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href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalences in</a>/<a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">of</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sub-quasi-category">sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">reflective localization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/fibrations+of+quasi-categories">fibrations</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+fibration">left/right fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivator">derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation to simplicial categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure for Kan complexes</a></li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#for_categories'>For <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> <li><a href='#RestInfGrpd'>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Groupoids</a></li> </ul> <li><a href='#models'>Models</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The <em>universal fibration of <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(∞,1)-categories</a> is the <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">generalized universal bundle</a> of <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 in that it is <a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>Cat</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> p \colon Z \to (\infty,1)Cat^{op} </annotation></semantics></math></div> <p>over the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-categories">(∞,1)-category of (∞,1)-categories</a> such that</p> <ul> <li> <p>its fiber <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>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p^{-1}(C)</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">C \in (\infty,1)Cat</annotation></semantics></math> is just 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>-category <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> itself;</p> </li> <li> <p>every <a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">p : C \to D</annotation></semantics></math> arises as the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of the universal fibration along an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>p</mi></msub><mo>:</mo><mi>D</mi><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>Cat</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">S_p : D \to (\infty,1)Cat^{op}</annotation></semantics></math>.</p> </li> </ul> <p>Recall from the discussion at <a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">generalized universal bundle</a> and at <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">stuff, structure, property</a> that for <a class="existingWikiWord" href="/nlab/show/n-category">n-categories</a> at least for low <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> the corresponding universal object was 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>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><msub><mi>Cat</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">n Cat_*</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> <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>-categories. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> should at least morally be <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><msub><mi>Cat</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">(\infty,1)Cat_*</annotation></semantics></math>.</p> <h2 id="definition">Definition</h2> <h3 id="for_categories">For <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>One description of the universal cartesian fibration is given in section 3.3.2 of <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT</a> as the contravariant <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a> applied to the identity functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>Cat</mi> <mi>op</mi></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">((\infty,1)Cat^{op})^{op} \to (\infty,1)Cat</annotation></semantics></math>.</p> <p>We can also give a more direct description:</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Z</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Z^{op}</annotation></semantics></math> is equivalent to the full subcategory of <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><msup><mi>Cat</mi> <mrow><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">(\infty,1)Cat^{[1]}</annotation></semantics></math> spanned by the morphisms of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>C</mi> <mrow><mi>x</mi><mo stretchy="false">/</mo></mrow></msub><mo>→</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C_{x/} \to C]</annotation></semantics></math> for small (∞,1)-categories <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 objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">x \in C</annotation></semantics></math>.</p> <p>The universal fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>Cat</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Z \to (\infty,1)Cat^{op}</annotation></semantics></math> is opposite to the target evaluation.</p> <p>Dually, the universal cocartesian fibration is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>′</mo><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Z' \to (\infty,1)Cat</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">Z'</annotation></semantics></math> is the (∞,1)-category of arrows of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>x</mi></mrow></msub><mo>→</mo><mi>C</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C_{/x} \to C]</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>This is the proof idea of <a href="#direct">this mathoverflow post</a>.</p> <p>By proposition 5.5.3.3 of Higher Topos Theory, there are presentable fibrations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RFib</mi><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">RFib \to (\infty,1)Cat</annotation></semantics></math> and <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><msup><mi>Cat</mi> <mrow><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup><mover><mo>→</mo><mi>tgt</mi></mover><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">(\infty,1)Cat^{[1]} \xrightarrow{tgt} (\infty,1)Cat</annotation></semantics></math> classifying functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>↦</mo><mi>𝒫</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C \mapsto \mathcal{P}(C)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>↦</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>Cat</mi> <mrow><mo stretchy="false">/</mo><mi>C</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C \mapsto (\infty,1)Cat_{/C}</annotation></semantics></math>.</p> <p>By proposition 5.3.6.2 of Higher Topos Theory, the yoneda embedding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>𝒫</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j : C \to \mathcal{P}(C)</annotation></semantics></math> is a natural transformation, and the covariant Grothendieck construction provides a cocartesian functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>′</mo><mo>→</mo><mi>RFib</mi></mrow><annotation encoding="application/x-tex">Z' \to RFib</annotation></semantics></math>. Since it is fiberwise fully faithful and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">(-)_!</annotation></semantics></math> preserves representable presheaves, we can identify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">Z'</annotation></semantics></math> with the full subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RFib</mi></mrow><annotation encoding="application/x-tex">RFib</annotation></semantics></math> consisting of the representable presheaves.</p> <p>The Grothendieck construction provides a fully faithful <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>Cat</mi> <mrow><mo stretchy="false">/</mo><mi>C</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{P}(C) \to (\infty,1)Cat_{/C}</annotation></semantics></math> whose essential image is the right fibrations. The contravariant Grothendieck construction a cartesian functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RFib</mi><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>Cat</mi> <mrow><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">RFib \to (\infty,1)Cat^{[1]}</annotation></semantics></math>. Since it is fiberwise fully faithful and pullbacks preserve right fibrations, we can identify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>RFib</mi></mrow><annotation encoding="application/x-tex">RFib</annotation></semantics></math> with the full subcategory of <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><msup><mi>Cat</mi> <mrow><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">( \infty,1)Cat^{[1]}</annotation></semantics></math> spanned by right fibrations.</p> <p>By the relationship between the covariant and contravariant Grothendieck constructions, the universal cartesian fibration is classified by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>op</mi><mo>:</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>Cat</mi> <mi>op</mi></msup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">op : ((\infty,1)Cat^{op})^{op} \to (\infty,1)Cat</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>A key point of this description is that for any small (∞,1)-category <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 functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><msub><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>x</mi></mrow></msub></mrow><annotation encoding="application/x-tex">x \mapsto C_{/x}</annotation></semantics></math> (where <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> acts by composition) is a fully faithful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>Cat</mi> <mrow><mo stretchy="false">/</mo><mi>C</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C \to (\infty,1)Cat_{/C}</annotation></semantics></math>. Dually, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>↦</mo><msub><mi>C</mi> <mrow><mi>x</mi><mo stretchy="false">/</mo></mrow></msub></mrow><annotation encoding="application/x-tex">x \mapsto C_{x/}</annotation></semantics></math> is a fully faithful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mi>Cat</mi> <mrow><mo stretchy="false">/</mo><mi>C</mi></mrow></msub></mrow><annotation encoding="application/x-tex">C^{op} \to (\infty,1)Cat_{/C}</annotation></semantics></math></p> </div> <p>The hom-spaces of the universal cocartesian fibration can be described as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>′</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>C</mi> <mrow><mo stretchy="false">/</mo><mi>x</mi></mrow></msub><mo>→</mo><mi>C</mi><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><msub><mi>D</mi> <mrow><mo stretchy="false">/</mo><mi>y</mi></mrow></msub><mo>→</mo><mi>D</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Core</mi><mo stretchy="false">(</mo><msub><mi>eval</mi> <mi>x</mi></msub><mo stretchy="false">↓</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Z'([C_{/x} \to C], [D_{/y} \to D]) \simeq Core(eval_x \downarrow y) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>eval</mi> <mi>x</mi></msub><mo>:</mo><msup><mi>D</mi> <mi>C</mi></msup><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">eval_x : D^C \to D</annotation></semantics></math>. This should be compared with the lax <a class="existingWikiWord" href="/nlab/show/slice+2-category">slice 2-category</a> construction. In fact, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">Z'</annotation></semantics></math> can be constructed by taking the underlying (∞,1) category of the lax coslice (or colax, depending on convention) over the point of the (∞,2)-category of (∞,1)-categories.</p> <h3 id="RestInfGrpd">For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Groupoids</h3> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>The universal fibration of <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 restricts to a <a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><msub><mo stretchy="false">|</mo> <mrow><mn>∞</mn><mi>Grpd</mi></mrow></msub><mo>→</mo><mn>∞</mn><msup><mi>Grpd</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Z|_{\infty Grpd} \to \infty Grpd^{op}</annotation></semantics></math> over <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> by <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> along the inclusion morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi><mo>↪</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">\infty Grpd \hookrightarrow (\infty,1)Cat</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>Z</mi><msub><mo stretchy="false">|</mo> <mrow><mn>∞</mn><mi>Grpd</mi></mrow></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Z</mi></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><mn>∞</mn><msup><mi>Grpd</mi> <mi>op</mi></msup></mtd> <mtd><mo>↪</mo></mtd> <mtd><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><msup><mi>Cat</mi> <mi>op</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Z|_{\infty Grpd} &\longrightarrow& Z \\ \big\downarrow && \big\downarrow \\ \infty Grpd^{op} &\hookrightarrow& (\infty,1)Cat^{op} } \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>The <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-functor">∞-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><msub><mo stretchy="false">|</mo> <mrow><mn>∞</mn><mi>Grpd</mi></mrow></msub><mo>→</mo><mn>∞</mn><msup><mi>Grpd</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Z|_{\infty Grpd} \to \infty Grpd^{op}</annotation></semantics></math> is even a <a class="existingWikiWord" href="/nlab/show/right+fibration">right fibration</a> and it is the <em>universal right fibration</em>. In fact it is (when restricted to small objects) the <a class="existingWikiWord" href="/nlab/show/object+classifier">object classifier</a> in the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a>, see at <a href="%28sub%29object+classifier+in+an+%28infinity%2C1%29-topos#ObjectClassifierInInfinityGroupoid">object classifier – In ∞Grpd</a>.</p> </div> <div class="num_prop" id="UniversalLeftFibration"> <h6 id="proposition_2">Proposition</h6> <p>The universal left fibration is the forgetful functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><msub><mi>Grpd</mi> <mo>*</mo></msub><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\infty Grpd_* \to \infty Grpd</annotation></semantics></math>. Its opposite is the universal right fibration.</p> </div> <p>(<a href="#Lurie09">Lurie 2009, Prop, 3.3.2.7</a>, <a href="#Cisinski19">Cisinski 2019, Sec. 5.2</a>, for the further restriction to the <a class="existingWikiWord" href="/nlab/show/universal+Kan+fibration">universal Kan fibration</a> see also <a href="#KapulkinLumsdaine12">Kapulkin & Lumsdaine 2021</a>)</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>The following are equivalent:</p> <ul> <li> <p>An <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-functor">∞-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">p : C \to D</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/right+Kan+fibration">right Kan fibration</a>.</p> </li> <li> <p>Every functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>p</mi></msub><mo>:</mo><mi>D</mi><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">S_p : D \to (\infty,1)Cat</annotation></semantics></math> that classifies <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> as a <a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a> factors through <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Grpd">∞-Grpd</a>.</p> </li> <li> <p>There is a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>p</mi></msub><mo>:</mo><mi>D</mi><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">G_p : D \to \infty Grpd</annotation></semantics></math> that classifies <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> as a <a class="existingWikiWord" href="/nlab/show/right+Kan+fibration">right Kan fibration</a>.</p> </li> </ul> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>This is proposition 3.3.2.5 in <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT</a>.</p> </div> <h2 id="models">Models</h2> <p>For concretely constructing the relation between <a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a>s <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 : E \to C</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-categories">(∞,1)-categories</a> and <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>p</mi></msub><mo>:</mo><mi>C</mi><mo>→</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">F_p : C \to (\infty,1)Cat</annotation></semantics></math> one may use a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between suitable <a class="existingWikiWord" href="/nlab/show/model+category">model categories</a> of <a class="existingWikiWord" href="/nlab/show/marked+simplicial+set">marked simplicial set</a>s.</p> <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> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> regarded as a <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> (i.e. as a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> with certain properties), the two model categories in question are</p> <ul> <li> <p>the projective <a class="existingWikiWord" href="/nlab/show/global+model+structure+on+simplicial+presheaves">global model structure on simplicial presheaves</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>SSet</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[C,SSet]</annotation></semantics></math> – this models the <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> <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><mi>Func</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)Func(C,(\infty,1)Cat)</annotation></semantics></math>.</p> </li> <li> <p>the <em>covariant model structure</em> on the <a class="existingWikiWord" href="/nlab/show/over+category">over category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SSet</mi><mo stretchy="false">/</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">SSet/C</annotation></semantics></math> – this models 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>-category of <a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a>s over <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>.</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between these is established by the <span class="newWikiWord">relative nerve<a href="/nlab/new/relative+nerve">?</a></span> construction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mo lspace="verythinmathspace" rspace="0em">−</mo></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>SSet</mi><mo stretchy="false">]</mo><mo>→</mo><mi>SSet</mi><mo stretchy="false">/</mo><mi>C</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N_{-}(C) : [C,SSet] \to SSet/C \,. </annotation></semantics></math></div> <p>By the <a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a> this functor has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mo lspace="verythinmathspace" rspace="0em">−</mo></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>:</mo><mi>SSet</mi><mo stretchy="false">/</mo><mi>C</mi><mo>→</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>SSet</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F_{-}(C) : SSet/C \to [C,SSet] \,. </annotation></semantics></math></div> <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 : E \to C</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/left+Kan+fibration">left Kan fibration</a> the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>:</mo><mi>C</mi><mo>→</mo><mi>SSet</mi></mrow><annotation encoding="application/x-tex">F_p(C) : C \to SSet</annotation></semantics></math> sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \in Obj(C)</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> <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>c</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>E</mi><msub><mo>×</mo> <mi>C</mi></msub><mo stretchy="false">{</mo><mi>c</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">p^{-1}(c) := E \times_C \{c\}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>p</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>:</mo><mi>c</mi><mo>↦</mo><msup><mi>p</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F_p(C) : c \mapsto p^{-1}(c) \,. </annotation></semantics></math></div> <p>(See remark 3.2.5.5 of <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+Kan+fibration">universal Kan fibration</a></p> </li> <li> <p><span class="newWikiWord">fibration of (∞,1)-categories<a href="/nlab/new/fibration+of+%28%E2%88%9E%2C1%29-categories">?</a></span></p> </li> </ul> <h2 id="references">References</h2> <p>Textbook accounts:</p> <ul> <li id="Lurie09"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, Section 3.3.2 of: <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em>, Annals of Mathematics Studies <strong>170</strong>, Princeton University Press 2009 (<a href="https://press.princeton.edu/titles/8957.html">pup:8957</a>, <a href="https://www.math.ias.edu/~lurie/papers/HTT.pdf">pdf</a>)</p> <blockquote> <p>(the concrete description in terms of model theory on marked simplicial sets is in section 3.2. A simpler version of this is in section 2.2.1)</p> </blockquote> </li> <li id="Cisinski19"> <p><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, Section 5.2 of: <em><a class="existingWikiWord" href="/nlab/show/Higher+Categories+and+Homotopical+Algebra">Higher Categories and Homotopical Algebra</a></em>, Cambridge University Press 2019 (<a href="https://doi.org/10.1017/9781108588737">doi:10.1017/9781108588737</a>, <a href="http://www.mathematik.uni-regensburg.de/cisinski/CatLR.pdf">pdf</a>)</p> <blockquote> <p>(the universal left fibration)</p> </blockquote> </li> </ul> <p>The direct description of the universal fibration is discussed at</p> <ul> <li id="direct">MathOverflow: <em><a href="https://mathoverflow.net/questions/383275/">A construction of the universal cocartesian fibration</a></em></li> </ul> <p>Discussion of the <a class="existingWikiWord" href="/nlab/show/universal+Kan+fibration">universal Kan fibration</a> as a <a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">categorical semantics</a> for a <a class="existingWikiWord" href="/nlab/show/univalence+axiom">univalent</a> <a class="existingWikiWord" href="/nlab/show/type+universe">type universe</a> in <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>:</p> <p>Discussion of fibrations via <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a> theory</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, chapter 5 <em><a href="https://kerodon.net/tag/01J2">Fibrations of ∞-Categories</a></em> (2021) in: <em><a class="existingWikiWord" href="/nlab/show/Kerodon">Kerodon</a></em></li> </ul> <p>On the <a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">categorical semantics</a> of <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> in <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>/<a class="existingWikiWord" href="/nlab/show/infinity-groupoids"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-groupoids</a>:</p> <ul> <li id="KapulkinLumsdaine12"><a class="existingWikiWord" href="/nlab/show/Chris+Kapulkin">Chris Kapulkin</a>, <a class="existingWikiWord" href="/nlab/show/Peter+LeFanu+Lumsdaine">Peter LeFanu Lumsdaine</a>, <em>The Simplicial Model of Univalent Foundations (after Voevodsky)</em>, Journal of the European Mathematical Society <strong>23</strong> (2021) 2071–2126 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/1211.2851">arXiv:1211.2851</a>, <a href="https://doi.org/10.4171/jems/1050">doi:10.4171/jems/1050</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/straightening+and+unstraightening">straightening and unstraightening</a> entirely within the context of <a class="existingWikiWord" href="/nlab/show/quasi-categories">quasi-categories</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <a class="existingWikiWord" href="/nlab/show/Hoang+Kim+Nguyen">Hoang Kim Nguyen</a>, <em>The universal coCartesian fibration</em> [<a href="https://arxiv.org/abs/2210.08945">arXiv:2210.08945</a>]</li> </ul> <p>which (along the lines of the discussion of the universal left fibration from <a href="#Cisinski19">Cisinski 2019</a>) allows to understand the <a class="existingWikiWord" href="/nlab/show/universal+coCartesian+fibration">universal coCartesian fibration</a> as <a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a> for the <a class="existingWikiWord" href="/nlab/show/univalence+axiom">univalent</a> <a class="existingWikiWord" href="/nlab/show/type+universe">type universe</a> in <a class="existingWikiWord" href="/nlab/show/directed+homotopy+type+theory">directed homotopy type theory</a>:</p> <ul> <li id="CisinskiEtAl23"><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <a class="existingWikiWord" href="/nlab/show/Hoang+Kim+Nguyen">Hoang Kim Nguyen</a>, Tashi Walde: <em>Univalent Directed Type Theory</em>, lecture series in the <em><a href="https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html">CMU Homotopy Type Theory Seminar</a></em> (13, 20, 27 Mar 2023) [<a href="https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html#230313">web</a>, video 1:<a href="https://www.youtube.com/watch?v=5YOltuTcBK8">YT</a>, 2:<a href="https://www.youtube.com/watch?v=xWmELBvHMPo">YT</a>, 3:<a href="https://www.youtube.com/watch?v=P0Cfb4eUJo4">YT</a>; slides 0:<a href="https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-intro_talk1.pdf">pdf</a>, 1:<a href="https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk1.pdf">pdf</a>, 2:<a href="https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk2.pdf">pdf</a>, 3:<a href="https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk3.pdf">pdf</a>]</li> </ul> <p>(see <a href="https://youtu.be/P0Cfb4eUJo4?t=4618">video 3 at 1:16:58</a> and <a href="https://www.cmu.edu/dietrich/philosophy/hott/seminars/slides/cisinski-nguyen-walde-talk3.pdf#page=33">slide 3.33</a>).</p> </body></html> </div> 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