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Prestack

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In <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, a prestack F over a category C equipped with some <a href="Grothendieck_topology" title="Grothendieck topology">Grothendieck topology</a> is a category together with a functor p: F → C satisfying a <a href="Fibered_category" class="mw-redirect" title="Fibered category">certain lifting condition</a> and such that (when the fibers are groupoids) locally isomorphic objects are isomorphic. A <a href="Stack_(mathematics)" title="Stack (mathematics)">stack</a> is a prestack with effective descents, meaning local objects may be patched together to become a global object.

Prestacks that appear in nature are typically stacks but some naively constructed prestacks (e.g., <a href="Groupoid_scheme" class="mw-redirect" title="Groupoid scheme">groupoid scheme</a> or the prestack of <a href="Projectivized_vector_bundle" class="mw-redirect" title="Projectivized vector bundle">projectivized vector bundles</a>) may not be stacks. Prestacks may be studied on their own or <a href="#Stacks_associated_to_prestacks">passed to stacks</a>.

Since a stack is a prestack, all the results on prestacks are valid for stacks as well. Throughout the article, we work with a fixed base category C; for example, C can be the category of all schemes over some fixed scheme equipped with some <a href="Grothendieck_topology" title="Grothendieck topology">Grothendieck topology</a>.

<div class="mw-heading mw-heading2"><h2 id="Informal_definition">Informal definition</h2></div>

Let F be a category and suppose it is <a href="Fibered_category" class="mw-redirect" title="Fibered category">fibered over C</a> through the functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:F\to C}">

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<annotation encoding="application/x-tex">{\displaystyle p:F\to C}</annotation>

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</math><img src="./c66d3f389313da08201d63cee44500df224b5fa3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.317ex; height:2.509ex;" alt="{\displaystyle p:F\to C}" loading="lazy">; this means that one can construct pullbacks along morphisms in C, up to canonical isomorphisms.

Given an object U in C and objects x, y in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(U)=p^{-1}(U)}">

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<annotation encoding="application/x-tex">{\displaystyle F(U)=p^{-1}(U)}</annotation>

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</math><img src="./e3b5969bee3fffb8bc15c881c4042f88bf43b622.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.525ex; height:3.176ex;" alt="{\displaystyle F(U)=p^{-1}(U)}" loading="lazy">, for each morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:V\to U}">

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<annotation encoding="application/x-tex">{\displaystyle f:V\to U}</annotation>

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</math><img src="./15330bffdda84438fcd2ab838c8437f0ac0e9d1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.4ex; height:2.509ex;" alt="{\displaystyle f:V\to U}" loading="lazy"> in C, after fixing pullbacks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}x,f^{*}y}">

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<annotation encoding="application/x-tex">{\displaystyle f^{*}x,f^{*}y}</annotation>

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</math><img src="./ebb02d26de0f07da75aad6f673cca570876a0fce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.269ex; height:2.676ex;" alt="{\displaystyle f^{*}x,f^{*}y}" loading="lazy">, we let<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\underline {\operatorname {Hom} }}(x,y)(V{\overset {f}{\to }}U)=[\operatorname {Hom} (f^{*}x,f^{*}y)]}</annotation>

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</math><img src="./d0e7de773177b85a60ea2affdc7365ae3f18d58c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:37.187ex; height:4.843ex;" alt="{\displaystyle {\underline {\operatorname {Hom} }}(x,y)(V{\overset {f}{\to }}U)=[\operatorname {Hom} (f^{*}x,f^{*}y)]}" loading="lazy"></dd></dl>

be the set of all morphisms from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}x}">

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<annotation encoding="application/x-tex">{\displaystyle f^{*}x}</annotation>

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</math><img src="./e2aa2e2c03e2800b59677064f1534455f3b82a11.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.704ex; height:2.676ex;" alt="{\displaystyle f^{*}x}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}y}">

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<annotation encoding="application/x-tex">{\displaystyle f^{*}y}</annotation>

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</math><img src="./acf49572c60fbc4d66d785fd5cf1028909a6de7e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.53ex; height:2.676ex;" alt="{\displaystyle f^{*}y}" loading="lazy">; here, the bracket means we canonically identify different Hom sets resulting from different choices of pullbacks. For each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g:W\to V}">

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</math><img src="./d7c63c562bbc8a34cbd882cc3d74fb8d61d7d315.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.89ex; height:2.509ex;" alt="{\displaystyle g:W\to V}" loading="lazy"> over U, define the restriction map from f to g: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\operatorname {Hom} }}(x,y)(V{\overset {f}{\to }}U)\to {\underline {\operatorname {Hom} }}(x,y)(W{\overset {f\circ g}{\to }}U)}">

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<annotation encoding="application/x-tex">{\displaystyle {\underline {\operatorname {Hom} }}(x,y)(V{\overset {f}{\to }}U)\to {\underline {\operatorname {Hom} }}(x,y)(W{\overset {f\circ g}{\to }}U)}</annotation>

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</math><img src="./c272d889a193902be7620b83fb66bf31c0a3bb49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:40.204ex; height:4.843ex;" alt="{\displaystyle {\underline {\operatorname {Hom} }}(x,y)(V{\overset {f}{\to }}U)\to {\underline {\operatorname {Hom} }}(x,y)(W{\overset {f\circ g}{\to }}U)}" loading="lazy">

to be the composition

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<annotation encoding="application/x-tex">{\displaystyle [\operatorname {Hom} (f^{*}x,f^{*}y)]{\overset {g^{*}}{\to }}[\operatorname {Hom} (g^{*}(f^{*}x),g^{*}(f^{*}y))]=[\operatorname {Hom} ((f\circ g)^{*}x,(f\circ g)^{*}y)]}</annotation>

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</math><img src="./d1976a1aa4f7de3d2431864dc01d68f35d62d896.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:72.181ex; height:4.343ex;" alt="{\displaystyle [\operatorname {Hom} (f^{*}x,f^{*}y)]{\overset {g^{*}}{\to }}[\operatorname {Hom} (g^{*}(f^{*}x),g^{*}(f^{*}y))]=[\operatorname {Hom} ((f\circ g)^{*}x,(f\circ g)^{*}y)]}" loading="lazy"></dd></dl>

where a canonical isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{*}\circ f^{*}\simeq (f\circ g)^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle g^{*}\circ f^{*}\simeq (f\circ g)^{*}}</annotation>

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</math><img src="./36d7bd127b3912891678b2a0af7021a2e941d620.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.293ex; height:2.843ex;" alt="{\displaystyle g^{*}\circ f^{*}\simeq (f\circ g)^{*}}" loading="lazy"> is used to get the = on the right. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\operatorname {Hom} }}(x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle {\underline {\operatorname {Hom} }}(x,y)}</annotation>

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</math><img src="./df87893de4d0e88c6789621e1987802776866d22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:10.172ex; height:3.343ex;" alt="{\displaystyle {\underline {\operatorname {Hom} }}(x,y)}" loading="lazy"> is a <a href="Presheaf_(category_theory)" title="Presheaf (category theory)">presheaf</a> on the <a href="Slice_category" class="mw-redirect" title="Slice category">slice category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{/U}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{/U}}</annotation>

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</math><img src="./0322cbe23a276f07a23c5f74619cfc1cd558cf3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.977ex; height:3.009ex;" alt="{\displaystyle C_{/U}}" loading="lazy">, the category of all morphisms in C with target U.

By definition, F is a prestack if, for each pair x, y, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\operatorname {Hom} }}(x,y)}">

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<annotation encoding="application/x-tex">{\displaystyle {\underline {\operatorname {Hom} }}(x,y)}</annotation>

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</math><img src="./df87893de4d0e88c6789621e1987802776866d22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:10.172ex; height:3.343ex;" alt="{\displaystyle {\underline {\operatorname {Hom} }}(x,y)}" loading="lazy"> is a <a href="Sheaf_of_sets" class="mw-redirect" title="Sheaf of sets">sheaf of sets</a> with respect to the induced <a href="Grothendieck_topology" title="Grothendieck topology">Grothendieck topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{/U}}">

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<annotation encoding="application/x-tex">{\displaystyle C_{/U}}</annotation>

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This definition can be equivalently phrased as follows.<a href="#cite_note-3">[3]</a> First, for each covering family <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{V_{i}\to U\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{V_{i}\to U\}}</annotation>

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</math><img src="./0e4663c8df988418dd20628b2de961ad72f55459.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.876ex; height:2.843ex;" alt="{\displaystyle \{V_{i}\to U\}}" loading="lazy">, we "define" the category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(\{V_{i}\to U\})}">

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<annotation encoding="application/x-tex">{\displaystyle F(\{V_{i}\to U\})}</annotation>

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</math><img src="./7fc6e6d8b1fa26825c1ebe6e7eef41c81ca96ca8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.427ex; height:2.843ex;" alt="{\displaystyle F(\{V_{i}\to U\})}" loading="lazy"> as a category where: writing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1}:V_{i}\times _{U}V_{j}\to V_{i},\,p_{12}:V_{i}\times _{U}V_{j}\times _{U}V_{k}\to V_{i}\times _{U}V_{j}}">

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<annotation encoding="application/x-tex">{\displaystyle p_{1}:V_{i}\times _{U}V_{j}\to V_{i},\,p_{12}:V_{i}\times _{U}V_{j}\times _{U}V_{k}\to V_{i}\times _{U}V_{j}}</annotation>

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</math><img src="./1ed98506616171246801adf3cdbdafa7aa8ffaf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; margin-left: -0.089ex; width:53.073ex; height:2.843ex;" alt="{\displaystyle p_{1}:V_{i}\times _{U}V_{j}\to V_{i},\,p_{12}:V_{i}\times _{U}V_{j}\times _{U}V_{k}\to V_{i}\times _{U}V_{j}}" loading="lazy">, etc.,

<ol><li>an object is a set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x_{i},\varphi _{ij})\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{(x_{i},\varphi _{ij})\}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle F(V_{i})}</annotation>

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</math><img src="./640cbf0910a80603fd4e20edf16bcd83de096d3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.705ex; height:2.843ex;" alt="{\displaystyle F(V_{i})}" loading="lazy"> and isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi _{ij}:p_{2}^{*}x_{j}{\overset {\sim }{\to }}p_{1}^{*}x_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi _{ij}:p_{2}^{*}x_{j}{\overset {\sim }{\to }}p_{1}^{*}x_{i}}</annotation>

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</math><img src="./bd7580dc30faf1fec506d6002d0b99cb2966fa59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.074ex; height:3.509ex;" alt="{\displaystyle \varphi _{ij}:p_{2}^{*}x_{j}{\overset {\sim }{\to }}p_{1}^{*}x_{i}}" loading="lazy"> that satisfy the cocycle condition: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{13}^{*}\varphi _{ik}=p_{12}^{*}\varphi _{ij}\circ p_{23}^{*}\varphi _{jk}}">

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<annotation encoding="application/x-tex">{\displaystyle p_{13}^{*}\varphi _{ik}=p_{12}^{*}\varphi _{ij}\circ p_{23}^{*}\varphi _{jk}}</annotation>

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<li>a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(x_{i},\varphi _{ij})\}\to \{(y_{i},\psi _{ij})\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{(x_{i},\varphi _{ij})\}\to \{(y_{i},\psi _{ij})\}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle F(V_{i})}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \psi _{ij}\circ p_{2}^{*}\alpha _{j}=p_{1}^{*}\alpha _{i}\circ \varphi _{ij}.}</annotation>

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</math><img src="./cffa8af39c03c3a08c3f6fc72f6733f751803de3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.254ex; height:2.843ex;" alt="{\displaystyle \psi _{ij}\circ p_{2}^{*}\alpha _{j}=p_{1}^{*}\alpha _{i}\circ \varphi _{ij}.}" loading="lazy"></li></ol>

An object of this category is called a descent datum. This category is not well-defined; the issue is that the pullbacks are determined only up to canonical isomorphisms; similarly fiber products are defined only up to canonical isomorphisms, despite the notational practice to the contrary. In practice, one simply makes some canonical identifications of pullbacks, their compositions, fiber products, etc.; up to such identifications, the above category is well-defined (in other words, it is defined up to a canonical equivalence of categories.)

There is an obvious functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(U)\to F(\{V_{i}\to U\})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(U)\to F(\{V_{i}\to U\})}</annotation>

</semantics>

</math><img src="./aae4438eaae210e9c13c6936f4c862f48faa6395.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.373ex; height:2.843ex;" alt="{\displaystyle F(U)\to F(\{V_{i}\to U\})}" loading="lazy"> that sends an object to the descent datum that it defines. One can then say: F is a prestack if and only if, for each covering family <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{V_{i}\to U\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{V_{i}\to U\}}</annotation>

</semantics>

</math><img src="./0e4663c8df988418dd20628b2de961ad72f55459.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.876ex; height:2.843ex;" alt="{\displaystyle \{V_{i}\to U\}}" loading="lazy">, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(U)\to F(\{V_{i}\to U\})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(U)\to F(\{V_{i}\to U\})}</annotation>

</semantics>

</math><img src="./aae4438eaae210e9c13c6936f4c862f48faa6395.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.373ex; height:2.843ex;" alt="{\displaystyle F(U)\to F(\{V_{i}\to U\})}" loading="lazy"> is fully faithful. A statement like this is independent of choices of canonical identifications mentioned early.

The essential image of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(U)\to F(\{V_{i}\to U\})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(U)\to F(\{V_{i}\to U\})}</annotation>

</semantics>

</math><img src="./aae4438eaae210e9c13c6936f4c862f48faa6395.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.373ex; height:2.843ex;" alt="{\displaystyle F(U)\to F(\{V_{i}\to U\})}" loading="lazy"> consists precisely of effective descent data (just the definition of "effective"). Thus, F is a stack if and only if, for each covering family <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{V_{i}\to U\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{V_{i}\to U\}}</annotation>

</semantics>

</math><img src="./0e4663c8df988418dd20628b2de961ad72f55459.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.876ex; height:2.843ex;" alt="{\displaystyle \{V_{i}\to U\}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(U)\to F(\{V_{i}\to U\})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(U)\to F(\{V_{i}\to U\})}</annotation>

</semantics>

</math><img src="./aae4438eaae210e9c13c6936f4c862f48faa6395.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.373ex; height:2.843ex;" alt="{\displaystyle F(U)\to F(\{V_{i}\to U\})}" loading="lazy"> is an equivalence of categories.

These reformulations of the definitions of prestacks and stacks make intuitive meanings of those concepts very explicit: (1) "fibered category" means one can construct a pullback (2) "prestack in groupoids" additionally means "locally isomorphic" implies "isomorphic" (3) "stack in groupoids" means, in addition to the previous properties, a global object can be constructed from local data subject to cocycle conditions. All these work up to canonical isomorphisms.

<div class="mw-heading mw-heading2"><h2 id="Morphisms">Morphisms</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Morphism_of_algebraic_stacks" title="Morphism of algebraic stacks">Morphism of algebraic stacks</a></div>

<div class="mw-heading mw-heading3"><h3 id="Definitions">Definitions</h3></div>

Given prestacks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:F\to C,q:G\to C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p:F\to C,q:G\to C}</annotation>

</semantics>

</math><img src="./314cafcf4d6d17fbb197dbfb88bc89c86443d04c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:21.565ex; height:2.509ex;" alt="{\displaystyle p:F\to C,q:G\to C}" loading="lazy"> over the fixed base category C, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:F\to G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:F\to G}</annotation>

</semantics>

</math><img src="./85edcca983d67148f3ef74994f52e81db9fa9a81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.397ex; height:2.509ex;" alt="{\displaystyle f:F\to G}" loading="lazy"> is a functor such that (1) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\circ f=p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\circ f=p}</annotation>

</semantics>

</math><img src="./0d6dcb460b2a8a39bbbc3e994812230e05c8ef0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.811ex; height:2.509ex;" alt="{\displaystyle q\circ f=p}" loading="lazy"> and (2) it maps cartesian morphisms to cartesian morphisms. Note (2) is automatic if G is fibered in groupoids; e.g., an algebraic stack (since all morphisms are cartesian then.)

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:F_{S}\to C}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p:F_{S}\to C}</annotation>

</semantics>

</math><img src="./0fd2c0c73ba5dd164a1fbc59cde8309ce0bce334.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:11.363ex; height:2.509ex;" alt="{\displaystyle p:F_{S}\to C}" loading="lazy"> is the <a href="Stack_associated_to_a_scheme" class="mw-redirect" title="Stack associated to a scheme">stack associated to a scheme</a> S in the base category C, then the fiber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{-1}(U)=F_{S}(U)}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p^{-1}(U)=F_{S}(U)}</annotation>

</semantics>

</math><img src="./d52df64d67aeaa12dd58cbc22cacd270b06bd413.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:16.661ex; height:3.176ex;" alt="{\displaystyle p^{-1}(U)=F_{S}(U)}" loading="lazy"> is, by construction, the set of all morphisms from U to S in C. Analogously, given a scheme U in C viewed as a stack (i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{U}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{U}}</annotation>

</semantics>

</math><img src="./3a1d028a6cbd245193ad5e16c7e1bb7c828a1ab0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.987ex; height:2.509ex;" alt="{\displaystyle F_{U}}" loading="lazy">) and a category F fibered in groupoids over C, the <a href="2-Yoneda_lemma" title="2-Yoneda lemma">2-Yoneda lemma</a> says: there is a natural equivalence of categories<a href="#cite_note-4">[4]</a>

<msub>

<mi>Funct</mi>

</mrow>

</msub>

<mover>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Funct} _{C}(U,F){\overset {\chi \mapsto \chi (1_{U})}{\to }}F(U)}</annotation>

</semantics>

</math><img src="./4535e96e0575609d0c8c8adc952995005e00f0de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.21ex; height:4.343ex;" alt="{\displaystyle \operatorname {Funct} _{C}(U,F){\overset {\chi \mapsto \chi (1_{U})}{\to }}F(U)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Funct} _{C}}">

<msub>

<mi>Funct</mi>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Funct} _{C}}</annotation>

</semantics>

</math><img src="./057d833a0e3a9d7702ecb122e3b00a636405b0ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.521ex; height:2.509ex;" alt="{\displaystyle \operatorname {Funct} _{C}}" loading="lazy"> refers to the relative <a href="Functor_category" title="Functor category">functor category</a>; the objects are the functors from U to F over C and the morphisms are the base-preserving natural transformations.<a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading3"><h3 id="Fiber_product">Fiber product</h3></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:F\to B,g:G\to B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:F\to B,g:G\to B}</annotation>

</semantics>

</math><img src="./29a043b58af812e5b3d277b0d2c532c67b59bc2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.627ex; height:2.509ex;" alt="{\displaystyle f:F\to B,g:G\to B}" loading="lazy"> be morphisms of prestacks. Then, by definition,<a href="#cite_note-6">[6]</a> the fiber product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\times _{B,f,g}G=F\times _{B}G}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\times _{B,f,g}G=F\times _{B}G}</annotation>

</semantics>

</math><img src="./1909e482bee01738e09c986a62d9d545531630b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.481ex; height:2.843ex;" alt="{\displaystyle F\times _{B,f,g}G=F\times _{B}G}" loading="lazy"> is the category where

<ol><li>an object is a triple <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,\psi )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y,\psi )}</annotation>

</semantics>

</math><img src="./fc45b223329d3e941be5d909ee30b1bd8c15883c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.876ex; height:2.843ex;" alt="{\displaystyle (x,y,\psi )}" loading="lazy"> consisting of an object x in F, an object y in G, both over the same object in C, and an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi :f(x){\overset {\sim }{\to }}g(y)}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi :f(x){\overset {\sim }{\to }}g(y)}</annotation>

</semantics>

</math><img src="./9fdd4a1d634aa72a5cf01b058b269a00de8eaf01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.272ex; height:3.343ex;" alt="{\displaystyle \psi :f(x){\overset {\sim }{\to }}g(y)}" loading="lazy"> in G over the identity morphism in C, and</li>

<li>a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,\psi )\to (x',y',\psi ')}">

<msup>

</msup>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y,\psi )\to (x',y',\psi ')}</annotation>

</semantics>

</math><img src="./841bd2c3e58c7fc932291cb85567fa3c7cc0efa2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.424ex; height:3.009ex;" alt="{\displaystyle (x,y,\psi )\to (x',y',\psi ')}" loading="lazy"> consists of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha :x\to x'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha :x\to x'}</annotation>

</semantics>

</math><img src="./ea5e5fe9941e97fb0808d3c495debb490d4bbe0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.383ex; height:2.509ex;" alt="{\displaystyle \alpha :x\to x'}" loading="lazy"> in F, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta :y\to y'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta :y\to y'}</annotation>

</semantics>

</math><img src="./200c32636becfa854220321f9ed1fe69f54dcc6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.884ex; height:2.843ex;" alt="{\displaystyle \beta :y\to y'}" loading="lazy"> in G, both over the same morphism in C, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(\beta )\circ \psi =\psi '\circ f(\alpha )}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(\beta )\circ \psi =\psi '\circ f(\alpha )}</annotation>

</semantics>

</math><img src="./07a5961b216f55fd837b2f8ede7b575ee17241ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.032ex; height:3.009ex;" alt="{\displaystyle g(\beta )\circ \psi =\psi '\circ f(\alpha )}" loading="lazy">.</li></ol>

It comes with the forgetful functors p, q from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\times _{B}G}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\times _{B}G}</annotation>

</semantics>

</math><img src="./3208c8bd72d68916c40c42125ab549c86d10f424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.887ex; height:2.509ex;" alt="{\displaystyle F\times _{B}G}" loading="lazy"> to F and G.

This fiber product behaves like a usual fiber product but up to natural isomorphisms. The meaning of this is the following. Firstly, the obvious square does not commute; instead, for each object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y,\psi )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y,\psi )}</annotation>

</semantics>

</math><img src="./fc45b223329d3e941be5d909ee30b1bd8c15883c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.876ex; height:2.843ex;" alt="{\displaystyle (x,y,\psi )}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\times _{B}G}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\times _{B}G}</annotation>

</semantics>

</math><img src="./3208c8bd72d68916c40c42125ab549c86d10f424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.887ex; height:2.509ex;" alt="{\displaystyle F\times _{B}G}" loading="lazy">:

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi :(f\circ p)(x,y,\psi )=f(x){\overset {\sim }{\to }}g(y)=(g\circ q)(x,y,\psi )}</annotation>

</semantics>

</math><img src="./2986ccab3e851335f932fca8f455663ef4460da5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.862ex; height:3.343ex;" alt="{\displaystyle \psi :(f\circ p)(x,y,\psi )=f(x){\overset {\sim }{\to }}g(y)=(g\circ q)(x,y,\psi )}" loading="lazy">.</dd></dl>

That is, there is an invertible <a href="Natural_transformation" title="Natural transformation">natural transformation</a> (= natural isomorphism)

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi :f\circ p{\overset {\sim }{\to }}g\circ q}</annotation>

</semantics>

</math><img src="./275d37639eb3959d87859f48a64333bee2c0323e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.092ex; height:3.176ex;" alt="{\displaystyle \Psi :f\circ p{\overset {\sim }{\to }}g\circ q}" loading="lazy">.</dd></dl>

Secondly, it satisfies the strict universal property: given a prestack H, morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u:H\to F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u:H\to F}</annotation>

</semantics>

</math><img src="./3db09044cf1562c055dd0aeffd57c9a36ede3ce5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.685ex; height:2.176ex;" alt="{\displaystyle u:H\to F}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v:H\to G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v:H\to G}</annotation>

</semantics>

</math><img src="./6253b60817c8b05f56e83ddfe21833cf8cec60da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.569ex; height:2.176ex;" alt="{\displaystyle v:H\to G}" loading="lazy">, a natural isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ u{\overset {\sim }{\to }}g\circ v}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\circ u{\overset {\sim }{\to }}g\circ v}</annotation>

</semantics>

</math><img src="./099f4ca6f5677ac01083f3fc1c823732d0f7292d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.565ex; height:3.176ex;" alt="{\displaystyle f\circ u{\overset {\sim }{\to }}g\circ v}" loading="lazy">, there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w:H\to F\times _{B}G}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w:H\to F\times _{B}G}</annotation>

</semantics>

</math><img src="./09f9ad60b54303774b514b557c45d06e3a2c2b6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.166ex; height:2.509ex;" alt="{\displaystyle w:H\to F\times _{B}G}" loading="lazy"> together with natural isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u{\overset {\sim }{\to }}p\circ w}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u{\overset {\sim }{\to }}p\circ w}</annotation>

</semantics>

</math><img src="./1891d2dd1d40a308393d40fe5e39d3a6525ef349.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.682ex; height:3.176ex;" alt="{\displaystyle u{\overset {\sim }{\to }}p\circ w}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q\circ w{\overset {\sim }{\to }}v}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q\circ w{\overset {\sim }{\to }}v}</annotation>

</semantics>

</math><img src="./756a7f6a709362f47adaf655e8062c08b33767bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.38ex; height:3.176ex;" alt="{\displaystyle q\circ w{\overset {\sim }{\to }}v}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ u{\overset {\sim }{\to }}g\circ v}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\circ u{\overset {\sim }{\to }}g\circ v}</annotation>

</semantics>

</math><img src="./099f4ca6f5677ac01083f3fc1c823732d0f7292d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.565ex; height:3.176ex;" alt="{\displaystyle f\circ u{\overset {\sim }{\to }}g\circ v}" loading="lazy"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ p\circ w{\overset {\sim }{\to }}g\circ q\circ w}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\circ p\circ w{\overset {\sim }{\to }}g\circ q\circ w}</annotation>

</semantics>

</math><img src="./e331070ee34cf05d64988adeef821776867520e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.064ex; height:3.176ex;" alt="{\displaystyle f\circ p\circ w{\overset {\sim }{\to }}g\circ q\circ w}" loading="lazy">. In general, a fiber product of F and G over B is a prestack canonically isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\times _{B}G}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\times _{B}G}</annotation>

</semantics>

</math><img src="./3208c8bd72d68916c40c42125ab549c86d10f424.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.887ex; height:2.509ex;" alt="{\displaystyle F\times _{B}G}" loading="lazy"> above.

When B is the base category C (the prestack over itself), B is dropped and one simply writes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F\times G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\times G}</annotation>

</semantics>

</math><img src="./eb3bfb0591b94327543bfd45bac70abb00f21717.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.408ex; height:2.176ex;" alt="{\displaystyle F\times G}" loading="lazy">. Note, in this case, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \psi }</annotation>

</semantics>

</math><img src="./45e5789e5d9c8f7c79744f43ecaaf8ba42a8553a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.513ex; height:2.509ex;" alt="{\displaystyle \psi }" loading="lazy"> in objects are all identities.

Example: For each prestack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p:X\to C}</annotation>

</semantics>

</math><img src="./0ca791deeac18606bfd4dc77405bb8976b24e4be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.556ex; height:2.509ex;" alt="{\displaystyle p:X\to C}" loading="lazy">, there is the diagonal morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta :X\to X\times X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta :X\to X\times X}</annotation>

</semantics>

</math><img src="./4303f3e72023d1af98eb3c227505fbba97ab720d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.268ex; height:2.176ex;" alt="{\displaystyle \Delta :X\to X\times X}" loading="lazy"> given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto (x,x,1_{p(x)})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\mapsto (x,x,1_{p(x)})}</annotation>

</semantics>

</math><img src="./87e27edcc5ee4798965394db84546b3b7c22228e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.922ex; height:3.176ex;" alt="{\displaystyle x\mapsto (x,x,1_{p(x)})}" loading="lazy">.

Example: Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{i}\to B_{i},G_{i}\to B_{i},\,i=1,2}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{i}\to B_{i},G_{i}\to B_{i},\,i=1,2}</annotation>

</semantics>

</math><img src="./89b3a2eacb18882c8715110ab8977f9ea514e7b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.991ex; height:2.509ex;" alt="{\displaystyle F_{i}\to B_{i},G_{i}\to B_{i},\,i=1,2}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F_{1}\times F_{2})\times _{B_{1}\times B_{2}}(G_{1}\times G_{2})\simeq (F_{1}\times _{B_{1}}G_{1})\times (F_{2}\times _{B_{2}}G_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

<msub>

</mrow>

</msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (F_{1}\times F_{2})\times _{B_{1}\times B_{2}}(G_{1}\times G_{2})\simeq (F_{1}\times _{B_{1}}G_{1})\times (F_{2}\times _{B_{2}}G_{2})}</annotation>

</semantics>

</math><img src="./fb5621df436fb46fc20c0a29a1bd4853266b71fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:59.388ex; height:3.009ex;" alt="{\displaystyle (F_{1}\times F_{2})\times _{B_{1}\times B_{2}}(G_{1}\times G_{2})\simeq (F_{1}\times _{B_{1}}G_{1})\times (F_{2}\times _{B_{2}}G_{2})}" loading="lazy">.<a href="#cite_note-7">[7]</a>

Example: Given <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:F\to B,g:G\to B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:F\to B,g:G\to B}</annotation>

</semantics>

</math><img src="./29a043b58af812e5b3d277b0d2c532c67b59bc2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.627ex; height:2.509ex;" alt="{\displaystyle f:F\to B,g:G\to B}" loading="lazy"> and the diagonal morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta :B\to B\times B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta :B\to B\times B}</annotation>

</semantics>

</math><img src="./58711224a5b6ed4d2565f6cb625adc049e45f65b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.62ex; height:2.176ex;" alt="{\displaystyle \Delta :B\to B\times B}" loading="lazy">,

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\times _{B}G\simeq (F\times G)\times _{B\times B,f\times g,\Delta }B}</annotation>

</semantics>

</math><img src="./67ede7799a1c7ecc98be09bbf1a880bb7ed7a2c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.068ex; height:3.009ex;" alt="{\displaystyle F\times _{B}G\simeq (F\times G)\times _{B\times B,f\times g,\Delta }B}" loading="lazy">;</dd></dl>

this isomorphism is constructed simply by hand.

<div class="mw-heading mw-heading3"><h3 id="Representable_morphisms">Representable morphisms</h3></div>

A morphism of prestacks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is said to be strongly representable if, for every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\to Y}</annotation>

</semantics>

</math><img src="./67557c9d8f71bd0bf4fa4d973eaf63aa8336c71b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.887ex; height:2.176ex;" alt="{\displaystyle S\to Y}" loading="lazy"> from a scheme S in C viewed as a prestack, the fiber product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times _{Y}S}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times _{Y}S}</annotation>

</semantics>

</math><img src="./2f5608b6262c7503f2bd03bdc12eb7bba64c6eb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.806ex; height:2.509ex;" alt="{\displaystyle X\times _{Y}S}" loading="lazy"> of prestacks is a scheme in C.

In particular, the definition applies to the structure map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p:X\to C}</annotation>

</semantics>

</math><img src="./0ca791deeac18606bfd4dc77405bb8976b24e4be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.556ex; height:2.509ex;" alt="{\displaystyle p:X\to C}" loading="lazy"> (the base category C is a prestack over itself via the identity). Then p is strongly representable if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\simeq X\times _{C}C}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\simeq X\times _{C}C}</annotation>

</semantics>

</math><img src="./01fd0ba946ade069e2a7763441adb7520f5a0d53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.146ex; height:2.509ex;" alt="{\displaystyle X\simeq X\times _{C}C}" loading="lazy"> is a scheme in C.

The definition applies also to the diagonal morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta :X\to X\times X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta :X\to X\times X}</annotation>

</semantics>

</math><img src="./4303f3e72023d1af98eb3c227505fbba97ab720d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.268ex; height:2.176ex;" alt="{\displaystyle \Delta :X\to X\times X}" loading="lazy">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

</semantics>

</math><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> is strongly representable, then every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\to X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\to X}</annotation>

</semantics>

</math><img src="./e0e8241962e6e1883ae50b5a471db0ecff377950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.377ex; height:2.176ex;" alt="{\displaystyle U\to X}" loading="lazy"> from a scheme U is strongly representable since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\times _{X}T\simeq (U\times T)\times _{X\times X}X}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\times _{X}T\simeq (U\times T)\times _{X\times X}X}</annotation>

</semantics>

</math><img src="./9f4aaad5bc293d10898fd67d04b62a8e94cb686c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.19ex; height:2.843ex;" alt="{\displaystyle U\times _{X}T\simeq (U\times T)\times _{X\times X}X}" loading="lazy"> is strongly representable for any T → X.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> is a strongly representable morphism, for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\to Y}</annotation>

</semantics>

</math><img src="./67557c9d8f71bd0bf4fa4d973eaf63aa8336c71b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.887ex; height:2.176ex;" alt="{\displaystyle S\to Y}" loading="lazy">, S a scheme viewed as a prestack, the projection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times _{Y}S\to S}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times _{Y}S\to S}</annotation>

</semantics>

</math><img src="./bf43c4956ef6346daa3169e4cccdd665e33750db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.919ex; height:2.509ex;" alt="{\displaystyle X\times _{Y}S\to S}" loading="lazy"> is a <a href="Morphism_of_schemes" title="Morphism of schemes">morphism of schemes</a>; this allows one to transfer many notions of properties on morphisms of schemes to the stack context. Namely, let P be a property on morphisms in the base category C that is stable under base changes and that is local on the topology of C (e.g., <a href="%C3%89tale_topology" title="Étale topology">étale topology</a> or <a href="Smooth_topology_(algebraic_geometry)" class="mw-redirect" title="Smooth topology (algebraic geometry)">smooth topology</a>). Then a strongly representable morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:X\to Y}</annotation>

</semantics>

</math><img src="./abd1e080abef4bbdab67b43819c6431e7561361c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.583ex; height:2.509ex;" alt="{\displaystyle f:X\to Y}" loading="lazy"> of prestacks is said to have the property P if, for every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\to Y}</annotation>

</semantics>

</math><img src="./4acfa7511f1a46baab3346d050df50b82f2ef141.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.024ex; height:2.176ex;" alt="{\displaystyle T\to Y}" loading="lazy">, T a scheme viewed as a prestack, the induced projection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X\times _{Y}T\to T}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times _{Y}T\to T}</annotation>

</semantics>

</math><img src="./04fa5231e9994e6ce9f7dc268a639a8889382e51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.193ex; height:2.509ex;" alt="{\displaystyle X\times _{Y}T\to T}" loading="lazy"> has the property P.

<div class="mw-heading mw-heading2"><h2 id="Example:_the_prestack_given_by_an_action_of_an_algebraic_group">Example: the prestack given by an action of an algebraic group</h2></div>

Let G be an <a href="Algebraic_group" title="Algebraic group">algebraic group</a> acting from the right on a scheme X of finite type over a field k. Then the group action of G on X determines a prestack (but not a stack) over the category C of k-schemes, as follows. Let F be the category where

<ol><li>an object is a pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (U,x)}</annotation>

</semantics>

</math><img src="./86bece9c0272e14e23c6c43e0164f292d329e19e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.956ex; height:2.843ex;" alt="{\displaystyle (U,x)}" loading="lazy"> consisting of a scheme U in C and x in the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(U)=\operatorname {Hom} _{C}(U,X)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X(U)=\operatorname {Hom} _{C}(U,X)}</annotation>

</semantics>

</math><img src="./57ae613928f6f7ffb39b45be93361f14c2c67e41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.599ex; height:2.843ex;" alt="{\displaystyle X(U)=\operatorname {Hom} _{C}(U,X)}" loading="lazy">,</li>

<li>a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,x)\to (V,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (U,x)\to (V,y)}</annotation>

</semantics>

</math><img src="./f17b01f63ba317d5a0ec93e5159aa1675e045440.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.356ex; height:2.843ex;" alt="{\displaystyle (U,x)\to (V,y)}" loading="lazy"> consists of an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\to V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\to V}</annotation>

</semantics>

</math><img src="./d8aade74fbc3d516467efd200969ce325f5425f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.184ex; height:2.176ex;" alt="{\displaystyle U\to V}" loading="lazy"> in C and an element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in G(U)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\in G(U)}</annotation>

</semantics>

</math><img src="./34ea73c8b8fed28e812ebefd8cd6c9c30606a46b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.375ex; height:2.843ex;" alt="{\displaystyle g\in G(U)}" loading="lazy"> such that xg = y' where we wrote <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y':U\to V{\overset {y}{\to }}X}">

<msup>

</msup>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y':U\to V{\overset {y}{\to }}X}</annotation>

</semantics>

</math><img src="./58e58f98b30410e1b713c378e068bb119614f1e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.27ex; height:3.676ex;" alt="{\displaystyle y':U\to V{\overset {y}{\to }}X}" loading="lazy">.</li></ol>

Through the forgetful functor to C, this category F is <a href="Fibered_category" class="mw-redirect" title="Fibered category">fibered</a> in <a href="Groupoid" title="Groupoid">groupoids</a> and is known as an action groupoid or a transformation groupoid. It may also be called the quotient prestack of X by G and be denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X/G]^{pre}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [X/G]^{pre}}</annotation>

</semantics>

</math><img src="./74f9977b2460dc6c247422e228c53417d12cb205.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.83ex; height:2.843ex;" alt="{\displaystyle [X/G]^{pre}}" loading="lazy">, since, as it turns out, the stackification of it is the <a href="Quotient_stack" title="Quotient stack">quotient stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X/G]}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [X/G]}</annotation>

</semantics>

</math><img src="./6c10708eac95ed8fcce0dd132f18c6d844536ebf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.263ex; height:2.843ex;" alt="{\displaystyle [X/G]}" loading="lazy">. The construction is a special case of forming <a href="#The_prestack_of_equivalence_classes">#The prestack of equivalence classes</a>; in particular, F is a prestack.

When X is a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle *=\operatorname {Spec} (k)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle *=\operatorname {Spec} (k)}</annotation>

</semantics>

</math><img src="./0bffa6eabbe687872d82961a1f02ba72fc90602f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.931ex; height:2.843ex;" alt="{\displaystyle *=\operatorname {Spec} (k)}" loading="lazy"> and G is affine, the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [*/G]^{pre}=BG^{pre}}">

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [*/G]^{pre}=BG^{pre}}</annotation>

</semantics>

</math><img src="./7fb7cff556ee3ac7ef369430ed9b13837c987f4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.268ex; height:2.843ex;" alt="{\displaystyle [*/G]^{pre}=BG^{pre}}" loading="lazy"> is the classifying prestack of G and its stackification is the <a href="Classifying_stack" class="mw-redirect" title="Classifying stack">classifying stack</a> of G.

One viewing X as a prestack (in fact a stack), there is the obvious canonical map

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi :X\to F}</annotation>

</semantics>

</math><img src="./9f3985f796cb0cba37c86912a3972c96c1a7f901.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.604ex; height:2.176ex;" alt="{\displaystyle \pi :X\to F}" loading="lazy"></dd></dl>

over C; explicitly, each object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,x:U\to X)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (U,x:U\to X)}</annotation>

</semantics>

</math><img src="./55527e15eeb2505374e7443284755b39d4ad2b88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.269ex; height:2.843ex;" alt="{\displaystyle (U,x:U\to X)}" loading="lazy"> in the prestack X goes to itself, and each morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,x)\to (V,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (U,x)\to (V,y)}</annotation>

</semantics>

</math><img src="./f17b01f63ba317d5a0ec93e5159aa1675e045440.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.356ex; height:2.843ex;" alt="{\displaystyle (U,x)\to (V,y)}" loading="lazy">, satisfying x equals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\to V{\overset {y}{\to }}X}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\to V{\overset {y}{\to }}X}</annotation>

</semantics>

</math><img src="./6fe2754bc685aba6be335db1a5f0987536c74fab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.488ex; height:3.343ex;" alt="{\displaystyle U\to V{\overset {y}{\to }}X}" loading="lazy"> by definition, goes to the identity group element of G(U).

Then the above canonical map fits into a 2-<a href="Coequalizer" title="Coequalizer">coequalizer</a> (a 2-quotient):

<mover>

</munder>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X\times G{\overset {s}{\underset {t}{\rightrightarrows }}}X{\overset {\pi }{\to }}F}</annotation>

</semantics>

</math><img src="./e4a3d796cf01a19b421c5d26abebe4b980dee7da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.015ex; height:5.176ex;" alt="{\displaystyle X\times G{\overset {s}{\underset {t}{\rightrightarrows }}}X{\overset {\pi }{\to }}F}" loading="lazy">,</dd></dl>

where t: (x, g) → xg is the given group action and s a projection. It is not 1-coequalizer since, instead of the equality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \circ s=\pi \circ t}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi \circ s=\pi \circ t}</annotation>

</semantics>

</math><img src="./4837b65d47ad841cb70090d1422a14e8513efa3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.082ex; height:2.009ex;" alt="{\displaystyle \pi \circ s=\pi \circ t}" loading="lazy">, one has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \circ s{\overset {\sim }{\to }}\pi \circ t}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi \circ s{\overset {\sim }{\to }}\pi \circ t}</annotation>

</semantics>

</math><img src="./df4a5a1acff9c3cc59c7b12ee8d828f51ed53563.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.307ex; height:2.843ex;" alt="{\displaystyle \pi \circ s{\overset {\sim }{\to }}\pi \circ t}" loading="lazy"> given by

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g:(\pi \circ s)(x,g)=\pi (x){\overset {\sim }{\to }}(\pi \circ t)(x,g)=\pi (xg).}</annotation>

</semantics>

</math><img src="./fa20d062962ca49b2713b75c7dfbd962ba8051e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.459ex; height:3.343ex;" alt="{\displaystyle g:(\pi \circ s)(x,g)=\pi (x){\overset {\sim }{\to }}(\pi \circ t)(x,g)=\pi (xg).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="The_prestack_of_equivalence_classes">The prestack of equivalence classes</h2></div>

Let X be a scheme in the base category C. By definition, an <a href="Quotient_by_an_equivalence_relation" title="Quotient by an equivalence relation">equivalence pre-relation</a> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\to X\times X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\to X\times X}</annotation>

</semantics>

</math><img src="./8352796d9e2dbae3dc9e4dea0219a5018982db46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.178ex; height:2.176ex;" alt="{\displaystyle R\to X\times X}" loading="lazy"> in C such that, for each scheme T in C, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(T):R(T)=\operatorname {Hom} (T,R)\to X(T)\times X(T)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(T):R(T)=\operatorname {Hom} (T,R)\to X(T)\times X(T)}</annotation>

</semantics>

</math><img src="./436854cd381c32e18f3f34bd1ac40cb7b406ca01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.36ex; height:2.843ex;" alt="{\displaystyle f(T):R(T)=\operatorname {Hom} (T,R)\to X(T)\times X(T)}" loading="lazy"> has the image that is an <a href="Equivalence_relation" title="Equivalence relation">equivalence relation</a>. The prefix "pre-" is because we do not require <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(T)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(T)}</annotation>

</semantics>

</math><img src="./71e79e53bc5161b01ee450c15644a4e9417fb0ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle f(T)}" loading="lazy"> to be an <a href="Injective_function" title="Injective function">injective function</a>.

Example: Let an algebraic group G act on a scheme X of finite type over a field k. Take <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=X\times _{k}G}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R=X\times _{k}G}</annotation>

</semantics>

</math><img src="./75865ebe0f91804760cef8f874fa022e825dd1d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.598ex; height:2.509ex;" alt="{\displaystyle R=X\times _{k}G}" loading="lazy"> and then for any scheme T over k let

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(T):R(T)\to X(T)\times X(T),\,(x,g)\mapsto (x,xg).}</annotation>

</semantics>

</math><img src="./dd8aebd1e77570a646dead28731273e66e050805.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.766ex; height:2.843ex;" alt="{\displaystyle f(T):R(T)\to X(T)\times X(T),\,(x,g)\mapsto (x,xg).}" loading="lazy"></dd></dl>

By <a href="Yoneda's_lemma" class="mw-redirect" title="Yoneda's lemma">Yoneda's lemma</a>, this determines a morphism f, which is clearly an equivalence pre-relation.

To each given equivalence pre-relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:R\to X\times X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:R\to X\times X}</annotation>

</semantics>

</math><img src="./c172c4669a538f246d4b7a41a3025e1482dc9643.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.394ex; height:2.509ex;" alt="{\displaystyle f:R\to X\times X}" loading="lazy"> (+ some more data), there is an associated prestack F defined as follows.<a href="#cite_note-8">[8]</a> Firstly, F is a category where: with the notations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=p_{1}\circ f,\,t=p_{2}\circ f}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s=p_{1}\circ f,\,t=p_{2}\circ f}</annotation>

</semantics>

</math><img src="./9fa2073597f13c62d42dcaea1f848906aa9b5f56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.942ex; height:2.509ex;" alt="{\displaystyle s=p_{1}\circ f,\,t=p_{2}\circ f}" loading="lazy">,

<div><ol><li>an object is a pair <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T,x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (T,x)}</annotation>

</semantics>

</math><img src="./43ea30dfe5b534691891ee68df7ebba033e14360.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.809ex; height:2.843ex;" alt="{\displaystyle (T,x)}" loading="lazy"> consisting of a scheme T and a morphism x: T → X in C</li><li>a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T,x)\to (S,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (T,x)\to (S,y)}</annotation>

</semantics>

</math><img src="./3f69e38ab8381d4dc2d7517dd49d8902f5522688.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.921ex; height:2.843ex;" alt="{\displaystyle (T,x)\to (S,y)}" loading="lazy"> consists of a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\to S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\to S}</annotation>

</semantics>

</math><img src="./07b18cbacd0b5f9bd8d9eb4759a1df899435f1b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.75ex; height:2.176ex;" alt="{\displaystyle T\to S}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta :T\to R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta :T\to R}</annotation>

</semantics>

</math><img src="./52fb07d0050503a35f84bf6cd35d9aac773863fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10ex; height:2.343ex;" alt="{\displaystyle \delta :T\to R}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\circ \delta =x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\circ \delta =x}</annotation>

</semantics>

</math><img src="./8c9e73692fd6ea618901659847f43e2ec468da29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.762ex; height:2.343ex;" alt="{\displaystyle s\circ \delta =x}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\circ \delta =y|_{T}:T\to S{\overset {y}{\to }}X}">

<msub>

</mrow>

</mrow>

</msub>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\circ \delta =y|_{T}:T\to S{\overset {y}{\to }}X}</annotation>

</semantics>

</math><img src="./dbe61d956fbc1eea775ec277deff984dbe3817ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.364ex; height:3.843ex;" alt="{\displaystyle t\circ \delta =y|_{T}:T\to S{\overset {y}{\to }}X}" loading="lazy"></li><li>the composition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (,\delta ):(T,x)\to (S,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (,\delta ):(T,x)\to (S,y)}</annotation>

</semantics>

</math><img src="./ff76c99c7af7f999fbde0059f7836c8b39f557d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.75ex; height:2.843ex;" alt="{\displaystyle (,\delta ):(T,x)\to (S,y)}" loading="lazy"> followed by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (,\delta '):(S,y)\to (U,z)}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (,\delta '):(S,y)\to (U,z)}</annotation>

</semantics>

</math><img src="./4f3d343441570f1161e4fda249b0bcb9b21e0457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.345ex; height:3.009ex;" alt="{\displaystyle (,\delta '):(S,y)\to (U,z)}" loading="lazy"> consists of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\to S\to U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\to S\to U}</annotation>

</semantics>

</math><img src="./55945ec96fc2fabd6d16c225f082719897ab005d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.146ex; height:2.176ex;" alt="{\displaystyle T\to S\to U}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta '':T\to R}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta '':T\to R}</annotation>

</semantics>

</math><img src="./9329bc541dd97a4e37387ef4d37b130f78c21831.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.142ex; height:2.509ex;" alt="{\displaystyle \delta '':T\to R}" loading="lazy"> obtained as follows: since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\circ \delta =y|_{T}=s\circ \delta '|_{T}}">

<msub>

</mrow>

</mrow>

</msub>

<msup>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\circ \delta =y|_{T}=s\circ \delta '|_{T}}</annotation>

</semantics>

</math><img src="./f2a38952c185e23f25ec6ca6107f5aac057a7d47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.531ex; height:3.009ex;" alt="{\displaystyle t\circ \delta =y|_{T}=s\circ \delta '|_{T}}" loading="lazy">, by the universal property, there is an induced map

<msup>

</msup>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (\delta ,\delta '|_{T}):T\to R\times _{t,s}R}</annotation>

</semantics>

</math><img src="./c73c90a0048f04e8736aa7ffdd90ade97f88c206.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.277ex; height:3.176ex;" alt="{\displaystyle (\delta ,\delta '|_{T}):T\to R\times _{t,s}R}" loading="lazy">.</dd></dl>

Then let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ''}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta ''}</annotation>

</semantics>

</math><img src="./e1c98b36e888b2dffe5c9a7bd7e29dda444adada.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.191ex; height:2.509ex;" alt="{\displaystyle \delta ''}" loading="lazy"> be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\to R\times _{t,s}R}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T\to R\times _{t,s}R}</annotation>

</semantics>

</math><img src="./110b8e7d13bc2366dbbfe688b872e96e545d4b3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.673ex; height:2.843ex;" alt="{\displaystyle T\to R\times _{t,s}R}" loading="lazy"> followed by the multiplication</li><li>the identity morphism for an object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (T,x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (T,x)}</annotation>

</semantics>

</math><img src="./43ea30dfe5b534691891ee68df7ebba033e14360.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.809ex; height:2.843ex;" alt="{\displaystyle (T,x)}" loading="lazy"> consists of the identity map T → T and δ that is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x:T\to X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x:T\to X}</annotation>

</semantics>

</math><img src="./bc1dc81ec58015f016e7d89645cc66757e92e7f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.497ex; height:2.176ex;" alt="{\displaystyle x:T\to X}" loading="lazy"> followed by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e:X\to R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e:X\to R}</annotation>

</semantics>

</math><img src="./b5196c44695a2af627cc7c966576663d095a9132.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.379ex; height:2.176ex;" alt="{\displaystyle e:X\to R}" loading="lazy">; the latter is obtained by factorizing the diagonal morphism through f, possible by reflexivity.</li></ol></div>

Via a forgetful functor, the category F is fibered in groupoids. Finally, we check F is a prestack;<a href="#cite_note-9">[9]</a> for that, notice: for objects x, y in F(U) and an object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:V\to U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:V\to U}</annotation>

</semantics>

</math><img src="./15330bffdda84438fcd2ab838c8437f0ac0e9d1e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.4ex; height:2.509ex;" alt="{\displaystyle f:V\to U}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{/U}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{/U}}</annotation>

</semantics>

</math><img src="./0322cbe23a276f07a23c5f74619cfc1cd558cf3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:3.977ex; height:3.009ex;" alt="{\displaystyle C_{/U}}" loading="lazy">,

<mtr>

<mtd>

</munder>

</mrow>

<mover>

</mover>

</mrow>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\underline {\operatorname {Hom} }}(x,y)(V{\overset {f}{\to }}U)&=[\operatorname {Hom} (f^{*}x,f^{*}y)]\\&=[\{\delta :V\to R|s\circ \delta =f^{*}x,t\circ \delta =f^{*}y\}]\\&=[\{\delta :V\to R|(s,t)\circ \delta =(x,y)\circ f\}].\end{aligned}}}</annotation>

</semantics>

</math><img src="./28ef55b7927e941e80817efcb46c80ebbaf52dec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:59.024ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}{\underline {\operatorname {Hom} }}(x,y)(V{\overset {f}{\to }}U)&=[\operatorname {Hom} (f^{*}x,f^{*}y)]\\&=[\{\delta :V\to R|s\circ \delta =f^{*}x,t\circ \delta =f^{*}y\}]\\&=[\{\delta :V\to R|(s,t)\circ \delta =(x,y)\circ f\}].\end{aligned}}}" loading="lazy"></dd></dl>

Now, this means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\operatorname {Hom} }}(x,y)}">

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {\operatorname {Hom} }}(x,y)}</annotation>

</semantics>

</math><img src="./df87893de4d0e88c6789621e1987802776866d22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:10.172ex; height:3.343ex;" alt="{\displaystyle {\underline {\operatorname {Hom} }}(x,y)}" loading="lazy"> is the fiber product of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s,t):R\to X\times X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (s,t):R\to X\times X}</annotation>

</semantics>

</math><img src="./291e7d200de8a44d47600a506c16fa469ebbe688.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.889ex; height:2.843ex;" alt="{\displaystyle (s,t):R\to X\times X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y):U\to X\times X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y):U\to X\times X}</annotation>

</semantics>

</math><img src="./bd3258e66f41077eb3e7dfe1c97e516672b8ede2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.463ex; height:2.843ex;" alt="{\displaystyle (x,y):U\to X\times X}" loading="lazy">. Since the fiber product of sheaves is a sheaf, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\operatorname {Hom} }}(x,y)}">

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {\operatorname {Hom} }}(x,y)}</annotation>

</semantics>

</math><img src="./df87893de4d0e88c6789621e1987802776866d22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-bottom: -0.755ex; width:10.172ex; height:3.343ex;" alt="{\displaystyle {\underline {\operatorname {Hom} }}(x,y)}" loading="lazy"> is a sheaf.

The prestack F above may be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X/\sim _{R}]^{pre}}">

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [X/\sim _{R}]^{pre}}</annotation>

</semantics>

</math><img src="./22a147e4e5d5ca8d43790af920a17cb11cd19b95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.936ex; height:2.843ex;" alt="{\displaystyle [X/\sim _{R}]^{pre}}" loading="lazy"> and the stackification of it is written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X/\sim _{R}]}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [X/\sim _{R}]}</annotation>

</semantics>

</math><img src="./0a61697b2c20b647491031ae8146dfd57f62560b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.369ex; height:2.843ex;" alt="{\displaystyle [X/\sim _{R}]}" loading="lazy">.

Note, when X is viewed as a stack, both X and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X/\sim _{R}]^{pre}}">

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [X/\sim _{R}]^{pre}}</annotation>

</semantics>

</math><img src="./22a147e4e5d5ca8d43790af920a17cb11cd19b95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.936ex; height:2.843ex;" alt="{\displaystyle [X/\sim _{R}]^{pre}}" loading="lazy"> have the same set of objects. On the morphism-level, while X has only identity morphisms as morphisms, the prestack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [X/\sim _{R}]^{pre}}">

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [X/\sim _{R}]^{pre}}</annotation>

</semantics>

</math><img src="./22a147e4e5d5ca8d43790af920a17cb11cd19b95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.936ex; height:2.843ex;" alt="{\displaystyle [X/\sim _{R}]^{pre}}" loading="lazy"> have additional morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>

</semantics>

</math><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> specified by the equivalence pre-relation f.

One importance of this construction is that it provides an atlas for an algebraic space: every <a href="Algebraic_space" title="Algebraic space">algebraic space</a> is of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [U/\sim _{R}]}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [U/\sim _{R}]}</annotation>

</semantics>

</math><img src="./6d99afd072438524e2ce3d7dcddf0948f89f2c9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.172ex; height:2.843ex;" alt="{\displaystyle [U/\sim _{R}]}" loading="lazy"> for some schemes U, R and an étale equivalence pre-relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:R\to U\times U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:R\to U\times U}</annotation>

</semantics>

</math><img src="./2e4f0dc9eb6e74ce0100af71021b982c4fbb629c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.999ex; height:2.509ex;" alt="{\displaystyle f:R\to U\times U}" loading="lazy"> such that, for each T, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(T):R(T)\to U(T)\times U(T)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(T):R(T)\to U(T)\times U(T)}</annotation>

</semantics>

</math><img src="./af6d6669109d62ee40ab8d46c4c4cc68d0a3236a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.782ex; height:2.843ex;" alt="{\displaystyle f(T):R(T)\to U(T)\times U(T)}" loading="lazy"> is an injective function ("étale" means the two possible maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s,t:R\to U\times U\to U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s,t:R\to U\times U\to U}</annotation>

</semantics>

</math><img src="./ba6e373cd43c5c8e91baeef4345475b6f72c5621.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.081ex; height:2.509ex;" alt="{\displaystyle s,t:R\to U\times U\to U}" loading="lazy"> are étale.)

Starting from a <a href="Deligne%E2%80%93Mumford_stack" title="Deligne–Mumford stack">Deligne–Mumford stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {X}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {X}}}</annotation>

</semantics>

</math><img src="./073868db564a5e806f61f8b2d526ba975cb5c077.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.671ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {X}}}" loading="lazy">, one can find an equivalence pre-relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:R\to U\times U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:R\to U\times U}</annotation>

</semantics>

</math><img src="./2e4f0dc9eb6e74ce0100af71021b982c4fbb629c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.999ex; height:2.509ex;" alt="{\displaystyle f:R\to U\times U}" loading="lazy"> for some schemes R, U so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {X}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {X}}}</annotation>

</semantics>

</math><img src="./073868db564a5e806f61f8b2d526ba975cb5c077.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.671ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {X}}}" loading="lazy"> is the stackification of the prestack associated to it: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {X}}\simeq [U/\sim _{R}]}">

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {X}}\simeq [U/\sim _{R}]}</annotation>

</semantics>

</math><img src="./cff3d7206ebceb82602b4ab76c99753404d6e1bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.941ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {X}}\simeq [U/\sim _{R}]}" loading="lazy">.<a href="#cite_note-10">[10]</a> This is done as follows. By definition, there is an étale surjective morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :U\to {\mathfrak {X}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi :U\to {\mathfrak {X}}}</annotation>

</semantics>

</math><img src="./bbb6276f28538ee061425baca10971f9480add29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.337ex; height:2.176ex;" alt="{\displaystyle \pi :U\to {\mathfrak {X}}}" loading="lazy"> from some scheme U. Since the diagonal is strongly representable, the fiber product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\times _{\mathfrak {X}}U=R}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\times _{\mathfrak {X}}U=R}</annotation>

</semantics>

</math><img src="./87b3f49d9d6ffd22f2e282ee2e54fff4c7f7b99a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.682ex; height:2.509ex;" alt="{\displaystyle U\times _{\mathfrak {X}}U=R}" loading="lazy"> is a scheme (that is, represented by a scheme) and then let

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s,t:R\rightrightarrows U}</annotation>

</semantics>

</math><img src="./1da0debe3cb75632315c5896cb99d31641278ea3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.062ex; height:2.509ex;" alt="{\displaystyle s,t:R\rightrightarrows U}" loading="lazy"></dd></dl>

be the first and second projections. Taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=(s,t):R\to U\times U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=(s,t):R\to U\times U}</annotation>

</semantics>

</math><img src="./3191580b86fed06ac8c8f4d9499032f54041e914.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.871ex; height:2.843ex;" alt="{\displaystyle f=(s,t):R\to U\times U}" loading="lazy">, we see <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is an equivalence pre-relation. We finish, roughly, as follows.

<ol><li>Extend <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :U\to {\mathfrak {X}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi :U\to {\mathfrak {X}}}</annotation>

</semantics>

</math><img src="./bbb6276f28538ee061425baca10971f9480add29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.337ex; height:2.176ex;" alt="{\displaystyle \pi :U\to {\mathfrak {X}}}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :[U/\sim _{R}]^{pre}\to {\mathfrak {X}}}">

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi :[U/\sim _{R}]^{pre}\to {\mathfrak {X}}}</annotation>

</semantics>

</math><img src="./558dcb363287c6f31c8eaf59bf5802c69aab4cc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.293ex; height:2.843ex;" alt="{\displaystyle \pi :[U/\sim _{R}]^{pre}\to {\mathfrak {X}}}" loading="lazy"> (nothing changes on the object-level; we only need to explain how to send <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta }</annotation>

</semantics>

</math><img src="./c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy">.)</li>

<li>By the universal property of stackification, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [U/\sim _{R}]\to {\mathfrak {X}}}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [U/\sim _{R}]\to {\mathfrak {X}}}</annotation>

</semantics>

</math><img src="./4e649ee760353f98cb803dcfc99928f8de2ef753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.457ex; height:2.843ex;" alt="{\displaystyle [U/\sim _{R}]\to {\mathfrak {X}}}" loading="lazy">.</li>

<li>Check the last map is an isomorphism.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Stacks_associated_to_prestacks">Stacks associated to prestacks</h2></div>

There is a way to associate a stack to a given prestack. It is similar to the <a href="Sheafification" class="mw-redirect" title="Sheafification">sheafification</a> of a presheaf and is called stackification. The idea of the construction is quite simple: given a prestack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:F\to C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p:F\to C}</annotation>

</semantics>

</math><img src="./c66d3f389313da08201d63cee44500df224b5fa3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.317ex; height:2.509ex;" alt="{\displaystyle p:F\to C}" loading="lazy">, we let HF be the category where an object is a descent datum and a morphism is that of descent data. (The details are omitted for now)

As it turns out, it is a stack and comes with a natural morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta :F\to HF}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta :F\to HF}</annotation>

</semantics>

</math><img src="./37ed150f1dd5ebeaf2799d2a6a1ed544d899139f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.187ex; height:2.176ex;" alt="{\displaystyle \theta :F\to HF}" loading="lazy"> such that F is a stack if and only if θ is an isomorphism.

In some special cases, the stackification can be described in terms of <a href="Torsor_(algebraic_geometry)" title="Torsor (algebraic geometry)">torsors</a> for affine group schemes or the generalizations. In fact, according to this point of view, a stack in groupoids is nothing but a category of torsors, and a prestack a category of trivial torsors, which are local models of torsors.

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFVistoli2005">Vistoli 2005</a>, § 3.7.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFBehrendConradEdidinFulton2006">Behrend et al. 2006</a>, Ch. 4., § 1.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFVistoli2005">Vistoli 2005</a>, Definition 4.6.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFVistoli2005">Vistoli 2005</a>, § 3.6.2.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFVistoli2005">Vistoli 2005</a>, Definition 3.33.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFBehrendConradEdidinFulton2006">Behrend et al. 2006</a>, Definition 2.25.

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFBehrendConradEdidinFulton2006">Behrend et al. 2006</a>, Example 2.29.

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFBehrendConradEdidinFulton2006">Behrend et al. 2006</a>, Definition 3.13.

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> The argument here is Lemma 25.6. of <a rel="nofollow" class="external text" href="https://stacky.net/files/written/Stacks/Stacks.pdf">M. Olsson's lecture notes on stacks</a>.

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <a href="#CITEREFBehrendConradEdidinFulton2006">Behrend et al. 2006</a>, Proposition 5.20. and <a href="#CITEREFBehrendConradEdidinFulton2006">Behrend et al. 2006</a>, Theorem 4.35.. Editorial note: the reference uses the language of groupoid schemes but a groupoid scheme they use is the same as an equivalence pre-relation used here; compare Proposition 3.6. and the verifications below.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFBehrendConradEdidinFulton2006" class="citation cs2">Behrend, Kai; Conrad, Brian; Edidin, Dan; Fulton, William; Fantechi, Barbara; Göttsche, Lothar; Kresch, Andrew (2006), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20080505043444/http://www.math.unizh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1#">Algebraic stacks</a>, archived from <a rel="nofollow" class="external text" href="http://www.math.unizh.ch/index.php?pr_vo_det&key1=1287&key2=580&no_cache=1">the original</a> on 2008-05-05, retrieved 2017-06-13</cite></li>

<li><cite id="CITEREFVistoli2005" class="citation cs2">Vistoli, Angelo (2005), "Grothendieck topologies, fibered categories and descent theory", Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.: Amer. Math. Soc., pp. 1–104, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0412512">math/0412512</a>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004math.....12512V">2004math.....12512V</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2223406">2223406</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFDai_Tamaki2019" class="citation web cs1">Dai Tamaki (August 7, 2019). <a rel="nofollow" class="external text" href="https://mathoverflow.net/q/3119">"Prestacks and fibered categories"</a>.</cite></li></ul></div><div><div class="zim-footer">

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