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class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+bicategory">enriched bicategory</a></p> </li> </ul> <p><strong>Transfors between 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+of+2-categories">equivalence of 2-categories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-natural+transformation">2-natural transformation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+natural+transformation">lax natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/icon">icon</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modification">modification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma+for+bicategories">Yoneda lemma for bicategories</a></p> </li> </ul> <p><strong>Morphisms in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fully+faithful+morphism">fully faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/faithful+morphism">faithful morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conservative+morphism">conservative morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonic+morphism">pseudomonic morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+morphism">discrete morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/eso+morphism">eso morphism</a></p> </li> </ul> <p><strong>Structures in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mate">mate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+object">cartesian object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration+in+a+2-category">fibration in a 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/codiscrete+cofibration">codiscrete cofibration</a></p> </li> </ul> <p><strong>Limits in 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-pullback">2-pullback</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inserter">inserter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverter">inverter</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equifier">equifier</a></p> </li> </ul> <p><strong>Structures on 2-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-monad">2-monad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/lax-idempotent+2-monad">lax-idempotent 2-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudomonad">pseudomonad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pseudoalgebra+for+a+2-monad">pseudoalgebra for a 2-monad</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gray+tensor+product">Gray tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proarrow+equipment">proarrow equipment</a></p> </li> </ul> </div></div> </div> </div> <h1 id="pullbacks"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-pullbacks</h1> <div class='maruku_toc'> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#equivalence_of_definitions'>Equivalence of definitions</a></li> <li><a href='#variations'>Variations</a></li> <ul> <li><a href='#StrictPullback'>Strict 2-pullbacks</a></li> <li><a href='#strict_weighted_limits'>Strict weighted limits</a></li> <li><a href='#lax_versions'>Lax versions</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <p>An ordinary <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> is a <a class="existingWikiWord" href="/nlab/show/limit">limit</a> over a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>C</mi><mo>←</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \to C \leftarrow B</annotation></semantics></math>. Accordingly, a <strong>2-pullback</strong> (or <strong>2-fiber product</strong>) is a <a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a> over such a diagram.</p> <h2 id="definition">Definition</h2> <p>Saying that “a 2-pullback is a 2-limit over a <a class="existingWikiWord" href="/nlab/show/cospan">cospan</a>” is in fact a sufficient definition, but we can simplify it and make it more explicit.</p> <p>A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-pullback</strong> in a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> is a square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><msup><mo></mo><mi>q</mi></msup></mrow></mpadded><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><mo>≅</mo></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mrow><msup><mo></mo><mi>f</mi></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><munder><mo>⟶</mo><mi>g</mi></munder></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P & \overset{p}{\to} & A \\ \mathllap{^q}\big\downarrow & \cong & \big\downarrow\mathrlap{^f} \\ B & \underset{g}{\longrightarrow} & C } </annotation></semantics></math></div> <p>which commutes up to <a class="existingWikiWord" href="/nlab/show/2-isomorphism">2-isomorphism</a>, and which is <a class="existingWikiWord" href="/nlab/show/universal+property">universal</a> among such squares in a <a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theoretic</a> sense. This means that</p> <ol> <li> <p>given any other such square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Z</mi></mtd> <mtd><mover><mo>⟶</mo><mi>v</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><msup><mo></mo><mi>w</mi></msup></mrow></mpadded><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><mo>≅</mo></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mrow><msup><mo></mo><mi>f</mi></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><munder><mo>⟶</mo><mi>g</mi></munder></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Z & \overset{v}{\longrightarrow} &A \\ \mathllap{^w}\big\downarrow & \cong & \big\downarrow\mathrlap{^f} \\ B & \underset{g}{\longrightarrow} & C } </annotation></semantics></math></div> <p>which commutes up to <a class="existingWikiWord" href="/nlab/show/2-isomorphism">2-isomorphism</a>, there exists a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">u \colon Z\to P</annotation></semantics></math> and isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mi>u</mi><mo>≅</mo><mi>v</mi></mrow><annotation encoding="application/x-tex">p u \cong v</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>u</mi><mo>≅</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">q u \cong w</annotation></semantics></math> which are <a class="existingWikiWord" href="/nlab/show/coherence">coherent</a> with the given ones above, and</p> </li> <li> <p>given any <a class="existingWikiWord" href="/nlab/show/pair">pair</a> of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>,</mo><mi>t</mi><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">u,t \colon Z\to P</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><mi>p</mi><mi>u</mi><mo>→</mo><mi>p</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\alpha \colon p u \to p t</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo lspace="verythinmathspace">:</mo><mi>q</mi><mi>u</mi><mo>→</mo><mi>q</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\beta \colon q u \to q t</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>α</mi><mo>=</mo><mi>g</mi><mi>β</mi></mrow><annotation encoding="application/x-tex">f \alpha = g \beta</annotation></semantics></math> (modulo the given isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>p</mi><mo>≅</mo><mi>g</mi><mi>q</mi></mrow><annotation encoding="application/x-tex">f p \cong g q</annotation></semantics></math>), there exists a unique 2-morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo lspace="verythinmathspace">:</mo><mi>u</mi><mo>→</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">\gamma \colon u\to t</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mi>γ</mi><mo>=</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">p \gamma = \alpha</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>γ</mi><mo>=</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">q \gamma = \beta</annotation></semantics></math>.</p> </li> </ol> <h2 id="equivalence_of_definitions">Equivalence of definitions</h2> <p>The simplification in the above explicit definition has to do with the omission of an unnecessary structure map. Note that an ordinary pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>f</mi></mover><mi>C</mi><mover><mo>←</mo><mi>g</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">A \overset{f}{\to} C \overset{g}{\leftarrow} B</annotation></semantics></math> comes equipped with maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mover><mo>→</mo><mi>p</mi></mover><mi>A</mi></mrow><annotation encoding="application/x-tex">P\overset{p}{\to} A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mover><mo>→</mo><mi>q</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">P\overset{q}{\to} B</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mover><mo>→</mo><mi>r</mi></mover><mi>C</mi></mrow><annotation encoding="application/x-tex">P\overset{r}{\to} C</annotation></semantics></math>, but since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>=</mo><mi>f</mi><mi>p</mi></mrow><annotation encoding="application/x-tex">r = f p</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>=</mo><mi>g</mi><mi>q</mi></mrow><annotation encoding="application/x-tex">r = g q</annotation></semantics></math>, the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> is superfluous data and is usually omitted. In the 2-categorical case, where identities are replaced by isomorphisms, it is, strictly speaking, different to give merely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math> with an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>p</mi><mo>≅</mo><mi>g</mi><mi>q</mi></mrow><annotation encoding="application/x-tex">f p \cong g q</annotation></semantics></math>, than to give <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> with isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>≅</mo><mi>f</mi><mi>p</mi></mrow><annotation encoding="application/x-tex">r \cong f p</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>≅</mo><mi>g</mi><mi>q</mi></mrow><annotation encoding="application/x-tex">r \cong g q</annotation></semantics></math>. However, when 2-limits are considered as only defined up to equivalence (as is the default on the nLab), the two resulting notions of “2-pullback” are the same. In much of the 2-categorical literature, the version with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> specified would be called a <strong>bipullback</strong> and the version with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> not specified would be called a <strong>bi-iso-comma-object</strong>.</p> <p>The unsimplified definition would be: a <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-pullback</strong> in a <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> is a diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mo></mo><mi>q</mi></msup><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>f</mi></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><munder><mo>→</mo><mi>g</mi></munder></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{P & \overset{p}{\to} &A \\ ^q\downarrow & \searrow & \downarrow^f\\ B& \underset{g}{\to} &C } </annotation></semantics></math></div> <p>in which each triangle <a class="existingWikiWord" href="/nlab/show/commuting+square">commutes</a> up to <a class="existingWikiWord" href="/nlab/show/2-isomorphism">2-isomorphism</a>, and which is <a class="existingWikiWord" href="/nlab/show/universal+property">universal</a> among such squares in a <a class="existingWikiWord" href="/nlab/show/2-category+theory">2-category theoretic</a> sense.</p> <p>This means that</p> <ol> <li id="AnyOther"> <p>given any other such square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Z</mi></mtd> <mtd><mover><mo>⟶</mo><mi>v</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><msup><mo></mo><mi>w</mi></msup></mrow></mpadded><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mpadded width="0"><mrow><msup><mo></mo><mi>f</mi></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><munder><mo>⟶</mo><mi>g</mi></munder></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ Z & \overset{v}{\longrightarrow} &A \\ \mathllap{^w}\big\downarrow & \searrow & \big\downarrow\mathrlap{^f} \\ B & \underset{g}{\longrightarrow} &C } </annotation></semantics></math></div> <p>in which the triangles commute up to <a class="existingWikiWord" href="/nlab/show/2-isomorphism">2-isomorphism</a>, there exists a <a class="existingWikiWord" href="/nlab/show/1-morphism">1-morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">u\colon Z \to P</annotation></semantics></math> and 2-isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mi>u</mi><mo>≅</mo><mi>v</mi></mrow><annotation encoding="application/x-tex">p u \cong v</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>u</mi><mo>≅</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">q u \cong w</annotation></semantics></math> which are <a class="existingWikiWord" href="/nlab/show/coherence">coherent</a> with the given ones above, and</p> </li> <li> <p>given any <a class="existingWikiWord" href="/nlab/show/pair">pair</a> of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>,</mo><mi>t</mi><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">u,t\colon Z \to P</annotation></semantics></math> and 2-morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo lspace="verythinmathspace">:</mo><mi>p</mi><mi>u</mi><mo>→</mo><mi>p</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\alpha\colon p u \to p t</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo lspace="verythinmathspace">:</mo><mi>q</mi><mi>u</mi><mo>→</mo><mi>q</mi><mi>t</mi></mrow><annotation encoding="application/x-tex">\beta\colon q u \to q t</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>α</mi><mo>=</mo><mi>g</mi><mi>β</mi></mrow><annotation encoding="application/x-tex">f \alpha = g \beta</annotation></semantics></math> (modulo the given isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mi>p</mi><mo>≅</mo><mi>g</mi><mi>q</mi></mrow><annotation encoding="application/x-tex">f p \cong g q</annotation></semantics></math>), there exists a unique 2-cell <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo lspace="verythinmathspace">:</mo><mi>u</mi><mo>→</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">\gamma\colon u \to t</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mi>γ</mi><mo>=</mo><mi>α</mi></mrow><annotation encoding="application/x-tex">p \gamma = \alpha</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>γ</mi><mo>=</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">q \gamma = \beta</annotation></semantics></math>.</p> </li> </ol> <p>To see that these definitions are equivalent, we observe that both assert the <a class="existingWikiWord" href="/nlab/show/representable+functor">representability</a> of some <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> (where “representability” is understood in the 2-categorical “up-to-equivalence” sense), and that the corresponding 2-functors are equivalent.</p> <ul> <li> <p>In the simplified case, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><msup><mi>K</mi> <mi>op</mi></msup><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">F_1\colon K^{op}\to Cat</annotation></semantics></math> sends an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> to the category whose</p> <ul> <li> <p>objects are squares commuting up to isomorphism, i.e. maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">v\colon Z\to A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">w\colon Z\to B</annotation></semantics></math> equipped with an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo lspace="verythinmathspace">:</mo><mi>f</mi><mi>v</mi><mo>≅</mo><mi>g</mi><mi>w</mi></mrow><annotation encoding="application/x-tex">\mu\colon f v \cong g w</annotation></semantics></math>, and whose</p> </li> <li> <p>morphisms from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>,</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v,w,\mu)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>′</mo><mo>,</mo><mi>w</mi><mo>′</mo><mo>,</mo><mi>μ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v',w',\mu')</annotation></semantics></math> are pairs <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>v</mi><mo>→</mo><mi>v</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\phi\colon v\to v'</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo lspace="verythinmathspace">:</mo><mi>w</mi><mo>→</mo><mi>w</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\psi\colon w\to w'</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>′</mo><mo>.</mo><mo stretchy="false">(</mo><mi>f</mi><mi>ϕ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>g</mi><mi>ψ</mi><mo stretchy="false">)</mo><mo>.</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu' . (f \phi) = (g \psi) . \mu</annotation></semantics></math>.</p> </li> </ul> </li> <li> <p>In the unsimplified case, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><msup><mi>K</mi> <mi>op</mi></msup><mo>→</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">F_2\colon K^{op}\to Cat</annotation></semantics></math> sends an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> to the category whose</p> <ul> <li> <p>objects consist of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">v\colon Z\to A</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">w\colon Z\to B</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo lspace="verythinmathspace">:</mo><mi>Z</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">x\colon Z\to C</annotation></semantics></math> equipped with isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi><mo lspace="verythinmathspace">:</mo><mi>f</mi><mi>v</mi><mo>≅</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">\kappa\colon f v \cong x</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo lspace="verythinmathspace">:</mo><mi>x</mi><mo>≅</mo><mi>g</mi><mi>w</mi></mrow><annotation encoding="application/x-tex">\lambda\colon x\cong g w</annotation></semantics></math>, and whose</p> </li> <li> <p>morphisms from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>w</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>κ</mi><mo>,</mo><mi>λ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v,w,x,\kappa,\lambda)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>v</mi><mo>′</mo><mo>,</mo><mi>w</mi><mo>′</mo><mo>,</mo><mi>x</mi><mo>′</mo><mo>,</mo><mi>κ</mi><mo>′</mo><mo>,</mo><mi>λ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v',w',x',\kappa',\lambda')</annotation></semantics></math> are triples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>v</mi><mo>→</mo><mi>v</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\phi\colon v\to v'</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo lspace="verythinmathspace">:</mo><mi>w</mi><mo>→</mo><mi>w</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\psi\colon w\to w'</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>χ</mi><mo lspace="verythinmathspace">:</mo><mi>x</mi><mo>→</mo><mi>x</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">\chi\colon x\to x'</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi><mo>′</mo><mo>.</mo><mo stretchy="false">(</mo><mi>f</mi><mi>ϕ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>χ</mi><mo>.</mo><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa' . (f \phi) = \chi . \kappa</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>′</mo><mo>.</mo><mi>χ</mi><mo>=</mo><mo stretchy="false">(</mo><mi>g</mi><mi>ψ</mi><mo stretchy="false">)</mo><mo>.</mo><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda' . \chi = (g \psi) . \lambda</annotation></semantics></math>.</p> </li> </ul> </li> </ul> <p>We have a canonical <a class="existingWikiWord" href="/nlab/show/pseudonatural+transformation">pseudonatural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">F_2\to F_1</annotation></semantics></math> that forgets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> and sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>=</mo><mi>λ</mi><mo>.</mo><mi>κ</mi></mrow><annotation encoding="application/x-tex">\mu = \lambda . \kappa</annotation></semantics></math>. This is easily seen to be an <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a>, so that any representing object for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">F_1</annotation></semantics></math> is also a representing object for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">F_2</annotation></semantics></math> and conversely. (Note, though, that in order to define an inverse equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>F</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">F_1\to F_2</annotation></semantics></math> we must choose whether to define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>f</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">x = f v</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>g</mi><mi>w</mi></mrow><annotation encoding="application/x-tex">x = g w</annotation></semantics></math>.)</p> <h2 id="variations">Variations</h2> <p>2-pullbacks can also be identified with <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a>, when the latter are interpreted in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>-enriched homotopy theory.</p> <h3 id="StrictPullback">Strict 2-pullbacks</h3> <p>If we are in a <a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a> and all the coherence isomorphisms (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>, etc.) are required to be identities, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> in property (1) is required to be unique, then we obtain the notion of a <strong>strict 2-pullback</strong>. This is an example of a <a class="existingWikiWord" href="/nlab/show/strict+2-limit">strict 2-limit</a>. Note that since we must have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>f</mi><mi>v</mi><mo>=</mo><mi>g</mi><mi>w</mi></mrow><annotation encoding="application/x-tex">x = f v = g w</annotation></semantics></math>, the two definitions above are still the same. In fact, they are now even isomorphic (and determined up to isomorphism, rather than equivalence).</p> <p>In literature where “2-limit” means “strict 2-limit,” of course “2-pullback” means “strict 2-pullback.”</p> <p>Obviously not every 2-pullback is a strict 2-pullback, but also not every strict 2-pullback is a 2-pullback, although the latter is true if either <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isofibration">isofibration</a> (and in particular if either is a <a class="existingWikiWord" href="/nlab/show/Grothendieck+fibration">Grothendieck fibration</a>). A strict 2-pullback is, in particular, an ordinary pullback in the underlying 1-category of our strict 2-category, but it has a stronger universal property than this, referring to 2-cells as well (namely, part (2) of the explicit definition).</p> <h3 id="strict_weighted_limits">Strict weighted limits</h3> <p>If the coherence isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> in the squares are retained, but in (1) the isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mi>u</mi><mo>≅</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">p u \cong r</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mi>u</mi><mo>≅</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">q u \cong s</annotation></semantics></math> are required to be identities and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> is required to be unique, then the simplified definition becomes that of a <strong><a class="existingWikiWord" href="/nlab/show/strict+iso-comma+object">strict iso-comma object</a></strong>, while the unsimplified definition becomes that of a <strong>strict pseudo-pullback</strong>. (Iso-comma objects are so named because if the isomorphisms in the squares are then replaced by mere morphisms, we obtain the notion of (strict) <a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a>).</p> <p>Every <a class="existingWikiWord" href="/nlab/show/strict+iso-comma+object">strict iso-comma object</a>, and every strict pseudo-pullback, is also a (non-strict) 2-pullback. In particular, if strict iso-comma objects and strict pseudo-pullbacks both exist, they are equivalent, but they are <em>not</em> isomorphic. (Note that their strict universal property determines them up to isomorphism, not just equivalence.) In many strict 2-categories, such as <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>, 2-pullbacks can naturally be constructed as either strict iso-comma objects or strict pseudo-pullbacks.</p> <h3 id="lax_versions">Lax versions</h3> <p>Replacing the isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> in the simplified definition by a mere transformation results in a <a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a>, while replacing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>κ</mi></mrow><annotation encoding="application/x-tex">\kappa</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> in the unsimplified definition by mere transformations results in a <a class="existingWikiWord" href="/nlab/show/lax+pullback">lax pullback</a>. In a <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a>, any <a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a> or <a class="existingWikiWord" href="/nlab/show/lax+pullback">lax pullback</a> is also a 2-pullback, but this is not true in a general 2-category. Note that comma objects are often misleadingly called lax pullbacks.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/comma+object">comma object</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 18, 2024 at 20:12:20. 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