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class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a></p> </li> </ul> <h2 id='sidebar_universal_constructions'>Universal constructions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal construction</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representable+functor'>representable functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor'>adjoint functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/limit'>limit</a>/<a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weighted+limit'>weighted limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/end'>end</a>/<a class='existingWikiWord' href='/nlab/show/diff/end'>coend</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Yoneda+lemma'>Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+functor+theorem'>adjoint functor theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monadicity+theorem'>monadicity theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+lifting+theorem'>adjoint lifting theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Tannaka+duality'>Tannaka duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Gabriel%E2%80%93Ulmer+duality'>Gabriel-Ulmer duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+object+argument'>small object argument</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Freyd-Mitchell+embedding+theorem'>Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/sheaf+and+topos+theory'>sheaf and topos theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+category+theory'>enriched category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></p> </li> </ul> <h2 id='sidebar_applications'>Applications</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#Definition'>Definition</a></li><li><a href='#fibrations_versus_pseudofunctors'>Fibrations versus pseudofunctors</a></li><li><a href='#fibrations_versus_presheaves_of_categories'>Fibrations versus presheaves of categories</a></li><li><a href='#remarks'>Remarks</a></li><li><a href='#examples'>Examples</a></li><li><a href='#properties'>Properties</a></li><li><a href='#variations'>Variations</a><ul><li><a href='#discrete_and_groupoidal_fibrations'>Discrete and groupoidal fibrations</a></li><li><a href='#opfibrations_and_bifibrations'>Opfibrations and bifibrations</a></li><li><a href='#StreetFibration'>Version respecting the Principle of equivalence</a></li><li><a href='#internal_version'>Internal version</a></li><li><a href='#twosided_version'>Two-sided version</a></li><li><a href='#higher_categorical_versions'>Higher categorical versions</a></li><li><a href='#versions_for_other_categorical_structures'>Versions for other categorical structures</a></li></ul></li><li><a href='#alternate_definitions'>Alternate definitions</a><ul><li><a href='#in_terms_of_adjoints'>In terms of adjoints</a></li><li><a href='#in_terms_of_pseudoalgebras'>In terms of pseudoalgebras</a></li></ul></li><li><a href='#discussions'>Discussions</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>A <em>Grothendieck fibration</em> (named after <a class='existingWikiWord' href='/nlab/show/diff/Alexander+Grothendieck'>Alexander Grothendieck</a>, also called a <em>fibered category</em> or just a <em>fibration</em>) is a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo lspace='verythinmathspace'>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p \colon E\to B</annotation></semantics></math> such that the <a class='existingWikiWord' href='/nlab/show/diff/fiber'>fibers</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mi>b</mi></msub><mo>=</mo><msup><mi>p</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E_b = p^{-1}(b)</annotation></semantics></math> depend (contravariantly) <a class='existingWikiWord' href='/nlab/show/diff/pseudofunctor'>pseudofunctorially</a> on <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>b\in B</annotation></semantics></math>.</p> <p>One also says that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> is a <em>fibered category</em> over <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>. One calls <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> the <em>total category</em> of the fibration (not to be confused with the independent notion of <em><a class='existingWikiWord' href='/nlab/show/diff/total+category'>total category</a></em>) and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> its <em>base category</em>.</p> <p><a class='existingWikiWord' href='/nlab/show/diff/duality'>Dually</a>, in a (Grothendieck) <em>opfibration</em> the dependence is <a class='existingWikiWord' href='/nlab/show/diff/functor'>covariant</a>.</p> <p>There is an <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalence</a> of <a class='existingWikiWord' href='/nlab/show/diff/strict+2-category'>2-categories</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Fib</mi><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo><mover><mo>↔</mo><mo>≃</mo></mover><mo stretchy='false'>[</mo><msup><mi>B</mi> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><mo stretchy='false'>]</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mstyle displaystyle='false'><mo>∫</mo></mstyle></mrow><annotation encoding='application/x-tex'> Fib(B) \overset {\simeq} {\leftrightarrow} [B^{op}, Cat] \;\colon\; \textstyle{\int} </annotation></semantics></math></div> <p>between the 2-category of fibrations, cartesian functors, and vertical natural transformations over <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, and the 2-category <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>B</mi> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>[B^{op},Cat]</annotation></semantics></math> of contravariant <a class='existingWikiWord' href='/nlab/show/diff/pseudofunctor'>pseudofunctor</a>s from <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> to <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a>, also called <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/indexed+category'>indexed categories</a>.</p> <p>The “<a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a>” <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∫</mo><mo lspace='verythinmathspace'>:</mo><mo stretchy='false'>[</mo><msup><mi>B</mi> <mi>op</mi></msup><mo>,</mo><mi>Cat</mi><mo stretchy='false'>]</mo><mo>→</mo><mi>Fib</mi><mo stretchy='false'>(</mo><mi>B</mi><mo stretchy='false'>)</mo><mo lspace='verythinmathspace'>:</mo><mi>F</mi><mo>↦</mo><mo>∫</mo><mi>F</mi></mrow><annotation encoding='application/x-tex'>\int \colon [B^{op}, Cat] \to Fib(B) \colon F \mapsto \int F</annotation></semantics></math> of a fibration from a pseudofunctor.</p> <p>Those fibrations corresponding to pseudofunctors that factor through <a class='existingWikiWord' href='/nlab/show/diff/Grpd'>Grpd</a> are called <strong><a class='existingWikiWord' href='/nlab/show/diff/fibration+fibered+in+groupoids'>categories fibered in groupoids</a></strong>.</p> <h2 id='Definition'>Definition</h2> <p>\begin{definition}\label{CartesianMorphism} <strong>(<a class='existingWikiWord' href='/nlab/show/diff/Cartesian+morphism'>cartesian morphism</a>)</strong> \linebreak Given a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mspace width='thinmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thinmathspace' /><mi>ℰ</mi><mo>⟶</mo><mi>ℬ</mi></mrow><annotation encoding='application/x-tex'>P \,\colon\, \mathcal{E} \longrightarrow \mathcal{B}</annotation></semantics></math>, a <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>f \colon x \to y</annotation></semantics></math> in its <a class='existingWikiWord' href='/nlab/show/diff/domain'>domain</a> <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding='application/x-tex'>\mathcal{E}</annotation></semantics></math> is called <em><a class='existingWikiWord' href='/nlab/show/diff/Cartesian+morphism'>cartesian</a></em> if for any morphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_17' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>z</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>g \colon z\to y</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding='application/x-tex'>\mathcal{E}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>w</mi><mo lspace='verythinmathspace'>:</mo><mi>P</mi><mo stretchy='false'>(</mo><mi>z</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>P</mi><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>w \colon P(z)\to P(x)</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℬ</mi></mrow><annotation encoding='application/x-tex'>\mathcal{B}</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>∘</mo><mi>w</mi><mo>=</mo><mi>P</mi><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>P(f)\circ w = P(g)</annotation></semantics></math>, there exists a <em>unique</em> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>w</mi><mo>^</mo></mover><mspace width='thinmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thinmathspace' /><mi>z</mi><mo>→</mo><mi>x</mi></mrow><annotation encoding='application/x-tex'>\widehat{w} \,\colon\, z\to x</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>=</mo><mi>f</mi><mo>∘</mo><mover><mi>w</mi><mo>^</mo></mover></mrow><annotation encoding='application/x-tex'>g = f \circ \widehat{w}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo stretchy='false'>(</mo><mover><mi>w</mi><mo>^</mo></mover><mo stretchy='false'>)</mo><mo>=</mo><mi>w</mi></mrow><annotation encoding='application/x-tex'>P(\widehat{w}) = w</annotation></semantics></math>:</p> <p>\begin{tikzcd}[sep=20pt] & z \ar[drrr, bend left=20, { \forall \, g }{description}] \ar[dr, dashed, { \exists! \, \widehat{w} }{description}] \ \mathcal{E} \ar[dd, { P }] && x \ar[rr, { f }{description}, { \mathrm{cartesian} }{swap, yshift=-1pt}]<br /> && y \ & P(z) \ar[drrr, bend left=20, { P(g) }{description}] \ar[dr, { \forall \, w }{description}] \ \mathcal{B} && P(x) \ar[rr, { P(f) }{description}] && P(y) \end{tikzcd}</p> <p>\end{definition}</p> <p>\begin{definition}\label{GrothendieckFibration} <strong>(Grothendieck fibration)</strong> \linebreak A <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo lspace='verythinmathspace'>:</mo><mi>ℰ</mi><mo>→</mo><mi>ℬ</mi></mrow><annotation encoding='application/x-tex'>p \colon \mathcal{E} \to \mathcal{B}</annotation></semantics></math> is a <em>fibration</em> if for all <a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi><mo>∈</mo><mi>ℰ</mi></mrow><annotation encoding='application/x-tex'>y \in \mathcal{E}</annotation></semantics></math> and <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mn>0</mn></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>x</mi> <mn>0</mn></msub><mo>→</mo><mi>P</mi><mo stretchy='false'>(</mo><mi>y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f_0 \colon x_0 \to P(y)</annotation></semantics></math>, there is a cartesian morphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>x</mi><mo>→</mo><mi>y</mi></mrow><annotation encoding='application/x-tex'>f \colon x \to y</annotation></semantics></math> (Def. \ref{CartesianMorphism}) such that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo>=</mo><msub><mi>f</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>P(f) = f_0</annotation></semantics></math>.<br />\end{definition}</p> <p>Such <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> is also called a “cartesian lifting” of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Pf</mi></mrow><annotation encoding='application/x-tex'>Pf</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math>, and a choice of cartesian lifting for every <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Pf</mi></mrow><annotation encoding='application/x-tex'>Pf</annotation></semantics></math> is called a <em><a class='existingWikiWord' href='/nlab/show/diff/cleavage'>cleavage</a></em>. Thus, assuming the <a class='existingWikiWord' href='/nlab/show/diff/axiom+of+choice'>axiom of choice</a>, a functor is a fibration iff it admits some cleavage.</p> <p>\begin{remark} <strong>(weakly cartesian morphisms)</strong> \linebreak We say that a morphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/prefibered+category'>weakly cartesian</a> if it has the property of a Cartesian morphism (Def. \ref{CartesianMorphism}) only when <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>w</mi></mrow><annotation encoding='application/x-tex'>w</annotation></semantics></math> is an <a class='existingWikiWord' href='/nlab/show/diff/identity+morphism'>identity morphism</a>.</p> <p>One can prove that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> is a fibration (Def. \ref{GrothendieckFibration}) if and only if firstly, it has the above property with “Cartesian” replaced by “weakly cartesian,” and secondly, the composite of weakly cartesian arrows is weakly cartesian.</p> <p>In a fibration, every weakly cartesian lifting <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Pf</mi></mrow><annotation encoding='application/x-tex'>Pf</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> is in fact cartesian (as one can show by combining the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal properties</a> of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> and of a given cartesian lifting to <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi></mrow><annotation encoding='application/x-tex'>e</annotation></semantics></math>), but this is not true in general.</p> <p>Some sources say “cartesian” and “strongly cartesian” instead of “weakly cartesian” and “cartesian,” respectively. If weakly cartesian liftings exist but weakly cartesian arrows are <em>not</em> necessarily closed under composition, one sometimes speaks of a <a class='existingWikiWord' href='/nlab/show/diff/prefibered+category'>prefibered category</a>. \end{remark}</p> <p>\begin{definition}\label{MorphismOfFibrations} <strong>(morphism of fibrations – <a class='existingWikiWord' href='/nlab/show/diff/fibred+functor'>fibered functor</a>)</strong> \linebreak For <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi></mrow><annotation encoding='application/x-tex'>P</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>P'</annotation></semantics></math> fibrations (Def. \ref{GrothendieckFibration}), a <a class='existingWikiWord' href='/nlab/show/diff/commutative+square'>commuting square</a> of <a class='existingWikiWord' href='/nlab/show/diff/functor'>functors</a> of the form</p> <div class='maruku-equation' id='eq:DiagramForFiberedFunctor'><span class='maruku-eq-number'>(1)</span><math class='maruku-mathml' display='block' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℰ</mi><mo>′</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>ℰ</mi></mtd></mtr> <mtr><mtd><mpadded lspace='-100%width' width='0'><mrow><msup><mrow /> <mrow><mi>P</mi><mo>′</mo></mrow></msup></mrow></mpadded><mo maxsize='1.2em' minsize='1.2em'>↓</mo></mtd> <mtd /> <mtd><mo maxsize='1.2em' minsize='1.2em'>↓</mo><mpadded width='0'><mrow><msup><mrow /> <mi>P</mi></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>ℬ</mi><mo>′</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>ℬ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ \mathcal{E}' & \longrightarrow & \mathcal{E} \\ \mathllap{{}^{P'}}\big\downarrow && \big\downarrow \mathrlap{{}^{P}} \\ \mathcal{B}' & \longrightarrow & \mathcal{B} } </annotation></semantics></math></div> <p>is a <em>morphism of fibrations</em> (also called a <em>cartesian square</em> or <em><a class='existingWikiWord' href='/nlab/show/diff/fibred+functor'>fibred functor</a></em>) if the top functor preserves Cartesian morphisms (Def. \ref{CartesianMorphism}).<br />\end{definition}</p> <p>\begin{definition}\label{TwoMorphismOfFibrations} <strong>(<a class='existingWikiWord' href='/nlab/show/diff/2-morphism'>2-morphisms</a> of fibrations – <a class='existingWikiWord' href='/nlab/show/diff/fibred+natural+transformation'>fibered natural transformation</a>)</strong> \linebreak A <a class='existingWikiWord' href='/nlab/show/diff/2-morphism'>2-morphism</a> (or <em><a class='existingWikiWord' href='/nlab/show/diff/fibred+natural+transformation'>fibered natural transformation</a></em>) between morphisms of fibrations (Def. \ref{MorphismOfFibrations}) is a pair of <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformations</a>, one lying over the other in <a class='maruku-eqref' href='#eq:DiagramForFiberedFunctor'>(1)</a>. \end{definition}</p> <p>\begin{remark} If the functor on base categories is an <a class='existingWikiWord' href='/nlab/show/diff/identity+functor'>identity</a>, then a 2-morphism (Def. \ref{TwoMorphismOfFibrations}) is a <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformations</a> between total categories which is “vertical” in that its component morphisms lie in <a class='existingWikiWord' href='/nlab/show/diff/fiber'>fibers</a>. \end{remark}</p> <h2 id='fibrations_versus_pseudofunctors'>Fibrations versus pseudofunctors</h2> <p>Given a fibration <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p:E\to B</annotation></semantics></math>, we obtain a pseudofunctor <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>B</mi> <mi>op</mi></msup><mo>→</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>B^{op}\to Cat</annotation></semantics></math> by sending each <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi><mo>∈</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>b\in B</annotation></semantics></math> to the category <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mi>b</mi></msub><mo>=</mo><msup><mi>p</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>E_b = p^{-1}(b)</annotation></semantics></math> of objects mapping onto <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math> and morphisms mapping onto <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mn>1</mn> <mi>b</mi></msub></mrow><annotation encoding='application/x-tex'>1_b</annotation></semantics></math>. To obtain the action on morphisms, given an <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>f:a\to b</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> and an object <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>∈</mo><msub><mi>E</mi> <mi>b</mi></msub></mrow><annotation encoding='application/x-tex'>e\in E_b</annotation></semantics></math>, we choose a cartesian arrow <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>e</mi><mo>′</mo><mo>→</mo><mi>e</mi></mrow><annotation encoding='application/x-tex'>\phi:e'\to e</annotation></semantics></math> over <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> and call its <a class='existingWikiWord' href='/nlab/show/diff/source'>source</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mi>e</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f^*(e)</annotation></semantics></math>. The universal factorization property of cartesian arrows then makes <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>f^*</annotation></semantics></math> into a functor <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mi>b</mi></msub><mo>→</mo><msub><mi>E</mi> <mi>a</mi></msub></mrow><annotation encoding='application/x-tex'>E_b \to E_a</annotation></semantics></math>, and it is easy to verify that it is a pseudofunctor. The functor in the other direction is called the <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a>. This yields a <a class='existingWikiWord' href='/nlab/show/diff/strict+2-equivalence+of+2-categories'>strict 2-equivalence of 2-categories</a> between</p> <ul> <li> <p>fibrations over <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, morphisms of fibrations over <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Id</mi> <mi>B</mi></msub></mrow><annotation encoding='application/x-tex'>Id_B</annotation></semantics></math>, and 2-cells over <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>id</mi> <mrow><msub><mi>Id</mi> <mi>B</mi></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>id_{Id_B}</annotation></semantics></math>, and</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pseudofunctor'>pseudofunctors</a> of the form <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>B</mi> <mi>op</mi></msup><mo>→</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>B^{op}\to Cat</annotation></semantics></math>, <a class='existingWikiWord' href='/nlab/show/diff/pseudonatural+transformation'>pseudonatural transformations</a>, and <a class='existingWikiWord' href='/nlab/show/diff/modification'>modifications</a>.</p> </li> </ul> <p>In fact, this is an instance of the general theory of representability for <a class='existingWikiWord' href='/nlab/show/diff/generalized+multicategory'>generalized multicategories</a>. There is a monad <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math> whose pseudoalgebras are pseudofunctors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>B</mi> <mi>op</mi></msup><mo>→</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>B^{op}\to Cat</annotation></semantics></math>, and whose “generalized multicategories” are functors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>E\to B</annotation></semantics></math>. Such a multicategory is “representable” precisely when it is a fibration, and moreover there is an induced monad <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>T</mi><mo stretchy='false'>^</mo></mover></mrow><annotation encoding='application/x-tex'>\hat{T}</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi><mo stretchy='false'>/</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>Cat/B</annotation></semantics></math> which is <a class='existingWikiWord' href='/nlab/show/diff/lax-idempotent+2-monad'>colax-idempotent</a> and whose pseudoalgebras are precisely the fibrations.</p> <p>This correspondence also generalizes to the correspondence between arbitrary functors over <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> and <a class='existingWikiWord' href='/nlab/show/diff/displayed+category'>displayed categories</a> over <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, i.e. normal lax functors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>→</mo><mi>Prof</mi></mrow><annotation encoding='application/x-tex'>B\to Prof</annotation></semantics></math>.</p> <h2 id='fibrations_versus_presheaves_of_categories'>Fibrations versus presheaves of categories</h2> <p>The <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a> is an equivalence of <a class='existingWikiWord' href='/nlab/show/diff/bicategory'>bicategories</a> from the bicategory of presheaves of categories to the bicategory of Grothendieck fibrations.</p> <p>As explained there, this equivalence, interpreted as a functor between 1-categories, has both a left and a right adjoint (which are equivalent in the bicategorical context). Roughly speaking, the left adjoint strictifies a Grothendieck fibration by adding formal pullbacks of objects, which themselves pullback strictly, whereas the right adjoint strictifies a Grothendieck fibration by equipping object with a functorial choice of a pullback for each possible morphism. See <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a> for more details.</p> <p>These two adjunctions can be turned into <a class='existingWikiWord' href='/nlab/show/diff/Quillen+equivalence'>Quillen equivalences</a> of <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model categories</a>. This can be deduced, for example, from two <a class='existingWikiWord' href='/nlab/show/diff/Quillen+equivalence'>Quillen equivalences</a> between <a class='existingWikiWord' href='/nlab/show/diff/Cartesian+fibration'>cartesian fibrations</a> and <a class='existingWikiWord' href='/nlab/show/diff/presheaf'>presheaves</a> of <a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+Cartesian+fibrations'>marked simplicial sets</a> on the <a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasicategory</a> given by the <a class='existingWikiWord' href='/nlab/show/diff/nerve'>nerve</a> of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>. See <a class='existingWikiWord' href='/nlab/show/diff/straightening+functor'>straightening functor</a> for more details.</p> <h2 id='remarks'>Remarks</h2> <ul> <li> <p>Fibrations are a “nonalgebraic” structure, since the base change functors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>f^*</annotation></semantics></math> are determined by universal properties, hence uniquely up to isomorphism. By contrast, pseudofunctors are an “algebraic” structure, since the functors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>f^*</annotation></semantics></math> are specified, together with the necessary coherence data and axioms; the latter come for free in a fibration because of the universal property.</p> </li> <li> <p>A <a class='existingWikiWord' href='/nlab/show/diff/stack'>stack</a>, being a particular type of pseudofunctor, can also be described as a particular sort of fibration. This was the original application for which Grothendieck introduced the notion.</p> </li> </ul> <h2 id='examples'>Examples</h2> <p>\begin{example} Let <a class='existingWikiWord' href='/nlab/show/diff/VectBund'>$\VBun$</a> denote the category whose <a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a> are <a class='existingWikiWord' href='/nlab/show/diff/pair'>pairs</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>M</mi><mo>,</mo><mi>V</mi><mo stretchy='false'>)</mo><mo>∈</mo><mo lspace='0em' rspace='thinmathspace'>Man</mo><mo>×</mo><mo lspace='0em' rspace='thinmathspace'>Man</mo></mrow><annotation encoding='application/x-tex'>(M,V)\in\Man\times\Man</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/manifold'>manifolds</a> where <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>q</mi><mo>:</mo><mi>V</mi><mo>⟶</mo><mi>M</mi></mrow><annotation encoding='application/x-tex'>q:V\longrightarrow M</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/vector+bundle'>vector bundle</a> over <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>. A morphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>M</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>V</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>⟶</mo><mo stretchy='false'>(</mo><msub><mi>M</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>V</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(M_{1},V_{1})\longrightarrow (M_{2},V_{2})</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/commutative+square'>commutative square</a>: \begin{center} \begin{tikzcd} V_{1} \arrow[r, g] \arrow[d, q_{1}] & V_{2} \arrow[d, q_{2}] \ M_{1} \arrow[r, f] & M_{2} <br />\end{tikzcd} \end{center} where <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> is a smooth map, and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> is a smooth map such that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><msub><mo lspace='mediummathspace' rspace='mediummathspace'>∣</mo> <mrow><msubsup><mi>q</mi> <mn>1</mn> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo stretchy='false'>(</mo><mi>m</mi><mo stretchy='false'>)</mo></mrow></msub><mo>:</mo><msubsup><mi>q</mi> <mn>1</mn> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo stretchy='false'>(</mo><mi>m</mi><mo stretchy='false'>)</mo><mo>⟶</mo><msubsup><mi>q</mi> <mn>2</mn> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>m</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>g\mid_{q_{1}^{-1}(m)}:q_{1}^{-1}(m)\longrightarrow q_{2}^{-1}(f(m))</annotation></semantics></math> is linear for every <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>m</mi><mo>∈</mo><msub><mi>M</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>m\in M_{1}</annotation></semantics></math>. Composition is given as in the <a class='existingWikiWord' href='/nlab/show/diff/arrow+category'>arrow category</a>. There is a forgetful functor <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒫</mi><mo>:</mo><mo lspace='0em' rspace='thinmathspace'>VBun</mo><mo>⟶</mo><mo lspace='0em' rspace='thinmathspace'>Man</mo></mrow><annotation encoding='application/x-tex'>\mathcal{P}:\VBun\longrightarrow\Man</annotation></semantics></math> which projects onto the base data. To see that this functor is a Grothendieck fibration, first consider a morphism of the form: \begin{center} \begin{tikzcd} f^{<em>}M_{2} \arrow[r] \arrow[d] & V_{2} \arrow[d, q_{2}] \ M_{1} \arrow[r, f] & M_{2} <br />\end{tikzcd} \end{center} given by the <a class='existingWikiWord' href='/nlab/show/diff/pullback+bundle'>pullback bundle</a> of a vector bundle <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mn>2</mn></msub><mo>⟶</mo><msub><mi>M</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>V_{2}\longrightarrow M_{2}</annotation></semantics></math> along a smooth map <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>M</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>M</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>M_{1}\longrightarrow M_{2}</annotation></semantics></math>. Certainly, the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of the <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> will supply the conditions for this square to define a <a class='existingWikiWord' href='/nlab/show/diff/Cartesian+morphism'>cartesian morphism</a> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='0em' rspace='thinmathspace'>VBun</mo></mrow><annotation encoding='application/x-tex'>\VBun</annotation></semantics></math> with respect to the the functor <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒫</mi></mrow><annotation encoding='application/x-tex'>\mathcal{P}</annotation></semantics></math>. We show that these are in fact all the cartesian morphisms in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='0em' rspace='thinmathspace'>VBun</mo></mrow><annotation encoding='application/x-tex'>\VBun</annotation></semantics></math>. Let <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>′</mo><mo>=</mo><mo stretchy='false'>(</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>M</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>e' = (V_{1},M_{1})</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>=</mo><mo stretchy='false'>(</mo><msub><mi>V</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>M</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>e = (V_{2},M_{2})</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>″</mo><mo>=</mo><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>M</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>e'' = (V,M)</annotation></semantics></math>, then by applying the definition from above directly, we see that a morphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>e</mi><mo>′</mo><mo>⟶</mo><mi>e</mi></mrow><annotation encoding='application/x-tex'>\phi:e'\longrightarrow e</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='0em' rspace='thinmathspace'>VBun</mo></mrow><annotation encoding='application/x-tex'>\VBun</annotation></semantics></math>, i.e., a square: \begin{center} \begin{tikzcd} V_{1} \arrow[r] \arrow[d, e] & V_{2} \arrow[d, e] \ M_{1} \arrow[r] & M_{2} <br />\end{tikzcd} \end{center} is cartesian, if given any morphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ψ</mi><mo>:</mo><mi>e</mi><mo>″</mo><mo>⟶</mo><mi>e</mi></mrow><annotation encoding='application/x-tex'>\psi:e''\longrightarrow e</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='0em' rspace='thinmathspace'>VBun</mo></mrow><annotation encoding='application/x-tex'>\VBun</annotation></semantics></math>, i.e., a square (which we draw around the first): \begin{center} \begin{tikzcd} V \arrow[rrd, bend left] \arrow[dd, e] & & \ & V_{1} \arrow[r] \arrow[d, e] & V_{2} \arrow[d, e] \ M \arrow[rr, bend right, shift right] & M_{1} \arrow[r] & M_{2} <br />\end{tikzcd} \end{center} and any morphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>:</mo><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>e</mi><mo>″</mo><mo stretchy='false'>)</mo><mo>⟶</mo><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>e</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>g:\mathcal{P}(e'')\longrightarrow \mathcal{P}(e')</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='0em' rspace='thinmathspace'>Man</mo></mrow><annotation encoding='application/x-tex'>\Man</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>ϕ</mi><mo stretchy='false'>)</mo><mo>∘</mo><mi>g</mi><mo>=</mo><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>ψ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{P}(\phi)\circ g = \mathcal{P}(\psi)</annotation></semantics></math>, i.e. a smooth map <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo>:</mo><mi>M</mi><mo>⟶</mo><msub><mi>M</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>g:M\longrightarrow M_{1}</annotation></semantics></math> that fills in the above diagram to a commuting diagram: \begin{center} \begin{tikzcd} V \arrow[rrd, bend left] \arrow[dd, e] & & \ & V_{1} \arrow[r] \arrow[d, e] & V_{2} \arrow[d, e] \ M \arrow[rr, bend right, shift right] \arrow[r, g] & M_{1} \arrow[r] & M_{2} <br />\end{tikzcd} \end{center} then there exists a unique morphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>χ</mi><mo>:</mo><mi>e</mi><mo>″</mo><mo>⟶</mo><mi>e</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>\chi:e''\longrightarrow e'</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='0em' rspace='thinmathspace'>VBun</mo></mrow><annotation encoding='application/x-tex'>\VBun</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ψ</mi><mo>=</mo><mi>ϕ</mi><mo>∘</mo><mi>χ</mi></mrow><annotation encoding='application/x-tex'>\psi = \phi\circ\chi</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>χ</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>g</mi></mrow><annotation encoding='application/x-tex'>\mathcal{P}(\chi) = g</annotation></semantics></math>, i.e., there exists a unique morphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi><mo>⟶</mo><msub><mi>V</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>V\longrightarrow V_{1}</annotation></semantics></math> that further fills the diagram to a commuting diagram: \begin{center} \begin{tikzcd} V \arrow[rrd, bend left] \arrow[dd, e] \arrow[rd, dashed] & & \ & V_{1} \arrow[r] \arrow[d, e] & V_{2} \arrow[d, e] \ M \arrow[rr, bend right, shift right] \arrow[r, g] & M_{1} \arrow[r] & M_{2} <br />\end{tikzcd} \end{center} It immediately follows from the above unfolding of the definition that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo lspace='0em' rspace='thinmathspace'>VBun</mo><mo stretchy='false'>(</mo><mi>e</mi><mo>″</mo><mo>,</mo><mi>e</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>≅</mo><mo lspace='0em' rspace='thinmathspace'>VBun</mo><mo stretchy='false'>(</mo><mi>e</mi><mo>″</mo><mo>,</mo><mi>p</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\VBun(e'',e')\cong\VBun(e'',p)</annotation></semantics></math> naturally in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>″</mo></mrow><annotation encoding='application/x-tex'>e''</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> is the pullback bundle of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mn>2</mn></msub><mo>⟶</mo><msub><mi>M</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>V_{2}\longrightarrow M_{2}</annotation></semantics></math> along <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>M</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>M</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>M_{1}\longrightarrow M_{2}</annotation></semantics></math>, and thus by the <a class='existingWikiWord' href='/nlab/show/diff/Yoneda+lemma#YonedaCorollaries'>Yoneda lemma</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>′</mo><mo>≅</mo><mi>p</mi></mrow><annotation encoding='application/x-tex'>e'\cong p</annotation></semantics></math>, and thus <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub><mo>≅</mo><mi>𝒫</mi><mo stretchy='false'>(</mo><mi>e</mi><mo>′</mo><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup><msub><mi>V</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>V_{1}\cong\mathcal{P}(e')^{*}V_{2}</annotation></semantics></math>. Note that the isomorphisms <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>′</mo><mo>≅</mo><mi>p</mi></mrow><annotation encoding='application/x-tex'>e'\cong p</annotation></semantics></math> preserve the commutativity of the diagrams we have drawn above by the universal property of the pullback, and by the definition of a cartesian morphism. Now that we have identified the cartesian morphisms, the statement that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒫</mi><mo>:</mo><mo lspace='0em' rspace='thinmathspace'>VBun</mo><mo>⟶</mo><mo lspace='0em' rspace='thinmathspace'>Man</mo></mrow><annotation encoding='application/x-tex'>\mathcal{P}:\VBun\longrightarrow\Man</annotation></semantics></math> is a Grothendieck fibration is equivalent to the statement that vector bundles can be pulled back along smooth maps. This is of course true, and thus <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒫</mi><mo>:</mo><mo lspace='0em' rspace='thinmathspace'>VBun</mo><mo>⟶</mo><mo lspace='0em' rspace='thinmathspace'>Man</mo></mrow><annotation encoding='application/x-tex'>\mathcal{P}:\VBun\longrightarrow\Man</annotation></semantics></math> is a Grothendieck fibration. The associated functors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>f^{*}</annotation></semantics></math> are the <a class='existingWikiWord' href='/nlab/show/diff/pullback#PullbackFunctor'>pullback psuedofunctors</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>:</mo><msub><mo lspace='0em' rspace='thinmathspace'>VBun</mo> <mrow><msub><mi>M</mi> <mn>2</mn></msub></mrow></msub><mo>⟶</mo><msub><mo lspace='0em' rspace='thinmathspace'>VBun</mo> <mrow><msub><mi>M</mi> <mn>1</mn></msub></mrow></msub></mrow><annotation encoding='application/x-tex'>f^{*}:\VBun_{M_{2}}\longrightarrow \VBun_{M_{1}}</annotation></semantics></math> that pullback vector bundles over a manifold <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>M</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>M_{2}</annotation></semantics></math> to vector bundles over a manifold <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>M</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>M_{1}</annotation></semantics></math> along a smooth map <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>M</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>M</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>f:M_{1}\longrightarrow M_{2}</annotation></semantics></math>. \end{example}</em></p> <p>\begin{example} Let <a class='existingWikiWord' href='/nlab/show/diff/Ring'>Ring</a> be the category of <a class='existingWikiWord' href='/nlab/show/diff/ring'>rings</a>, and <a class='existingWikiWord' href='/nlab/show/diff/Mod'>Mod</a> the category of pairs <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>R</mi><mo>,</mo><mi>M</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(R,M)</annotation></semantics></math> where <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> is a ring and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math> is a (left) <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-module. Then the evident forgetful functor <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Mod</mi><mo>→</mo><mi>Ring</mi></mrow><annotation encoding='application/x-tex'>Mod\to Ring</annotation></semantics></math> is a fibration and an opfibration. The functors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>f^*</annotation></semantics></math> are given by restriction of scalars, <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow><annotation encoding='application/x-tex'>f_!</annotation></semantics></math> is extension of scalars, and the right adjoint <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>f_*</annotation></semantics></math> is coextension of scalars. \end{example}</p> <p>\begin{example} Let <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> be any category with <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullbacks</a> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mstyle mathvariant='bold'><mn>2</mn></mstyle></mrow><annotation encoding='application/x-tex'>\mathbf{2}</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/walking+structure'>free-living</a> arrow, so that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mstyle mathvariant='bold'><mn>2</mn></mstyle></msup></mrow><annotation encoding='application/x-tex'>C^{\mathbf{2}}</annotation></semantics></math> is the category of arrows and commutative squares in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>. Then the “codomain” functor <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mstyle mathvariant='bold'><mn>2</mn></mstyle></msup><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>C^{\mathbf{2}} \to C</annotation></semantics></math> is a fibration and opfibration. The cartesian arrows are precisely the pullback squares, and the functors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow><annotation encoding='application/x-tex'>f_!</annotation></semantics></math> are just given by composition. The right adjoints <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>f_*</annotation></semantics></math> exist iff <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is <a class='existingWikiWord' href='/nlab/show/diff/locally+cartesian+closed+category'>locally cartesian closed</a>. The term “cartesian” is motivated by this example, which is usually called the <strong><a class='existingWikiWord' href='/nlab/show/diff/codomain+fibration'>codomain fibration</a></strong> over <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>. \end{example}</p> <p>\begin{example} \label{SimpleFibration} Let <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> be a category with binary <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>cartesian products</a>. Then for each object <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Γ</mi></mrow><annotation encoding='application/x-tex'>\Gamma</annotation></semantics></math>, we can construct an indexed comonad <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>→</mo><mi>Comonad</mi></mrow><annotation encoding='application/x-tex'>C \to Comonad</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/diff/coreader+comonad'>environment comonad</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Γ</mi><mo>×</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo></mrow><annotation encoding='application/x-tex'>\Gamma \times -</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, and taking <a class='existingWikiWord' href='/nlab/show/diff/Kleisli+category'>Kleisli categories</a> we get a functor <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>C^{op} \to Cat</annotation></semantics></math>. The corresponding fibration is called the <span class='newWikiWord'>simple fibration<a href='/nlab/new/simple+fibration'>?</a></span> (e.g., in <a href='#Jacobs98'>Jacobs 1998</a>) over <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> because it can be used to give semantics to simply-typed <a class='existingWikiWord' href='/nlab/show/diff/lambda-calculus'>lambda calculus</a>. The simple fibration can be seen as a full subfibration of the <a class='existingWikiWord' href='/nlab/show/diff/codomain+fibration'>codomain fibration</a>, with objects being the <a class='existingWikiWord' href='/nlab/show/diff/projection'>projections</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Γ</mi><mo>×</mo><mi>A</mi><mo>→</mo><mi>Γ</mi></mrow><annotation encoding='application/x-tex'>\Gamma \times A \to \Gamma</annotation></semantics></math>. \end{example}</p> <p>\begin{example}\label{FreeCoproductCompletion} <strong>(<a class='existingWikiWord' href='/nlab/show/diff/family'>families</a>, <a class='existingWikiWord' href='/nlab/show/diff/free+coproduct+completion'>free coproduct completion</a>)</strong> \linebreak Let <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> be any category and let <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Fam</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Fam(C)</annotation></semantics></math> be the category of set-indexed <a class='existingWikiWord' href='/nlab/show/diff/family'>families</a> of objects of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> (the <a class='existingWikiWord' href='/nlab/show/diff/free+coproduct+completion'>free coproduct completion</a> of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, which is a <a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a> by the discussion <a href='free+coproduct+completion#AsAGrothendieckConstruction'>there</a>). The forgetful functor <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Fam</mi><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Fam(C)\to Set</annotation></semantics></math> taking a family to its indexing set is a fibration; the functors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>f^*</annotation></semantics></math> are given by reindexing. They have left adjoints iff <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> has small coproducts, and right adjoints iff <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> has small products. \end{example}</p> <h2 id='properties'>Properties</h2> <ul> <li> <p>It is easy to verify that fibrations are closed under <a class='existingWikiWord' href='/nlab/show/diff/pullback'>pullback</a> in <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a>, and that the composite of fibrations is a fibration. This latter property is notably difficult to even express in the language of pseudofunctors.</p> </li> <li> <p>Every fibration (or opfibration) is a <a class='existingWikiWord' href='/nlab/show/diff/exponentiable+functor'>Conduché functor</a>, and therefore an <a class='existingWikiWord' href='/nlab/show/diff/exponential+object'>exponentiable morphism</a> in <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a>.</p> </li> <li> <p>Every fibration or opfibration is an <a class='existingWikiWord' href='/nlab/show/diff/isofibration'>isofibration</a>. In particular, <a class='existingWikiWord' href='/nlab/show/diff/strict+2-limit'>strict 2-pullbacks</a> of fibrations are also <a class='existingWikiWord' href='/nlab/show/diff/2-pullback'>2-pullbacks</a>.</p> </li> <li id='FiberedLimit'> <p>Suppose that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p\colon A\to B</annotation></semantics></math> is a fibration, and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> has <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math>-indexed <a class='existingWikiWord' href='/nlab/show/diff/limit'>limits</a>. Then <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> has <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math>-indexed limits and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> strictly preserves them if and only if <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> has <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi></mrow><annotation encoding='application/x-tex'>I</annotation></semantics></math>-indexed <a class='existingWikiWord' href='/nlab/show/diff/fibered+limit'>fibred limits</a>. This is in <a href='#Gray'>Gray 66</a>; for a general perspective, see <a href='#Hermida'>Hermida 99, Corollary 4.9</a>. Concretely, for instance, from right to left, given a <a class='existingWikiWord' href='/nlab/show/diff/diagram'>diagram</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>I</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>f\colon I\to A</annotation></semantics></math>, let <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>L</mi></mrow><annotation encoding='application/x-tex'>L</annotation></semantics></math> be the limit of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>I</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p f\colon I\to B</annotation></semantics></math>, with projections <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>L</mi><mo>→</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\phi_i\colon L \to p(f(i))</annotation></semantics></math>. Then for each <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding='application/x-tex'>i\in I</annotation></semantics></math>, let <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>)</mo><mo>=</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mo>*</mo></msubsup><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>i</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>∈</mo><msup><mi>p</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>g(i) = \phi_i^*(f(i)) \in p^{-1}(L)</annotation></semantics></math>; these objects form a diagram <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo lspace='verythinmathspace'>:</mo><mi>I</mi><mo>→</mo><msup><mi>p</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>L</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>g\colon I\to p^{-1}(L)</annotation></semantics></math> whose limit is the limit of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math>.</p> </li> <li> <p>Dually, if <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p\colon A\to B</annotation></semantics></math> is an opfibration, then <a class='existingWikiWord' href='/nlab/show/diff/colimit'>colimits</a> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> can be constructed out of those in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> and in the fiber categories.</p> </li> <li> <p>Suppose that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p\colon A\to B</annotation></semantics></math> is a fibred <a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+category'>cartesian closed category</a> with <span class='newWikiWord'>simple products<a href='/nlab/new/simple+products'>?</a></span>. Then if <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> is cartesian closed, so is <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_174' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_175' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> preserves the structure. (See <a href='#Hermida'>Hermida 99, Corollary 4.12</a>.) But the converse does not hold, see Section 4.3.2 of <a href='#HermidaThesis'>Hermida’s thesis</a>.</p> </li> <li> <p>If <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_176' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p\colon A\to B</annotation></semantics></math> is a fibration and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_177' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> admits an <a class='existingWikiWord' href='/nlab/show/diff/orthogonal+factorization+system'>orthogonal factorization system</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_178' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>E</mi><mo>,</mo><mi>M</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(E,M)</annotation></semantics></math>, then there is a factorization system <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_179' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>E</mi><mo>′</mo><mo>,</mo><mi>M</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(E',M')</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_180' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_181' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>M'</annotation></semantics></math> is the class of cartesian arrows whose image in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_182' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> lies in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_183' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_184' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>E'</annotation></semantics></math> is the class of all arrows whose image in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_185' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> lies in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_186' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>. A dual construction is possible if <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_187' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> is an opfibration. If it is a bifibration (or more generally, an <span class='newWikiWord'>ambifibration<a href='/nlab/new/ambifibration'>?</a></span> over <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_188' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>E</mi><mo>,</mo><mi>M</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(E,M)</annotation></semantics></math>), then these together form a <a class='existingWikiWord' href='/nlab/show/diff/ternary+factorization+system'>ternary factorization system</a>.</p> </li> <li> <p>Under suitable hypotheses, versions of the preceding fact can be shown <a class='existingWikiWord' href='/nlab/show/diff/weak+factorization+system'>weak factorization systems</a> and <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model structures</a> as well.</p> </li> <li> <p>Generalizing in a different direction, if <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_189' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p\colon A\to B</annotation></semantics></math> is a fibration and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_190' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>E</mi><mo>,</mo><mi>M</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(E,M)</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/factorization+structure'>factorization structure for sinks</a> on <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_191' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_192' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> admits a factorization structure for sinks <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_193' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>E</mi><mo>′</mo><mo>,</mo><mi>M</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(E',M')</annotation></semantics></math>, where <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_194' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>M'</annotation></semantics></math> is the class of cartesian arrows whose image in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_195' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> lies in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_196' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>M</mi></mrow><annotation encoding='application/x-tex'>M</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_197' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>′</mo></mrow><annotation encoding='application/x-tex'>E'</annotation></semantics></math> is the class of all arrows whose image in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_198' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> lies in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_199' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>. Similarly, we can lift factorization structures for <em>cosinks</em> along an opfibration. To lift in the “opposite” way we require more of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_200' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math>; see <a class='existingWikiWord' href='/nlab/show/diff/topological+concrete+category'>topological concrete category</a>.</p> </li> </ul> <h2 id='variations'>Variations</h2> <h3 id='discrete_and_groupoidal_fibrations'>Discrete and groupoidal fibrations</h3> <p>One important special case of a fibration is when each fiber is a <a class='existingWikiWord' href='/nlab/show/diff/groupoid'>groupoid</a>; these correspond to pseudofunctors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_201' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>B</mi> <mi>op</mi></msup><mo>→</mo><mi>Grpd</mi></mrow><annotation encoding='application/x-tex'>B^{op}\to Grpd</annotation></semantics></math>. These are also called <em>categories fibered in groupoids</em>. A fibration <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_202' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>E\to B</annotation></semantics></math> is fibered in groupoids precisely when <em>every</em> morphism in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_203' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> is cartesian.</p> <p>Another important special case is when each fiber is a <a class='existingWikiWord' href='/nlab/show/diff/discrete+category'>discrete category</a>; these correspond to functors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_204' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>B</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>B^{op}\to Set</annotation></semantics></math>. These are also called <em><a class='existingWikiWord' href='/nlab/show/diff/discrete+fibration'>discrete fibrations</a></em>. A functor <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_205' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo lspace='verythinmathspace'>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p\colon E\to B</annotation></semantics></math> is a discrete fibration precisely when for every <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_206' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>∈</mo><mi>E</mi></mrow><annotation encoding='application/x-tex'>e\in E</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_207' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>b</mi><mo>→</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>e</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f\colon b\to p(e)</annotation></semantics></math> there is a <em>unique</em> lifting of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_208' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> to a morphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_209' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>′</mo><mo>→</mo><mi>e</mi></mrow><annotation encoding='application/x-tex'>e'\to e</annotation></semantics></math>.</p> <h3 id='opfibrations_and_bifibrations'>Opfibrations and bifibrations</h3> <p>We say that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_210' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo lspace='verythinmathspace'>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p\colon E\to B</annotation></semantics></math> is an <strong>opfibration</strong> if <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_211' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>p</mi> <mi>op</mi></msup><mo>:</mo><msup><mi>E</mi> <mi>op</mi></msup><mo>→</mo><msup><mi>B</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>p^{op}:E^{op}\to B^{op}</annotation></semantics></math> is a fibration. These correspond to covariant pseudofunctors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_212' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>B\to Cat</annotation></semantics></math>. A functor that is both a fibration and an opfibration is called a <strong><a class='existingWikiWord' href='/nlab/show/diff/bifibration'>bifibration</a></strong>. It is not hard to see that a fibration is a bifibration iff each functor <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_213' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>f^*</annotation></semantics></math> has a left adjoint, written <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_214' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mo>!</mo></msub></mrow><annotation encoding='application/x-tex'>f_!</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_215' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Σ</mi> <mi>f</mi></msub></mrow><annotation encoding='application/x-tex'>\Sigma_f</annotation></semantics></math>. In many cases <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_216' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>f^*</annotation></semantics></math> also has a right adjoint, written <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_217' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow><annotation encoding='application/x-tex'>f_*</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_218' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Π</mi> <mi>f</mi></msub></mrow><annotation encoding='application/x-tex'>\Pi_f</annotation></semantics></math>, but this is not as easily expressible in fibrational language.</p> <p>Grothendieck originally called an opfibration a <em>cofibered category</em>, and if the fibers are groupoids a <a class='existingWikiWord' href='/nlab/show/diff/fibration+fibered+in+groupoids'>category cofibered in groupoids</a> (cf. <a class='existingWikiWord' href='/nlab/show/diff/SGA1'>SGA I</a>, <a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>Higher Topos Theory</a>). However, that term has fallen out of favor in the homotopy-theory and category-theory communities (though still used sometimes in algebraic geometry), because an opfibration still has a <em>lifting</em> property, as is characteristic of other notions of <a class='existingWikiWord' href='/nlab/show/diff/fibration'>fibration</a>, as opposed to the <em>extension</em> property exhibited by <a class='existingWikiWord' href='/nlab/show/diff/cofibration'>cofibrations</a> in <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a>.</p> <p>Note that an opfibration is the same as an internal fibration in the 2-category <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_219' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Cat</mi> <mi>co</mi></msup></mrow><annotation encoding='application/x-tex'>Cat^{co}</annotation></semantics></math>, while it is the fibrations in the 2-category <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_220' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Cat</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>Cat^{op}</annotation></semantics></math> which are more deserving of the name “cofibration.”</p> <p>Note also that a given pseudofunctor <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_221' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>B</mi> <mi>op</mi></msup><mo>→</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>B^{op}\to Cat</annotation></semantics></math> can be represented both by a fibration <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_222' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>E_1\to B</annotation></semantics></math> and by an opfibration <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_223' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub><mo>→</mo><msup><mi>B</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>E_2\to B^{op}</annotation></semantics></math>. However <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_224' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>E_2</annotation></semantics></math> is <em>not</em> the opposite category of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_225' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>E_1</annotation></semantics></math>.</p> <h3 id='StreetFibration'>Version respecting the Principle of equivalence</h3> <p>There is something apparently in violation of the <a class='existingWikiWord' href='/nlab/show/diff/principle+of+equivalence'>principle of equivalence</a> about the notion of fibration, namely the requirement that for every <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_226' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>f:a\to b</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_227' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>∈</mo><msub><mi>E</mi> <mi>b</mi></msub></mrow><annotation encoding='application/x-tex'>e\in E_b</annotation></semantics></math> there exists a <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_228' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>e</mi><mo>′</mo><mo>→</mo><mi>e</mi></mrow><annotation encoding='application/x-tex'>\phi:e'\to e</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_229' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo stretchy='false'>(</mo><mi>e</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>p(e')</annotation></semantics></math> is <em><a class='existingWikiWord' href='/nlab/show/diff/equality'>equal</a></em>, rather than merely <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphic</a>, to <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_230' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math>. This is connected with the fact that we use strict fibers, rather than <a class='existingWikiWord' href='/nlab/show/diff/essential+fiber'>essential fiber</a>s, and that fibrations and pseudofunctors can be recovered from each other up to isomorphism rather than merely equivalence.</p> <p>Note that almost any fibration between “concrete” categories that arises in practice does satisfy this strict property. However, even stating the strict property requires our categories to be <a class='existingWikiWord' href='/nlab/show/diff/strict+category'>strict categories</a> (i.e. to have a notion of equality of objects), so when working in a context where not all categories are strict (such as <a class='existingWikiWord' href='/nlab/show/diff/univalent+category'>internal categories in homotopy type theory</a>, or with objects in a <a class='existingWikiWord' href='/nlab/show/diff/bicategory'>bicategory</a>) it is problematic. Sometimes (such as in type theory) this can be avoided by working with <a class='existingWikiWord' href='/nlab/show/diff/displayed+category'>displayed categories</a> instead, but in some cases (such as internally in a bicategory) one does not have classifying objects either, so it is sometimes useful to have a version which manifestly accords to the <a class='existingWikiWord' href='/nlab/show/diff/principle+of+equivalence'>principle of equivalence</a>.</p> <p>The correct modification, first given by <a class='existingWikiWord' href='/nlab/show/diff/Ross+Street'>Ross Street</a>, is simply to require that for any <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_231' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>f:a\to b</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_232' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>∈</mo><msub><mi>E</mi> <mi>b</mi></msub></mrow><annotation encoding='application/x-tex'>e\in E_b</annotation></semantics></math> there exists a cartesian <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_233' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>e</mi><mo>′</mo><mo>→</mo><mi>e</mi></mrow><annotation encoding='application/x-tex'>\phi:e'\to e</annotation></semantics></math> and an <em>isomorphism</em> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_234' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>h</mi><mo>:</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>e</mi><mo>′</mo><mo stretchy='false'>)</mo><mo>≅</mo><mi>a</mi></mrow><annotation encoding='application/x-tex'>h:p(e') \cong a</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_235' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>∘</mo><mi>h</mi><mo>=</mo><mi>p</mi><mo stretchy='false'>(</mo><mi>ϕ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f\circ h = p(\phi)</annotation></semantics></math>; the definition of “cartesian” is unchanged; this gives the notion of <a class='existingWikiWord' href='/nlab/show/diff/Street+fibration'>Street fibration</a>. Every <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalence of categories</a> is a Street fibration, which is not true for the concept of Grothendieck fibrations according to the <a class='existingWikiWord' href='/nlab/show/diff/principle+of+equivalence'>principle of equivalence</a>, but every Street fibration can be factored as an equivalence of categories followed by a Grothendieck fibration.</p> <p>We might also think that it violates the <a class='existingWikiWord' href='/nlab/show/diff/principle+of+equivalence'>principle of equivalence</a> to say that the target of the cartesian arrow <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_236' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding='application/x-tex'>\phi</annotation></semantics></math> is equal to the given object <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_237' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi></mrow><annotation encoding='application/x-tex'>e</annotation></semantics></math>, akin to the topological distinction between <a class='existingWikiWord' href='/nlab/show/diff/Serre+fibration'>Serre fibrations</a> and <a class='existingWikiWord' href='/nlab/show/diff/Dold+fibration'>Dold fibrations</a>, where the initial point of a lifted path can only be specified up to homotopy. However, this part of the definition is really better regarded as a typing assertion, akin to saying, in the definition of a <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> of two objects <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_238' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_239' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, that the target of the two projections <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_240' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>A\times B\to A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_241' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>×</mo><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>A\times B \to B</annotation></semantics></math> are <em>equal</em> to <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_242' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_243' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>. Moreover, any weakening along these lines would actually be equivalent to the version above: if we demand only that for any <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_244' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>a</mi><mo>→</mo><mi>b</mi></mrow><annotation encoding='application/x-tex'>f\colon a\to b</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_245' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_246' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mo>∈</mo><msub><mi>E</mi> <mi>b</mi></msub></mrow><annotation encoding='application/x-tex'>e\in E_b</annotation></semantics></math>, there exists a cartesian <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_247' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi><mo lspace='verythinmathspace'>:</mo><mi>e</mi><mo>′</mo><mo>→</mo><mover><mi>e</mi><mo stretchy='false'>^</mo></mover></mrow><annotation encoding='application/x-tex'>\phi\colon e' \to \hat{e}</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_248' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo stretchy='false'>(</mo><mi>ϕ</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>p(\phi)=f</annotation></semantics></math> and an isomorphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_249' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>e</mi><mo stretchy='false'>^</mo></mover><mo>≅</mo><mi>e</mi></mrow><annotation encoding='application/x-tex'>\hat{e}\cong e</annotation></semantics></math> in the fiber <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_250' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mi>b</mi></msub></mrow><annotation encoding='application/x-tex'>E_b</annotation></semantics></math>, then you can just compose <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_251' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding='application/x-tex'>\phi</annotation></semantics></math> with the isomorphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_252' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>e</mi><mo stretchy='false'>^</mo></mover><mo>≅</mo><mi>e</mi></mrow><annotation encoding='application/x-tex'>\hat{e}\cong e</annotation></semantics></math> to get a cartesian arrow <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_253' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mi>ϕ</mi><mo stretchy='false'>^</mo></mover><mo lspace='verythinmathspace'>:</mo><mi>e</mi><mo>′</mo><mo>→</mo><mi>e</mi></mrow><annotation encoding='application/x-tex'>\hat{\phi}\colon e'\to e</annotation></semantics></math> with <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_254' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo stretchy='false'>(</mo><mi>ϕ</mi><mo stretchy='false'>)</mo><mo>=</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>p(\phi)=f</annotation></semantics></math> still. The reason this doesn’t work in topology is that paths come with parametrizations, and requiring the lower triangle (in the square drawn at <a class='existingWikiWord' href='/nlab/show/diff/Dold+fibration'>Dold fibration</a>) to commute strictly prevents the reparametrization necessary to compose with a vertical homotopy.</p> <p>The idea of <a class='existingWikiWord' href='/nlab/show/diff/proto-fibration'>proto-fibration</a> is closely related to this.</p> <h3 id='internal_version'>Internal version</h3> <p>In a <a class='existingWikiWord' href='/nlab/show/diff/strict+2-category'>strict 2-category</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_255' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>, a morphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_256' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p:E\to B</annotation></semantics></math> is called a fibration if for every object <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_257' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_258' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>K</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>K(X,E)\to K(X,B)</annotation></semantics></math> is a fibration of categories, and for every morphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_259' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>f:Y\to X</annotation></semantics></math>, the square</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_260' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>K</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>E</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>K</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>E</mi><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>K</mi><mo stretchy='false'>(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>K</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{ K(X,E) & \to & K(Y,E) \\ \downarrow && \downarrow \\ K(X,B) & \to & K(Y,B)}</annotation></semantics></math></div> <p>is a morphism of fibrations. There is an alternate characterization in terms of <a class='existingWikiWord' href='/nlab/show/diff/comma+object'>comma object</a>s and adjoints, see <a class='existingWikiWord' href='/nlab/show/diff/fibration+in+a+2-category'>fibration in a 2-category</a>. The same definition works in a <a class='existingWikiWord' href='/nlab/show/diff/bicategory'>bicategory</a>, as long as we use the version in accord with the <a class='existingWikiWord' href='/nlab/show/diff/principle+of+equivalence'>principle of equivalence</a> above. Interpreted in <a class='existingWikiWord' href='/nlab/show/diff/Cat'>Cat</a> we obtain the explicit notion we started with.</p> <h3 id='twosided_version'>Two-sided version</h3> <p>A <strong>two-sided fibration</strong>, or a <strong>fibration from <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_261' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_262' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math></strong>, is a span <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_263' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>←</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>A \leftarrow E \rightarrow B</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_264' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>E\to A</annotation></semantics></math> is a fibration, <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_265' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>E\to B</annotation></semantics></math> is an opfibration, and the structure “commutes” in a natural way. Such two-sided fibrations correspond to pseudofunctors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_266' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>A</mi> <mi>op</mi></msup><mo>×</mo><mi>B</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>A^{op}\times B \to Cat</annotation></semantics></math>. If they are discrete, they correspond to functors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_267' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>A</mi> <mi>op</mi></msup><mo>×</mo><mi>B</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding='application/x-tex'>A^{op}\times B\to Set</annotation></semantics></math>, i.e. to <a class='existingWikiWord' href='/nlab/show/diff/profunctor'>profunctors</a> from <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_268' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_269' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>.</p> <p>See <a class='existingWikiWord' href='/nlab/show/diff/two-sided+fibration'>two-sided fibration</a> for more details.</p> <p>Note that a pseudofunctor <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_270' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>A</mi> <mi>op</mi></msup><mo>×</mo><mi>B</mi><mo>→</mo><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>A^{op}\times B \to Cat</annotation></semantics></math> can also be represented by an opfibration <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_271' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><mo>→</mo><msup><mi>A</mi> <mi>op</mi></msup><mo>×</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>E_1\to A^{op}\times B</annotation></semantics></math> and by a fibration <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_272' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub><mo>→</mo><mi>A</mi><mo>×</mo><msup><mi>B</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>E_2\to A\times B^{op}</annotation></semantics></math>, but there is no simple relationship between the three categories <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_273' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_274' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>E_1</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_275' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>E_2</annotation></semantics></math>.</p> <h3 id='higher_categorical_versions'>Higher categorical versions</h3> <p>There is also a notion of fibration for 2-categories that has been studied by Hermida. See <a class='existingWikiWord' href='/nlab/show/diff/n-fibration'>n-fibration</a> for a general version.</p> <p>For <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-categories</a> the notion of fibered category is modeled by the notion of <a class='existingWikiWord' href='/nlab/show/diff/Cartesian+fibration'>Cartesian fibration</a> of simplicial sets. The corresponding analog of the Grothendieck construction is discussed at <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-Grothendieck+construction'>(∞,1)-Grothendieck construction</a>.</p> <h3 id='versions_for_other_categorical_structures'>Versions for other categorical structures</h3> <ul> <li> <p>There is an analog for <a class='existingWikiWord' href='/nlab/show/diff/multicategory'>multicategories</a>. See <a class='existingWikiWord' href='/nlab/show/diff/fibration+of+multicategories'>fibration of multicategories</a>.</p> </li> <li> <p>For higher multicategories, see <a class='existingWikiWord' href='/nlab/show/diff/Cartesian+fibration+of+dendroidal+sets'>Cartesian fibration of dendroidal sets</a>.</p> </li> <li> <p>For <a class='existingWikiWord' href='/nlab/show/diff/double+category'>double categories</a>, see <a class='existingWikiWord' href='/nlab/show/diff/double+fibration'>double fibration</a>.</p> </li> </ul> <h2 id='alternate_definitions'>Alternate definitions</h2> <p>There are several alternate characterizations of when a functor is a fibration, some of which are more convenient for <a class='existingWikiWord' href='/nlab/show/diff/fibration+in+a+2-category'>internalization</a>. Here we mention a few.</p> <h3 id='in_terms_of_adjoints'>In terms of adjoints</h3> <p>Since Grothendieck fibrations are a <a class='existingWikiWord' href='/nlab/show/diff/strict+category'>strict</a> notion, in what follows we denote by <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_276' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo stretchy='false'>↓</mo><mi>p</mi></mrow><annotation encoding='application/x-tex'>B\downarrow p</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/strict+category'>strict</a> <a class='existingWikiWord' href='/nlab/show/diff/comma+category'>comma category</a> (i.e. determined up to <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a>, not merely up to <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalence</a>) and by <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_277' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi><mo stretchy='false'>/</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>Cat/B</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/slice+2-category'>strict slice 2-category</a>.</p> <div class='un_lemma'> <h6 id='lemma'>Lemma</h6> <p>A functor <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_278' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo lspace='verythinmathspace'>:</mo><mi>E</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>p \colon E \to B</annotation></semantics></math> is a cloven fibration if and only if the canonical functor <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_279' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo lspace='verythinmathspace'>:</mo><mi>E</mi><mo>→</mo><mi>B</mi><mo stretchy='false'>↓</mo><mi>p</mi></mrow><annotation encoding='application/x-tex'>i \colon E \to B\downarrow p</annotation></semantics></math> has a right adjoint <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_280' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>r</mi></mrow><annotation encoding='application/x-tex'>r</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_281' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi><mo stretchy='false'>/</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>Cat / B</annotation></semantics></math>.</p> </div> <div class='proof'> <h6 id='proof'>Proof</h6> <p>First, recall that the strict slice 2-category <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_282' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi><mo stretchy='false'>/</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>Cat/X</annotation></semantics></math> has objects the functors <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_283' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding='application/x-tex'>C \to X</annotation></semantics></math>, as morphisms the commuting triangles</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_284' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>C</mi></mtd> <mtd><mover><mo>→</mo><mi>h</mi></mover></mtd> <mtd><mi>C</mi><mo>′</mo></mtd></mtr> <mtr><mtd /> <mtd><mi>f</mi><mo>↘</mo><mo>↙</mo><mi>g</mi></mtd> <mtd /></mtr> <mtr><mtd /> <mtd><mi>X</mi><mo>,</mo></mtd> <mtd /></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'>\array{C & \stackrel{h}{\to} & C' \\ & f \searrow \swarrow g & \\ & X, & }</annotation></semantics></math></div> <p>and as 2-cells the natural transformations <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_285' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>α</mi><mo>:</mo><msub><mi>h</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>h</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>\alpha : h_1 \to h_2</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_286' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mi>α</mi><mo>=</mo><msub><mi>id</mi> <mi>f</mi></msub></mrow><annotation encoding='application/x-tex'>g\alpha = id_f</annotation></semantics></math>.</p> <p>Next, recall that the <a class='existingWikiWord' href='/nlab/show/diff/comma+category'>comma category</a> <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_287' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo stretchy='false'>↓</mo><mi>p</mi></mrow><annotation encoding='application/x-tex'>B\downarrow p</annotation></semantics></math> has objects the triples <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_288' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(x, e, k)</annotation></semantics></math>, with <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_289' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo lspace='verythinmathspace'>:</mo><mi>x</mi><mo>→</mo><mi>p</mi><mi>e</mi></mrow><annotation encoding='application/x-tex'>k \colon x \to p e</annotation></semantics></math>. Let <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_290' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>π</mi><mo lspace='verythinmathspace'>:</mo><mi>B</mi><mo stretchy='false'>↓</mo><mi>p</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>\pi \colon B\downarrow p \to B</annotation></semantics></math> denote the projection <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_291' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>x</mi><mo>,</mo><mi>e</mi><mo>,</mo><mi>k</mi><mo stretchy='false'>)</mo><mo>↦</mo><mi>x</mi></mrow><annotation encoding='application/x-tex'>(x, e, k) \mapsto x</annotation></semantics></math>.</p> <p>The canonical morphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_292' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>:</mo><mi>E</mi><mo>→</mo><mi>B</mi><mo stretchy='false'>↓</mo><mi>p</mi></mrow><annotation encoding='application/x-tex'>i:E \to B\downarrow p</annotation></semantics></math> is simply the inclusion functor of identity maps <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_293' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mi>e</mi><mo>=</mo><msub><mn>1</mn> <mrow><mi>p</mi><mi>e</mi></mrow></msub><mo lspace='verythinmathspace'>:</mo><mi>p</mi><mi>e</mi><mo>→</mo><mi>p</mi><mi>e</mi></mrow><annotation encoding='application/x-tex'>i e = 1_{p e} \colon p e \to p e</annotation></semantics></math>.</p> <p>Somewhat imprecisely, seeing both categories <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_294' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_295' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo stretchy='false'>↓</mo><mi>p</mi></mrow><annotation encoding='application/x-tex'>B\downarrow p</annotation></semantics></math> as sitting over <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_296' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> means that functors between those should be the identity on the <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_297' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math> component, and natural transformations should have the identity as their <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_298' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>b</mi></mrow><annotation encoding='application/x-tex'>b</annotation></semantics></math> component.</p> <p>To give an adjunction <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_299' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>⊣</mo><mi>r</mi></mrow><annotation encoding='application/x-tex'>i \dashv r</annotation></semantics></math> it suffices to give, for each <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_300' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo lspace='verythinmathspace'>:</mo><mi>x</mi><mo>→</mo><mi>p</mi><mi>e</mi></mrow><annotation encoding='application/x-tex'>k \colon x \to p e</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_301' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo stretchy='false'>↓</mo><mi>p</mi></mrow><annotation encoding='application/x-tex'>B\downarrow p</annotation></semantics></math>, an object <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_302' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>r</mi><mi>k</mi></mrow><annotation encoding='application/x-tex'>r k</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_303' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_304' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mi>r</mi><mi>k</mi><mo>=</mo><mi>x</mi></mrow><annotation encoding='application/x-tex'>p r k = x</annotation></semantics></math> and an arrow <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_305' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mi>r</mi><mi>k</mi><mo>=</mo><msub><mn>1</mn> <mi>x</mi></msub><mo>→</mo><mi>k</mi></mrow><annotation encoding='application/x-tex'>i r k = 1_x \to k</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_306' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo stretchy='false'>↓</mo><mi>p</mi></mrow><annotation encoding='application/x-tex'>B\downarrow p</annotation></semantics></math> that is <a class='existingWikiWord' href='/nlab/show/diff/reflection+along+a+functor'>universal</a> from <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_307' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_308' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>. For the adjunction to live in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_309' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi><mo stretchy='false'>/</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>Cat / B</annotation></semantics></math> we must have that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_310' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>π</mi><mo>∘</mo><mi>i</mi><mi>r</mi><mi>k</mi><mo>=</mo><msub><mn>1</mn> <mrow><mi>p</mi><mi>r</mi><mi>k</mi></mrow></msub><mo>=</mo><msub><mn>1</mn> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>\pi \circ i r k = 1_{p r k} = 1_x</annotation></semantics></math>, so the universal arrow must be of the form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_311' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>x</mi></mtd> <mtd><mover><mo>→</mo><mn>1</mn></mover></mtd> <mtd><mi>x</mi></mtd></mtr> <mtr><mtd><mpadded lspace='-100%width' width='0'><mn>1</mn></mpadded><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo><mpadded width='0'><mrow><mi>p</mi><msub><mi>ϵ</mi> <mi>k</mi></msub></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><munder><mo>→</mo><mi>k</mi></munder></mtd> <mtd><mi>p</mi><mi>e</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ x & \overset{1}{\to} & x \\ \mathllap{1} \downarrow & & \downarrow \mathrlap{p \epsilon_k} \\ x & \underset{k}{\to} & p e } </annotation></semantics></math></div> <p>and thus amounts to a choice of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_312' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϵ</mi> <mi>k</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>r</mi><mi>k</mi><mo>→</mo><mi>e</mi></mrow><annotation encoding='application/x-tex'>\epsilon_k \colon r k \to e</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_313' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>E</mi></mrow><annotation encoding='application/x-tex'>E</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_314' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><msub><mi>ϵ</mi> <mi>k</mi></msub><mo>=</mo><mi>k</mi></mrow><annotation encoding='application/x-tex'>p \epsilon_k = k</annotation></semantics></math>.</p> <p>The universal property of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_315' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϵ</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>\epsilon_k</annotation></semantics></math> tells us that for any other morphism in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_316' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo stretchy='false'>↓</mo><mi>p</mi></mrow><annotation encoding='application/x-tex'>B\downarrow p</annotation></semantics></math> from some <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_317' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mi>y</mi></mrow><annotation encoding='application/x-tex'>i y</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_318' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>, i.e., for any <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_319' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>y</mi></mrow><annotation encoding='application/x-tex'>y</annotation></semantics></math> and any pair <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_320' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f,g)</annotation></semantics></math> making the square</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_321' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>p</mi><mi>y</mi></mtd> <mtd><mover><mo>→</mo><mn>1</mn></mover></mtd> <mtd><mi>p</mi><mi>y</mi></mtd></mtr> <mtr><mtd><mpadded lspace='-100%width' width='0'><mi>f</mi></mpadded><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo><mpadded width='0'><mrow><mi>p</mi><mi>g</mi></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><munder><mo>→</mo><mi>k</mi></munder></mtd> <mtd><mi>p</mi><mi>e</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ p y & \stackrel{1}{\to} & p y \\ \mathllap{f} \downarrow & & \downarrow \mathrlap{p g} \\ x & \underset{k}{\to} & p e } </annotation></semantics></math></div> <p>commute, there is a unique map <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_322' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>h</mi><mo lspace='verythinmathspace'>:</mo><mi>y</mi><mo>→</mo><mi>r</mi><mi>k</mi></mrow><annotation encoding='application/x-tex'>h \colon y \to r k</annotation></semantics></math> in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_323' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> such that the above square factors in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_324' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mo stretchy='false'>↓</mo><mi>p</mi></mrow><annotation encoding='application/x-tex'>B\downarrow p</annotation></semantics></math> as</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_325' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>p</mi><mi>y</mi></mtd> <mtd><mover><mo>→</mo><mn>1</mn></mover></mtd> <mtd><mi>p</mi><mi>y</mi></mtd></mtr> <mtr><mtd><mpadded lspace='-100%width' width='0'><mrow><mi>p</mi><mi>h</mi></mrow></mpadded><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo><mpadded width='0'><mrow><mi>p</mi><mi>h</mi></mrow></mpadded></mtd></mtr> <mtr><mtd><mpadded lspace='-100%width' width='0'><mrow><mi>p</mi><mi>r</mi><mi>k</mi><mo>=</mo></mrow></mpadded><mi>x</mi></mtd> <mtd><mover><mo>→</mo><mn>1</mn></mover></mtd> <mtd><mi>x</mi><mpadded width='0'><mrow><mo>=</mo><mi>p</mi><mi>r</mi><mi>k</mi></mrow></mpadded></mtd></mtr> <mtr><mtd><mpadded lspace='-100%width' width='0'><mn>1</mn></mpadded><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd><mo stretchy='false'>↓</mo><mpadded width='0'><mrow><mi>p</mi><msub><mi>ϵ</mi> <mi>k</mi></msub></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>x</mi></mtd> <mtd><munder><mo>→</mo><mi>k</mi></munder></mtd> <mtd><mi>p</mi><mi>e</mi><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ p y & \stackrel{1}{\to} & p y \\ \mathllap{p h} \downarrow & & \downarrow \mathrlap{p h} \\ \mathllap{p r k =} x & \stackrel{1}{\to} & x \mathrlap{= p r k}\\ \mathllap{1} \downarrow & & \downarrow \mathrlap{p \epsilon_k} \\ x & \underset{k}{\to} & p e. } </annotation></semantics></math></div> <p>In other words, the universal property provides a unique <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_326' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>h</mi></mrow><annotation encoding='application/x-tex'>h</annotation></semantics></math> such that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_327' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϵ</mi> <mi>k</mi></msub><mi>h</mi><mo>=</mo><mi>g</mi></mrow><annotation encoding='application/x-tex'>\epsilon_k h = g</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_328' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mi>h</mi><mo>=</mo><mi>f</mi></mrow><annotation encoding='application/x-tex'>p h = f</annotation></semantics></math>, which exactly asserts that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_329' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϵ</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>\epsilon_k</annotation></semantics></math> is a cartesian lift of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_330' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>.</p> <p>So the existence of a right adjoint to <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_331' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> means precisely that for each morphism <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_332' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo lspace='verythinmathspace'>:</mo><mi>x</mi><mo>→</mo><mi>p</mi><mi>e</mi></mrow><annotation encoding='application/x-tex'>k \colon x \to p e</annotation></semantics></math> a choice is given of a cartesian lift of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_333' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>, which means in turn that <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_334' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> is a cloven fibration.</p> </div> <h3 id='in_terms_of_pseudoalgebras'>In terms of pseudoalgebras</h3> <p>Fibrations are pseudoalgebras for a <a class='existingWikiWord' href='/nlab/show/diff/lax-idempotent+2-monad'>lax-idempotent pseudomonad</a> (see there for more details).</p> <h2 id='discussions'>Discussions</h2> <p>The following discussion brings out some interesting points about the equivalence between fibrations and pseudofunctors.</p> <div class='query'> <p><a class='existingWikiWord' href='/nlab/show/diff/Sridhar+Ramesh'>Sridhar Ramesh</a>: I have a (possibly stupid) question about the nature of this equivalence. I assume the idea here is that moving from a cloven fibration to the corresponding pseudofunctor is in some sense “inverse” to carrying out the Grothendieck construction in the other direction. But, in trying to get a good intuition for the nuances of non-splittable fibrations, I seem to be stumbling upon just in what sense this is so. For example, consider the nontrivial group homomorphism from Z (integer addition) to Z_2 (integer addition modulo 2); this gives us a non-splittable fibration (and, for that matter, an opfibration), for which a cleavage can be readily selected. No matter what cleavage is selected, the corresponding (contravariant) pseudofunctor from Z_2 to Cat, it would appear to me, is the one which sends the unique object in Z_2 to the subcategory of Z containing only even integers (let us call this 2Z), and which sends both of Z_2’s morphisms to identity; thus, it is actually a genuine functor, and indeed, a “constant” functor. Applying the Grothendieck construction now, I would seem to get back the projection from Z_2 X 2Z onto Z_2. But can this really be equivalent to the fibration I started with? After all, Z and Z_2 X 2Z are very different groups. So either “equivalence” means something trickier here than I realize, or I keep making a mistake somewhere along the line. Either way, it’d be great if someone could help me see the light.</p> <p><a class='existingWikiWord' href='/nlab/show/diff/Mike+Shulman'>Mike Shulman</a>: Good question! I think the missing subtlety is that a pseudofunctor is not uniquely determined by its action on objects and morphisms, even if its domain is a mere category (or a mere group); there are also natural coherence isomorphisms <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_335' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>g</mi> <mo>*</mo></msup><msup><mi>f</mi> <mo>*</mo></msup><mo>≅</mo><mo stretchy='false'>(</mo><mi>g</mi><mi>f</mi><msup><mo stretchy='false'>)</mo> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>g^* f^* \cong (g f)^*</annotation></semantics></math> to take into account. For instance, if <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_336' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> is the nonidentity element of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_337' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/2</annotation></semantics></math>, then <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_338' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mi>g</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>g g = 1</annotation></semantics></math>, so even if <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_339' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> acts by the identity on <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_340' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>2\mathbb{Z}</annotation></semantics></math>, a pseudofunctor also contains the additional data of a natural automorphism of the identity of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_341' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>2\mathbb{Z}</annotation></semantics></math>, i.e. a (central) element of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_342' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>2\mathbb{Z}</annotation></semantics></math>. If you start from <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_343' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}</annotation></semantics></math>, then depending on your cleavage your element can be anything that’s 2 mod 4, while if you start from <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_344' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn><mo>×</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/2\times 2\mathbb{Z}</annotation></semantics></math>, your element can be anything that’s 0 mod 4. Finally, there is a pseudonatural equivalence between two such pseudofunctors just when their corresponding elements differ by a multiple of 4, so you get exactly two equivalence classes of pseudofunctors, corresponding to the two groups <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_345' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_346' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn><mo>×</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}/2\times 2\mathbb{Z}</annotation></semantics></math>. Of course we are reproducing the classification of group extensions via group cohomology.</p> <p>By the way, this sort of thing (by which I mean, the cohomology class that classifies some categorical structure arising as the trace of a coherence isomorphism) happens in lots of other places too. For instance, a <a class='existingWikiWord' href='/nlab/show/diff/2-group'>2-group</a> is classified by a group <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_347' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>, an abelian group <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_348' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>, an action of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_349' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> on <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_350' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math>, and an element in <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_351' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>H</mi> <mn>3</mn></msup><mo stretchy='false'>(</mo><mi>G</mi><mo>;</mo><mi>H</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>H^3(G;H)</annotation></semantics></math>. If you replace a 2-group by a <a class='existingWikiWord' href='/nlab/show/diff/skeleton'>skeletal</a> one, then <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_352' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> is the group of objects (which is strictly associative and unital, by skeletality), <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_353' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi></mrow><annotation encoding='application/x-tex'>H</annotation></semantics></math> is the group of endomorphisms of the unit, and the action is defined by “whiskering”. The cohomology class comes from the <a class='existingWikiWord' href='/nlab/show/diff/associator'>associator</a> isomorphism, which can (and often must) still be nontrivial even though the multiplication is “strictly associative” at the level of objects (by skeletality).</p> <p><em>Toby</em>: So the multiplication is strictly associative, but the <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_354' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>-group itself is not a strict <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_355' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn></mrow><annotation encoding='application/x-tex'>2</annotation></semantics></math>-group, since it uses a different associator than the identity. As in the example of the pseudofunctor from <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_356' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Z</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>Z_2</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_357' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Cat</mi></mrow><annotation encoding='application/x-tex'>Cat</annotation></semantics></math>, there is some additional structure here which is not trivial, even though it seems like it could be.</p> <p><a class='existingWikiWord' href='/nlab/show/diff/Sridhar+Ramesh'>Sridhar Ramesh</a>: Ah, of course, that’s what I was missing. Thanks, both of you; that clears it all up.</p> </div> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+fibration'>opfibration</a>, <a class='existingWikiWord' href='/nlab/show/diff/bifibration'>bifibration</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/two-sided+fibration'>two-sided fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+fibration'>monoidal fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/n-fibration'>fibration of 2-categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/n-fibration'>n-fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cartesian+fibration'>Cartesian fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/foliated+category'>foliated category</a></p> </li><ins class='diffins'> </ins><ins class='diffins'><li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction'>Grothendieck construction</a>, <a class='existingWikiWord' href='/nlab/show/diff/category+of+elements'>category of elements</a></p> </li></ins> </ul> <h2 id='References'>References</h2> <p>The concept was introduced in the context of <a class='existingWikiWord' href='/nlab/show/diff/descent'>descent theory</a> and in the guise now known as <a class='existingWikiWord' href='/nlab/show/diff/indexed+category'>indexed categories</a>, in</p> <ul> <li id='Grothendieck60'><a class='existingWikiWord' href='/nlab/show/diff/Alexander+Grothendieck'>Alexander Grothendieck</a>, p. 300 in: <em>Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats</em>, Séminaire <a class='existingWikiWord' href='/nlab/show/diff/Bourbaki'>N. Bourbaki</a> exp. no190 (1960) 299-327 [[numdam:SB_1958-1960__5__299_0](http://www.numdam.org/item/?id=SB_1958-1960__5__299_0)]</li> </ul> <p>and then elaborated on in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Alexander+Grothendieck'>Alexander Grothendieck</a>, exposé VI of: <em>Revêtements Etales et Groupe Fondamental - Séminaire de Géometrie Algébrique du Bois Marie 1960/61</em> (<a class='existingWikiWord' href='/nlab/show/diff/SGA1'>SGA 1</a>) , LNM <strong>224</strong> Springer (1971) [updated version with comments by M. Raynaud: arxiv.0206203]</li> </ul> <p>Another important early reference:</p> <ul> <li id='Gray'><a class='existingWikiWord' href='/nlab/show/diff/John+W.+Gray'>John W. Gray</a>, <em>Fibred and Cofibred Categories</em>, pp. 21-83 in: <a class='existingWikiWord' href='/nlab/show/diff/Samuel+Eilenberg'>S. Eilenberg</a>, <a class='existingWikiWord' href='/nlab/show/diff/David+K.+Harrison'>D. K. Harrison</a>, <a class='existingWikiWord' href='/nlab/show/diff/Saunders+Mac+Lane'>S. MacLane</a>, <a class='existingWikiWord' href='/nlab/show/diff/Helmut+R%C3%B6hrl'>H. Röhrl</a> (eds.): <em><a class='existingWikiWord' href='/nlab/show/diff/Proceedings+of+the+Conference+on+Categorical+Algebra+-+La+Jolla+1965'>Proceedings of the Conference on Categorical Algebra - La Jolla 1965</a></em>, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]</li> </ul> <p>Early development of the theory:</p> <ul> <li id='Bénabou1975'> <p><a class='existingWikiWord' href='/nlab/show/diff/Jean+B%C3%A9nabou'>Jean Bénabou</a>, <em>Fibrations petites et localement petites</em>, C. R. Acad. Sci. Paris <strong>281</strong> Série A (1975) 897-900 [[gallica](http://gallica.bnf.fr/ark:/12148/bpt6k6228235m/f171.image#)]</p> </li> <li id='Bénabou1985'> <p><a class='existingWikiWord' href='/nlab/show/diff/Jean+B%C3%A9nabou'>Jean Bénabou</a>, <em>Fibered Categories and the Foundations of Naive Category Theory</em>, The Journal of Symbolic Logic, Vol. <strong>50</strong> 1 (1985) 10-37 [[doi:10.2307/2273784](http://dx.doi.org/10.2307/2273784)]</p> </li> </ul> <p>reviewed in:</p> <ul> <li id='Streicher18'><a class='existingWikiWord' href='/nlab/show/diff/Thomas+Streicher'>Thomas Streicher</a>, <em>Fibred Categories à la Jean Bénabou</em>, [[arXiv:1801.02927](https://arxiv.org/abs/1801.02927)]</li> </ul> <p>Further review with emphasis on <a class='existingWikiWord' href='/nlab/show/diff/descent'>descent</a> and Grothendieck fibrations as incarnations of <a class='existingWikiWord' href='/nlab/show/diff/stack'>stacks</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Angelo+Vistoli'>Angelo Vistoli</a>, <em>Grothendieck topologies, fibered categories and descent theory</em> [[math.AG/0412512](http://arxiv.org/abs/math/0412512), <a href='http://www.ams.org/mathscinet-getitem?mr=2223406'>MR2223406</a>] in: Fantechi et al. (eds.), <em>Fundamental algebraic geometry. Grothendieck’s <a class='existingWikiWord' href='/nlab/show/diff/FGA+explained'>FGA explained</a></em>, Mathematical Surveys and Monographs <strong>123</strong>, Amer. Math. Soc. (2005) 1-104 [[ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), <a href='http://www.ams.org/mathscinet-getitem?mr=2007f:14001'>MR2007f:14001</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Barbara+Fantechi'>Barbara Fantechi</a>, <a class='existingWikiWord' href='/nlab/show/diff/Lothar+G%C3%B6ttsche'>Lothar Göttsche</a>, <a class='existingWikiWord' href='/nlab/show/diff/Luc+Illusie'>Luc Illusie</a>, <a class='existingWikiWord' href='/nlab/show/diff/Steven+L.+Kleiman'>Steven L. Kleiman</a>, <a class='existingWikiWord' href='/nlab/show/diff/Nitin+Nitsure'>Nitin Nitsure</a>, <a class='existingWikiWord' href='/nlab/show/diff/Angelo+Vistoli'>Angelo Vistoli</a>, Chapter 3 of: <em>Fundamental algebraic geometry. Grothendieck’s <a class='existingWikiWord' href='/nlab/show/diff/FGA+explained'>FGA explained</a></em>, Mathematical Surveys and Monographs <strong>123</strong>, Amer. Math. Soc. (2005) [[MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001), <a href='https://bookstore.ams.org/surv-123-s'>ISBN:978-0-8218-4245-4</a>, <a href='http://indico.ictp.it/event/a0255/other-view?view=ictptimetable'>lecture notes</a>]</p> </li> <li id='Warner12'> <p><a class='existingWikiWord' href='/nlab/show/diff/Garth+Warner'>Garth Warner</a>: <em>Fibrations and Sheaves</em>, EPrint Collection, University of Washington (2012) [[hdl:1773/20977](http://hdl.handle.net/1773/20977), <a href='https://sites.math.washington.edu//~warner/Warner_FIBRATIONS%20AND%20SHEAVES.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/Warner-FibrationsAndSheaves.pdf' title='pdf'>pdf</a>]</p> </li> </ul> <p>Textbook accounts:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael+Barr'>Michael Barr</a>, <a class='existingWikiWord' href='/nlab/show/diff/Charles+Wells'>Charles Wells</a>, Chapter 12 of: <em>Category Theory for Computing Science</em> , Prentice Hall 1995³. (<a href='http://www.tac.mta.ca/tac/reprints/articles/22/tr22abs.html'>TAC reprints no.22 (2012)</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Francis+Borceux'>Francis Borceux</a>, Chapter 8 of: <em><a class='existingWikiWord' href='/nlab/show/diff/Handbook+of+Categorical+Algebra'>Handbook of Categorical Algebra</a> vol. 2</em>, Cambridge UP 1994.</p> </li> <li id='Jacobs98'> <p><a class='existingWikiWord' href='/nlab/show/diff/Bart+Jacobs'>Bart Jacobs</a>, Chapters 1 and 9 in: <em>Categorical Logic and Type Theory</em>, Studies in Logic and the Foundations of Mathematics <strong>141</strong>, Elsevier (1998) [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), <a href='https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf'>pdf</a>]</p> <blockquote> <p>(in the context of <a class='existingWikiWord' href='/nlab/show/diff/categorical+semantics+of+dependent+type+theory'>categorical semantics of dependent types</a>)</p> </blockquote> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Peter+Johnstone'>Peter Johnstone</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Sketches+of+an+Elephant'>Sketches of an Elephant</a> vol.1, Oxford UP 2002. (Part B)</em></p> <blockquote> <p>(beware that this uses the non-standard terms “prone” and “supine” for “cartesian” and “opcartesian” morphisms)</p> </blockquote> </li> <li id='JohnsonYau20'> <p><a class='existingWikiWord' href='/nlab/show/diff/Niles+Johnson'>Niles Johnson</a>, <a class='existingWikiWord' href='/nlab/show/diff/Donald+Yau'>Donald Yau</a>, Chapter 9 of: <em>2-Dimensional Categories</em>, Oxford University Press 2021 (<a href='http://arxiv.org/abs/2002.06055'>arXiv:2002.06055</a>, <a href='https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378'>doi:10.1093/oso/9780198871378.001.0001</a>)</p> </li> </ul> <p>See also:</p> <ul> <li> <p>André Joyal, <em><a class='existingWikiWord' href='/joyalscatlab/published/Grothendieck+fibrations' title='joyalscatlab'>Grothendieck fibrations</a></em></p> </li> <li> <p>Wikipedia, <em><a href='http://en.wikipedia.org/wiki/Fibred_category'>Fibered category</a></em></p> </li> </ul> <p>On the <a class='existingWikiWord' href='/nlab/show/diff/2-category'>2-category</a> of fibrations and application to <a class='existingWikiWord' href='/nlab/show/diff/computer+science'>computer science</a>:</p> <ul> <li id='Hermida'> <p><a class='existingWikiWord' href='/nlab/show/diff/Claudio+Hermida'>Claudio Hermida</a>, <em>Some properties of <math class='maruku-mathml' display='inline' id='mathml_1c42d63f0194a376c013530dc1553ebaaae16e3c_358' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Fib</mi></mrow><annotation encoding='application/x-tex'>Fib</annotation></semantics></math> as a fibred 2-category</em>, Journal of Pure and Applied Algebra <strong>134</strong> (1999) 83-109 []</p> </li> <li id='HermidaThesis'> <p><a class='existingWikiWord' href='/nlab/show/diff/Claudio+Hermida'>Claudio Hermida</a>, <a href='https://era.ed.ac.uk/bitstream/handle/1842/14057/Hermida1993.Pdf'>PhD thesis, University of Edinburgh</a></p> </li> </ul> <p>A <a class='existingWikiWord' href='/nlab/show/diff/2-monad'>2-comonad</a> characterizing Grothendieck fibrations:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Jacopo+Emmenegger'>Jacopo Emmenegger</a>, <a class='existingWikiWord' href='/nlab/show/diff/Luca+Mesiti'>Luca Mesiti</a>, <a class='existingWikiWord' href='/nlab/show/diff/Giuseppe+Rosolini'>Giuseppe Rosolini</a>, <a class='existingWikiWord' href='/nlab/show/diff/Thomas+Streicher'>Thomas Streicher</a>, <em>A comonad for Grothendieck fibrations</em> [[arXiv:2305.01474](https://arxiv.org/abs/2305.01474)]</li> </ul> <p>On the connection between Grothendieck fibrations and <a class='existingWikiWord' href='/nlab/show/diff/factorization+system'>factorisation systems</a> (see also <a class='existingWikiWord' href='/nlab/show/diff/stable+factorization+system'>stable factorisation system</a>):</p> <ul> <li> <p>J. Hughes and <a class='existingWikiWord' href='/nlab/show/diff/Bart+Jacobs'>Bart Jacobs</a>, <em>Factorization systems and fibrations: Toward a fibred Birkhoff variety theorem</em>, Electr. Notes in Theor. Comp. Sci., 69 (2002)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Ji%C5%99%C3%AD+Rosick%C3%BD'>Jiří Rosický</a>, <a class='existingWikiWord' href='/nlab/show/diff/Walter+Tholen'>Walter Tholen</a>, <em>Factorization, fibration and torsion</em>, <a href='http://arxiv.org/abs/0801.0063'>arxiv/0801.0063</a>, Journal of homotopy and Related Structures</p> </li> <li> <p>Miloslav Štěpán, <em>Factorization systems and double categories</em>, <a href='https://arxiv.org/abs/2305.06714'>arXiv:2305.06714</a>.</p> </li> </ul> <p>See also:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Ronnie+Brown'>R. Brown</a>, R. Sivera, <em>Algebraic colimit calculations in homotopy theory using fibred and cofibred categories</em>, TAC <strong>22</strong> (2009) pp.222-251. (<a href='http://www.tac.mta.ca/tac/volumes/22/8/22-08.pdf'>pdf</a>)</li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on September 25, 2024 at 21:00:32. 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