mi></msub></mrow><annotation encoding="application/x-tex">X_{pf}</annotation></semantics></math> is the (homotopy) pullback of <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> along the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mi>pf</mi></msub><mo>:</mo><mstyle mathvariant="bold"><mn>2</mn></mstyle><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">c_{pf}:\mathbf{2}\to Y</annotation></semantics></math> classifying the arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>pf</mi><mo>:</mo><mi>px</mi><mo>→</mo><msup><mi>px</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup></mrow><annotation encoding="application/x-tex">pf:px\to px^\prime</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>.</p> </div> </p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p><strong>(Grothendieck fibration)</strong></p> <p>If for every morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> and every lift of its target there is at least one lift which has as its target the chosen one and is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-cartesian morphism in the strong sense, one says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a <strong>fibered category</strong> (also called <a class="existingWikiWord" href="/nlab/show/Grothendieck+fibration">Grothendieck fibration</a>). Equivalently,</p> <ul> <li>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/prefibered+category">prefibered category</a>) for every morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> and every lift of its target there is at least one lift through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> which has as its target the chosen one and is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-cartesian morphism in the weak sense</li> </ul> <p><em>and</em></p> <ul> <li>the composition of every two composable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-cartesian morphisms (in the weak sense) is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-cartesian morphism (in the weak sense).</li> </ul> </div> <h4 id="CartInOrdCatReformulation">Reformulations</h4> <p>We discuss equivalent reformulations of the above definition of Cartesian morphism that lend themselves better to generalization to <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>.</p> <p>For the following, we need this notation: let</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X/x_2</annotation></semantics></math> by the <a class="existingWikiWord" href="/nlab/show/overcategory">overcategory</a> of <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> over the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y/p(x_2)</annotation></semantics></math> the corresponding <a class="existingWikiWord" href="/nlab/show/overcategory">overcategory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(x_2)</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">X/f</annotation></semantics></math> the category whose objects</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Obj</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">/</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>a</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>x</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd></mtd> <mtd><msub><mi>x</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> Obj(X/f) = \left\{ \array{ && a \\ &\swarrow && \searrow \\ x_1 &&\stackrel{f}{\to}&& x_2 } \right\} </annotation></semantics></math></div> <p>are objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> of <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> equipped with morphisms to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math> such that the obvious triangle commutes, and whose morphisms are morphisms between these tip objects such that all diagrams in sight commute.</p> </li> <li> <p>similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y/p(f)</annotation></semantics></math>.</p> </li> </ul> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The condition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>Mor</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">f \in Mor X</annotation></semantics></math> is a Cartesian morphism with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y </annotation></semantics></math> is equivalent to the condition that the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>X</mi><mo stretchy="false">/</mo><mi>f</mi><mo>→</mo><mi>X</mi><mo stretchy="false">/</mo><mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow><msub><mo>×</mo> <mrow><mi>Y</mi><mo stretchy="false">/</mo><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mrow></msub><mi>Y</mi><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \phi : X/f \to X/{x_2} \times_{Y/{p(x_2)}} Y/p(f) </annotation></semantics></math></div> <p>into the (strict) <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of the obvious projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow><mo>→</mo><mi>Y</mi><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X/{x_2} \to Y/p(x_2)</annotation></semantics></math> along the projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y/p(f) \to Y/p(x_2)</annotation></semantics></math> induced by the commutativity of</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>X</mi><mo stretchy="false">/</mo><mi>f</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mi>Y</mi><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ϕ</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi><mo stretchy="false">/</mo><mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi><mo stretchy="false">/</mo><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X/f &\stackrel{\phi_2}{\to}& Y/p(f) \\ {}^{\mathllap{\phi_1}}\downarrow && \downarrow \\ X/{x_2} &\to& Y/{p(x_2)} } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/k-surjective+functor">surjective equivalence</a>, and this in turn is equivalent to it being an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of categories.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>It is immediate 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> being an isomorphism of categories is equivalent to the condition that <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 a Cartesian morphism. We discuss that just the condition 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 a surjective equivalence already implies that it is an isomorphism of categories.</p> <p>So assume now 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 a surjective equivalence.</p> <p>Notice that objects in the pullback category are compatible pairs</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><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>a</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>x</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd></mtd> <mtd><msub><mi>x</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>∈</mo><mi>X</mi><msub><mo stretchy="false">/</mo> <mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><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><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>∈</mo><mi>Y</mi><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \left( \array{ && a \\ &&& \searrow \\ x_1 &&\stackrel{f}{\to}&& x_2 } \right) \in X/_{x_2} \;\;\;\;\;\;\,,\;\;\;\;\;\; \left( \array{ && b \\ & \swarrow && \searrow \\ p(x_1) &&\stackrel{p(f)}{\to}&& p(x_2) } \right) \in Y/p(f) \right) \,. </annotation></semantics></math></div> <p>We have 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> being <em>surjective</em> on object means that every such pair is in the image of some object</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>a</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>x</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd></mtd> <mtd><msub><mi>x</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>∈</mo><msub><mi>X</mi> <mi>f</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \left( \array{ && a \\ &{}^{\mathllap{g}}\swarrow&& \searrow \\ x_1 &&\stackrel{f}{\to}&& x_2 } \right) \in X_{f} \,, </annotation></semantics></math></div> <p>and hence that every filler <em>exists</em> . Assume two such fillers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g'</annotation></semantics></math>. Then by the fact that an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> is a surjection (even an isomorphism) on corresponding hom-sets, it follows that there exists (even uniquely) a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">X/f</annotation></semantics></math> connecting them</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></mtd> <mtd></mtd> <mtd><mi>a</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msup><mo>↙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>h</mi></mpadded></msup></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>a</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>g</mi><mo>′</mo></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>x</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd></mtd> <mtd><msub><mi>x</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && a \\ &{}^{\mathllap{g}}\swarrow & \downarrow^{\mathrlap{h}} & \searrow \\ && a \\ \downarrow & {}^{\mathllap{g'}}\swarrow && \searrow & \downarrow \\ x_1 &&\stackrel{f}{\to}&& x_2 } </annotation></semantics></math></div> <p>such that this maps under <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> to the identity morphism in the pullback category. But in particular this maps to the morphism</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></mtd> <mtd></mtd> <mtd><mi>a</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>h</mi></mpadded></msup></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>a</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>x</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && a \\ && \downarrow^{\mathrlap{h}} & \searrow \\ && a \\ & && \searrow & \downarrow \\ &&&& x_2 } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow></mrow><annotation encoding="application/x-tex">X/{x_2}</annotation></semantics></math> and evidently is the identity there if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> is the identity. Hence this maps also to the identity in the pullback category if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> is the identity. So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> must be the identity. So if two lifts of an object through the surjective equivalence <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> exist, they must already be equal. Hence the surjective equivalence <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 even an isomorphism on objects and hence an isomorphism of categories.</p> </div> <h3 id="in_categories_2">In <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</h3> <p>The notion of cartesian morphism generalizes from <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>. The definition given above can be rephrased as a pullback relation between homsets, which we can take as an abstract definition</p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p><strong>(cartesian edge in 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>-category)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> be a functor of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a>. Then, a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>x</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">f : x \to x'</annotation></semantics></math> in <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> is <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-cartesian</strong> if and only if the commutative square induced by the distributivity of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation 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y="78.713"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#YxCLr74adyplMZ6NkUsVBPWRv-Q=-glyph-4-1" x="112.7235" y="78.713"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#YxCLr74adyplMZ6NkUsVBPWRv-Q=-glyph-3-2" x="116.83975" y="78.713"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#YxCLr74adyplMZ6NkUsVBPWRv-Q=-glyph-4-2" x="122.46225" y="78.713"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#YxCLr74adyplMZ6NkUsVBPWRv-Q=-glyph-3-3" x="126.5785" y="78.713"></use> <use xlink:href="#YxCLr74adyplMZ6NkUsVBPWRv-Q=-glyph-3-5" x="129.518471" y="78.713"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#YxCLr74adyplMZ6NkUsVBPWRv-Q=-glyph-4-1" x="134.84725" y="78.713"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#YxCLr74adyplMZ6NkUsVBPWRv-Q=-glyph-3-4" x="138.96475" y="78.713"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use 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href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a>. 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xlink:href="#V59gojjEC6Gv85OWh-2YuOPGmLA=-glyph-3-2" x="79.01875" y="77.689"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#V59gojjEC6Gv85OWh-2YuOPGmLA=-glyph-4-1" x="83.135" y="79.4465"></use> </g> </svg> </div> <p>is a pullback square.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>The definition of being <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-cartesian can be described as a pullback square of hom-functors. By the contravariant <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(-, x)</annotation></semantics></math> classifies the <a class="existingWikiWord" href="/nlab/show/right+fibration">right fibration</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mo stretchy="false">/</mo><mi>x</mi></mrow></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X_{/x} \to X</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi>p</mi><mo>−</mo><mo>,</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y(p-, p(x))</annotation></semantics></math> classifies the pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Y</mi> <mrow><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></msub><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y_{/p(x)} \to Y</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>; that is by the <a class="existingWikiWord" href="/nlab/show/comma+category">comma category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">↓</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">(p \downarrow p(x)) \to Y</annotation></semantics></math>. 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<path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 132.34525 -9.452531 L 132.34525 -54.354875 " transform="matrix(1, 0, 0, -1, 145.436, 84.692)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.486449 2.867867 C -2.033324 1.149117 -1.021605 0.336617 0.0018325 0.00068 C -1.021605 -0.335258 -2.033324 -1.147758 -2.486449 -2.870414 " transform="matrix(0, 1, 1, 0, 277.78057, 139.28723)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#VXVMTC9pEf7YBUl16pcFVs0GmNM=-glyph-1-3" x="281.295" y="118.012"></use> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -84.588344 -67.02675 L -1.693813 -67.02675 " transform="matrix(1, 0, 0, -1, 145.436, 84.692)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.486996 2.868777 C -2.033871 1.146121 -1.018246 0.333621 0.001285 0.00159 C -1.018246 -0.334348 -2.033871 -1.146848 -2.486996 -2.869504 " transform="matrix(1, 0, 0, -1, 143.98309, 151.72034)"></path> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#VXVMTC9pEf7YBUl16pcFVs0GmNM=-glyph-1-3" x="91.038" y="145.713"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#VXVMTC9pEf7YBUl16pcFVs0GmNM=-glyph-6-1" x="96.3655" y="145.713"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#VXVMTC9pEf7YBUl16pcFVs0GmNM=-glyph-1-4" x="100.483" y="145.713"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#VXVMTC9pEf7YBUl16pcFVs0GmNM=-glyph-6-2" x="106.6655" y="145.713"></use> </g> <g fill="rgb(0%, 0%, 0%)" fill-opacity="1"> <use xlink:href="#VXVMTC9pEf7YBUl16pcFVs0GmNM=-glyph-7-1" x="110.783" y="147.4705"></use> </g> <path fill="none" stroke-width="0.47818" stroke-linecap="butt" stroke-linejoin="miter" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M 49.435094 -67.02675 L 119.314 -67.02675 " transform="matrix(1, 0, 0, -1, 145.436, 84.692)"></path> <path fill="none" stroke-width="0.47818" stroke-linecap="round" stroke-linejoin="round" stroke="rgb(0%, 0%, 0%)" stroke-opacity="1" stroke-miterlimit="10" d="M -2.48716 2.868777 C -2.034035 1.146121 -1.01841 0.333621 0.00112125 0.00159 C -1.01841 -0.334348 -2.034035 -1.146848 -2.48716 -2.869504 " transform="matrix(1, 0, 0, -1, 264.98716, 151.72034)"></path> </svg> </div> <p>The bottom-right square and wide rectangle are pullbacks by construction. By the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>, the bottom-left square is a pullback, and thus the top-left square is a pullback if and only if the tall rectangle is.</p> <p>By the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a>, the top-left square is a pullback iff <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-cartesian, so the proposition follows.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f : x \to y</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mo stretchy="false">/</mo><mi>f</mi></mrow></msub><mo>→</mo><msub><mi>X</mi> <mrow><mo stretchy="false">/</mo><mi>x</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{/f} \to X_{/x}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a>, and composing its inverse with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mo stretchy="false">/</mo><mi>f</mi></mrow></msub><mo>→</mo><msub><mi>X</mi> <mrow><mo stretchy="false">/</mo><mi>x</mi></mrow></msub></mrow><annotation encoding="application/x-tex">X_{/f} \to X_{/x}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub><mo>:</mo><msub><mi>X</mi> <mrow><mo stretchy="false">/</mo><mi>x</mi></mrow></msub><mo>→</mo><msub><mi>X</mi> <mrow><mo stretchy="false">/</mo><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">f_! : X_{/x} \to X_{/y}</annotation></semantics></math>.</p> </div> <p>To make this concrete, we can also discuss adaptations of the abstract idea to two different models of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> theory: <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a> and <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+categories">sSet categories</a>.</p> <h4 id="in_quasicategories">In quasi-categories</h4> <p>We formulate a notion <strong>cartesian edge</strong> or <em>cartesian morphism</em> in a <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>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> relative to a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> of simplicial sets. In the case that these simplicial sets are <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-categories</a> – i.e. simplicial set incarnations of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-categories</a> – this yields a notion of cartesian morphisms in <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> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> be a morphism of <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>s. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f : x_1 \to x_2</annotation></semantics></math> be an edge in <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>, i.e. a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f : \Delta^1 \to X</annotation></semantics></math>.</p> <p>Recall the notion of <a class="existingWikiWord" href="/nlab/show/over+quasi-category">over quasi-category</a> obtained from the notion of <a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a>. Using this we obtain <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">X/f</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">/</mo><mrow><msub><mi>x</mi> <mn>2</mn></msub></mrow></mrow><annotation encoding="application/x-tex">X/{x_2}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S/p(f)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S/p(x_2)</annotation></semantics></math> in generalization of the categories considered in the above definition of cartesian morphisms in categories.</p> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p><strong>(cartesian edge in a simplicial set)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/inner+Kan+fibration">inner Kan fibration</a> of simplicial sets.</p> <p>Then a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f : x \to y</annotation></semantics></math> in <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> is <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-cartesian</strong> if the induced morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mo stretchy="false">/</mo><mi>f</mi></mrow></msub><mo>→</mo><msub><mi>X</mi> <mrow><mo stretchy="false">/</mo><mi>y</mi></mrow></msub><msub><mo>×</mo> <mrow><msub><mi>Y</mi> <mrow><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msub><msub><mi>Y</mi> <mrow><mo stretchy="false">/</mo><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> X_{/f} \to X_{/y} \times_{Y_{/p(y)}} Y_{/p(f)} </annotation></semantics></math></div> <p>into the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> is an acyclic <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>.</p> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, def 2.4.1.1</a>.</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>The morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f : x \to y</annotation></semantics></math> as above, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a>, is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-cartesian precisely if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \geq 2</annotation></semantics></math> and all right outer <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><mtr><mtd><msup><mi>Δ</mi> <mrow><mo stretchy="false">{</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><mi>Λ</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mi>n</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Delta^{\{n-1,n\}} \\ \downarrow & \searrow^f \\ \Lambda[n]_n &\to& X \\ \downarrow && \downarrow^{\mathrlap{p}} \\ \Delta[n] &\to& Y } </annotation></semantics></math></div> <p>(with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\Lambda[n]_n</annotation></semantics></math> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th <a class="existingWikiWord" href="/nlab/show/horn">horn</a> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>) such that the last edge of the horn is the given edge <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>, a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math></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><msup><mi>Δ</mi> <mrow><mo stretchy="false">{</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><mi>Λ</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mi>n</mi></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mi>σ</mi></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Delta^{\{n-1,n\}} \\ \downarrow & \searrow^f \\ \Lambda[n]_n &\to& X \\ \downarrow &{}^\sigma\nearrow& \downarrow^{\mathrlap{p}} \\ \Delta[n] &\to& Y } </annotation></semantics></math></div> <p>exists.</p> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT remark 2.4.1.4</a>.</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>This means that an <a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> with a collection of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-cartesian morphisms in <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> specified satisfies the same kind of condition as a <em><a class="existingWikiWord" href="/nlab/show/right+fibration">right fibration</a></em> , the only difference being that not <em>all</em> right outer horns inclusion are required to have lifts, but only those where the last edge of the horn maps to a cartesian morphism.</p> <p>In this sense a <a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a> is a generalization of a <a class="existingWikiWord" href="/nlab/show/right+fibration">right fibration</a>.</p> </div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a> of <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a> then a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f : x \to y</annotation></semantics></math> in <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> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-Cartesian precisely if for all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> in <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> the 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>Hom</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>Hom</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>Hom</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>y</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><msub><mi>Hom</mi> <mi>Y</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>Hom</mi> <mi>Y</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>Hom</mi> <mi>Y</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>p</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Hom_X(a,x) &\stackrel{Hom_X(a,f)}{\to}& Hom_X(a,y) \\ \downarrow && \downarrow \\ Hom_Y(p(a), p(x)) &\stackrel{Hom_Y(p(a), p(f))}{\to}& Hom_Y(p(a), p(y)) } </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-objects in a quasi-category</a> is a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> square (in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> equipped with its <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">standard model structure</a>).</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. 2.4.4.3</a>.</p> </div> <h4 id="in_categories_3">In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math>-categories</h4> <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">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+categories">simplicially enriched categories</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F : C \to D</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>.</p> <div class="num_defn"> <h6 id="definition_6">Definition</h6> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">f : x \to y \in C</annotation></semantics></math> is <strong><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>-cartesian</strong> if it is so under the <a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>:</mo><mi>sSet</mi><mi>Cat</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">N : sSet Cat \to sSet</annotation></semantics></math> in the sense of quasi-categories above, i.e. if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>f</mi></mrow></msub><mo>→</mo><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>y</mi></mrow></msub><msub><mo>×</mo> <mrow><mi>N</mi><mo stretchy="false">(</mo><mi>D</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msub><mi>N</mi><mo stretchy="false">(</mo><mi>D</mi><msub><mo stretchy="false">)</mo> <mrow><mo stretchy="false">/</mo><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> N(C)_{/f} \to N(C)_{/y} \times_{N(D)_{/F(y)}} N(D)_{/F(f)} </annotation></semantics></math></div> <p>is an acyclic <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>.</p> </div> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <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">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> are enriched in <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es and if <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 hom-wise a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, then</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>:</mo><mi>N</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>N</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(F) : N(C) \to N(D)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a>;</p> </li> <li> <p>a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">f :x \to y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N(F)</annotation></semantics></math>-cartesian morphism precisely if for all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> 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> the 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><mi>C</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>C</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>y</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>D</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>D</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ C(a,x) &\stackrel{C(a,f)}{\to}& C(a,y) \\ \downarrow && \downarrow \\ D(F(a), F(x)) &\stackrel{C(F(a), F(f))}{\to}& D(F(a), F(y)) } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a> square in <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> equipped with its <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">standard model structure</a>.</p> </li> </ul> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. 2.4.1.10</a>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="pullback_along_cartesian_morphisms">Pullback along Cartesian morphisms</h3> <div class="num_lemma"> <h6 id="lemma">Lemma</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>ℰ</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">p : \mathcal{E} \to \mathcal{C}</annotation></semantics></math> a functor, if in a 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><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>h</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><mover><mo>→</mo><mi>k</mi></mover></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\stackrel{f}{\to}& B \\ {}^{\mathllap{g}}\downarrow && \downarrow^{\mathrlap{h}} \\ C &\stackrel{k}{\to}& D } </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> the two vertical morphisms are vertical with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> (meaning that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Id</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(g) = Id_p(A)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Id</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(h) = Id(B)</annotation></semantics></math>) and if the two horizontal morphisms are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-Cartesian morphisms, then this square is a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> square.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>If</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>Q</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>h</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><mover><mo>→</mo><mi>k</mi></mover></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Q &\stackrel{}{\to}& B \\ \downarrow && \downarrow^{\mathrlap{h}} \\ C &\stackrel{k}{\to}& D } </annotation></semantics></math></div> <p>is another cone over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>D</mi><mo>←</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">C \to D \leftarrow B</annotation></semantics></math>, then its image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is</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>p</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ p(Q) \\ \downarrow & \searrow \\ p(C) &\stackrel{p(k)}{\to}& D } \,. </annotation></semantics></math></div> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(f) = p(k)</annotation></semantics></math>, another lift of the right horn of this is given by</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>Q</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Q \\ & \searrow \\ A &\stackrel{f}{\to}& B } </annotation></semantics></math></div> <p>which gives a unique filler <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">Q \to A</annotation></semantics></math> by the fact that <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 Cartesian.</p> <p>But this produces now two fillers – namely the original <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">Q \to C</annotation></semantics></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>→</mo><mi>A</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">Q \to A \to C</annotation></semantics></math> just obtained – of the horn</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>Q</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><mover><mo>→</mo><mi>k</mi></mover></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Q &\to& B \\ && \downarrow \\ C &\stackrel{k}{\to}& D } </annotation></semantics></math></div> <p>over</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>p</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ p(Q) \\ \downarrow & \searrow \\ p(C) &\stackrel{p(k)}{\to}& D } \,. </annotation></semantics></math></div> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is Cartesian, these two fillers must be equal. This means that the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">Q \to A</annotation></semantics></math> is a cone morphism and unique as such. Hence the original square is a pullback.</p> </div> <p>This appears as <a class="existingWikiWord" href="/nlab/show/Elephant">Elephant, lemma 1.3.3</a>.</p> <h3 id="cartesian_morphisms_and_equivalences">Cartesian morphisms and equivalences</h3> <div class="num_lemma"> <h6 id="observation">Observation</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/category">category</a>, a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> 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> is cartesian with respect to the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">C \to *</annotation></semantics></math> precisely if it is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <p>In particular all <a class="existingWikiWord" href="/nlab/show/identity">identity</a> morphisms are cartesian.</p> </div> <p>This is trivial to see. The analog statement holds also for <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a>, where it is rather more nontrivial and quite useful:</p> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a>, a morphism 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> is cartesian with respect to the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal</a> morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">C \to *</annotation></semantics></math> precisely if it is an <a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalence</a>.</p> <p>More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a>, a morphism <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> in <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> is an <a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalence</a> precisely if it is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-cartesian and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(f)</annotation></semantics></math> is an equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>The first statement is a proposition of <a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a>, slightly reformulated in the language of cartesian morphisms. It appears as <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop 1.2.4.3</a>. A proof appears below <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, corollary 2.1.2.2</a>.</p> <p>The second statement is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. 2.4.1.5</a>.</p> </div> <h2 id="related_pages">Related pages</h2> <ul> <li>A Cartesian morphism is the special case of a <a class="existingWikiWord" href="/nlab/show/strictly+final+lift">strictly final lift</a> of a structured <a class="existingWikiWord" href="/nlab/show/sink">sink</a> when the sink is a singleton.</li> </ul> <h2 id="references">References</h2> <p>Original reference:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, §VI.5 of: <em>Revêtements Étales et Groupe Fondamental - Séminaire de Géometrie Algébrique du Bois Marie 1960/61</em> (<a class="existingWikiWord" href="/nlab/show/SGA+1">SGA 1</a>) , LNM <strong>224</strong> Springer (1971) [updated version with comments by M. Raynaud: <a href="http://arxiv.org/abs/math/0206203">arxiv.0206203</a>]</li> </ul> <p>Review:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Angelo+Vistoli">Angelo Vistoli</a>, Def. 3.1 in: <em>Grothendieck topologies, fibered categories and descent theory</em>, in: <em><a class="existingWikiWord" href="/nlab/show/Fundamental+algebraic+geometry+--+Grothendieck%27s+FGA+explained">Fundamental algebraic geometry – Grothendieck's FGA explained</a></em>, Mathematical Surveys and Monographs <strong>123</strong>, Amer. Math. Soc. (2005) 1-104 [<a href="https://bookstore.ams.org/surv-123-s">ISBN:978-0-8218-4245-4</a>, <a href="http://arxiv.org/abs/math/0412512">math.AG/0412512</a>]</li> </ul> <p>For the 1-categorical case see for instance section B1.3 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Peter+Johnstone">Peter Johnstone</a>, <em><a class="existingWikiWord" href="/nlab/show/Sketches+of+an+Elephant">Sketches of an Elephant</a></em></li> </ul> <p>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 version is in section 2.4 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em></li> </ul> <p>See also the references at <a class="existingWikiWord" href="/nlab/show/Grothendieck+fibration">Grothendieck fibration</a>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on December 28, 2023 at 11:40:03. 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