Do disjoint unions of stacks commute with finite fibre products?

Question

Choose a big $\mathit{fppf}$-site $(\mathbf{Sch})_{\mathit{fppf}}$ and let $S$ be a scheme in that site. Let $\{\mathcal{X}_i\mid i\in I\}$ be a family of stacks in groupoids over $S$ and let $\mathcal{Y}\to\mathcal{Z}$ be a morphism of stacks in groupoids over $S$.
Let $\mathcal{X}\colon\!\!=\coprod_{i\in I}\mathcal{X}_i$ be the disjoint union as described in [Champs algébriques, G.Laumon/ L.Moret-Bailly, (3.3)]. Let $-\times-$ denote the $2$-fibre product of stacks in groupoids over $S$. Is there a canonical morphism of stacks $$\coprod_{i\in I}(\mathcal{X}_i\times_{\mathcal{Z}}\mathcal{Y})\to\Big(\coprod_{i\in I}\mathcal{X}_i\Big)\times_{\mathcal{Z}}\mathcal{Y}\quad$$ an isomorphism?

Answer 1

I would split this problem up into two parts (here, 'sheaf (of groupoids)' is used instead of stack in order to disambiguate between Algebraic stacks (geometric objects) and mere (pseudo-)functors satisfying descent). :

1.) Show that the inclusion of algebraic stacks into the category of fppf sheaves of groupoids on Sch preserves coproducts. This follows immediately from the fact that algebraic stacks are a full (2,1)-subcategory of fppf sheaves and that if $F,G$ are two algebraic stacks, their sheafy coproduct is representable by an algebraic stack. If D is a diagram landing in a full subcategory whose limit or colimit exists in the ambient category and is in the full subcategory, this is also a limit or colimit of the diagram landing in the full subcategory without reference to the ambient category. I think the proof here is immediate by taking a disjoint union of the atlases.

2.) Show that colimits are universal in (2,1)-stack topoi. This follows from the left-exactness of the stackification (2,1)-functor together with the altogether more obvious version of this fact for (2,1)-topoi of groupoid fibrations (also called (2,1)-presheaf topoi), where one can immediately reduce to proving the statement for groupoids pointwise.

Answer 2

An object over a scheme $T$ on the left is given by a decomposition of $T$ into a parametrized disjoint union $T_i$ of schemes, and a parametrized family of triples $(x_i, y_i, \phi_i)$, where $x_i$ is an object of $X_i$ over $T_i$, $y_i$ is an object of $Y$ over $T_i$, and $\phi_i$ is an isomorphism $\rho_{X_i}(x_i) \to \rho_Y(y_i)$ in $Z$ over $T_i$. A morphism over $id_T$ is a parametrized family of pairs of maps $(f_i: x_i \to x'_i, g_i: y_i \to y'_i)$ that satisfy suitable commutative square relations. In particular, if two objects come from unequal decompositions of $T$, then there are no morphisms between them. Let us omit discussion of other morphisms, and pretend the "fibered category" property takes care of them.

An object over a scheme $T$ on the right is given by a decomposition of $T$ into a parametrized disjoint union $T_i$ of schemes, and a tuple $((x_i), y, \phi)$, where $x_i$ is an object of $X_i$ over $T_i$, $y$ is an object of $Y$ over $T$, and $\phi$ is an isomorphism $\rho_{\coprod X_i}((x_i)) \to \rho_Y(y)$ in $Z$ over $T$. A morphism over $id_T$ is a pair $((f_i: x_i \to x'_i), g:y \to y')$ that satisfies conditions that I won't describe.

In order to match these data, we need to identify $y$ with the parametrized family $(y_i)$, and $\phi$ with $(\phi_i)$ for objects, and $g$ with $(g_i)$ for morphisms. This is just using the fibered category property: pulling back along the isomorphism $\coprod T_i \to T$ yields a decomposition of $y$ that is unique up to unique isomorphism. It might be helpful to check that the object $\rho_{\coprod X_i}((x_i))$ in $Z$ over $T$ is identified with the tuple $(\rho_{X_i}(x_i))$ of objects in $Z$ over $T_i$.