Lemma 97.12.1 (05XN)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 97: Criteria for Representability
Section 97.12: Finite Hilbert stacks
Lemma 97.12.1 (cite)

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Lemma 97.12.1. Consider a $2$-commutative diagram

\[ \xymatrix{ \mathcal{X}' \ar[r]_ G \ar[d]_{F'} & \mathcal{X} \ar[d]^ F \\ \mathcal{Y}' \ar[r]^ H & \mathcal{Y} } \]

of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$ with a given $2$-isomorphism $\gamma : H \circ F' \to F \circ G$. In this situation we obtain a canonical $1$-morphism $\mathcal{H}_ d(\mathcal{X}'/\mathcal{Y}') \to \mathcal{H}_ d(\mathcal{X}/\mathcal{Y})$. This morphism is compatible with the forgetful $1$-morphisms of Examples of Stacks, Equation (95.18.2.1).

Proof. We map the object $(U, Z, y', x', \alpha ')$ to the object $(U, Z, H(y'), G(x'), \gamma \star \text{id}_ H \star \alpha ')$ where $\star $ denotes horizontal composition of $2$-morphisms, see Categories, Definition 4.28.1. To a morphism $(f, g, b, a) : (U_1, Z_1, y_1', x_1', \alpha _1') \to (U_2, Z_2, y_2', x_2', \alpha _2')$ we assign $(f, g, H(b), G(a))$. We omit the verification that this defines a functor between categories over $(\mathit{Sch}/S)_{fppf}$. $\square$

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