Editing Algebraic stack (section)

===== Representable diagonal =====
By definition, a 1-morphism <math>f:\mathcal{X} \to \mathcal{Y}</math> of categories fibered in groupoids is '''representable by algebraic spaces'''<ref>{{Cite web|title=Section 92.9 (04SX): Morphisms representable by algebraic spaces—The Stacks project|url=https://stacks.math.columbia.edu/tag/04SX|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> if for any fppf morphism <math>U \to S</math> of schemes and any 1-morphism <math>y: (Sch/U)_{fppf} \to \mathcal{Y}</math>, the associated category fibered in groupoids<blockquote><math>(Sch/U)_{fppf}\times_{\mathcal{Y}} \mathcal{X}</math></blockquote>is '''representable as an algebraic space''',<ref>{{Cite web|title=Section 92.7 (04SU): Split categories fibred in groupoids—The Stacks project|url=https://stacks.math.columbia.edu/tag/04SU|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref><ref>{{Cite web|title=Section 92.8 (02ZV): Categories fibred in groupoids representable by algebraic spaces—The Stacks project|url=https://stacks.math.columbia.edu/tag/02ZV|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> meaning there exists an algebraic space<blockquote><math>F:(Sch/S)^{op}_{fppf} \to Sets</math></blockquote>such that the associated fibered category <math>\mathcal{S}_F \to (Sch/S)_{fppf}</math><ref><math>Sets \to Cat</math> is the embedding sending a set <math>S</math> to the category of objects <math>S</math> and only identity morphisms. Then, the Grothendieck construction can be applied to give a category fibered in groupoids</ref> is equivalent to <math>(Sch/U)_{fppf}\times_{\mathcal{Y}} \mathcal{X}</math>. There are a number of equivalent conditions for representability of the diagonal<ref>{{Cite web|title=Lemma 92.10.11 (045G)—The Stacks project|url=https://stacks.math.columbia.edu/tag/045G|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref> which help give intuition for this technical condition, but one of main motivations is the following: for a scheme <math>U</math> and objects <math>x, y \in \operatorname{Ob}(\mathcal{X}_U)</math> the sheaf <math>\operatorname{Isom}(x,y)</math> is representable as an algebraic space. In particular, the stabilizer group for any point on the stack <math>x : \operatorname{Spec}(k) \to \mathcal{X}_{\operatorname{Spec}(k)}</math> is representable as an algebraic space.

Another important equivalence of having a representable diagonal is the technical condition that the intersection of any two algebraic spaces in an algebraic stack is an algebraic space. Reformulated using fiber products<blockquote><math>\begin{matrix}
Y \times_{\mathcal{X}}Z & \to & Y \\
\downarrow & & \downarrow \\
Z & \to & \mathcal{X}
\end{matrix}</math></blockquote>the representability of the diagonal is equivalent to <math>Y \to \mathcal{X}</math> being representable for an algebraic space <math>Y</math>. This is because given morphisms <math>Y \to \mathcal{X}, Z \to \mathcal{X}</math> from algebraic spaces, they extend to maps <math>\mathcal{X}\times\mathcal{X}</math> from the diagonal map. There is an analogous statement for algebraic spaces which gives representability of a sheaf on <math>(F/S)_{fppf}</math> as an algebraic space.<ref>{{Cite web|title=Section 78.5 (046I): Bootstrapping the diagonal—The Stacks project|url=https://stacks.math.columbia.edu/tag/046I|access-date=2020-08-29|website=stacks.math.columbia.edu}}</ref>

Note that an analogous condition of representability of the diagonal holds for some formulations of [[higher stacks]]<ref>{{cite arXiv|last=Simpson|first=Carlos|date=1996-09-17|title=Algebraic (geometric) ''n''-stacks|eprint=alg-geom/9609014}}</ref> where the fiber product is an <math>(n-1)</math>-stack for an <math>n</math>-stack <math>\mathcal{X}</math>.