Lemma 101.41.1 (0E8L)—The Stacks project

Lemma 101.41.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Consider a $2$-commutative solid diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[rr]_-x \ar[d]_ j & & \mathcal{X} \ar[d]^{\Delta _ f} \\ \mathop{\mathrm{Spec}}(A) \ar[rr]^{(a_1, a_2, \varphi )} \ar@{..>}[rru] & & \mathcal{X} \times _\mathcal {Y} \mathcal{X} } \]

where $A$ is a valuation ring with field of fractions $K$. Let $\gamma : (a_1, a_2, \varphi ) \circ j \longrightarrow \Delta _ f \circ x$ be a $2$-morphism witnessing the $2$-commutativity of the diagram. Then

Writing $\gamma = (\alpha _1, \alpha _2)$ with $\alpha _ i : a_ i \circ j \to x$ we obtain two dotted arrows $(a_1, \alpha _1, \text{id})$ and $(a_2, \alpha _2, \varphi )$ in the diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-x \ar[d]_ j & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r]^-{f \circ a_1} \ar@{..>}[ru] & \mathcal{Y} } \]
The category of dotted arrows for the original diagram and $\gamma $ is a setoid whose set of isomorphism classes of objects equal to the set of morphisms $(a_1, \alpha _1, \text{id}) \to (a_2, \alpha _2, \varphi )$ in the category of dotted arrows.

Proof. Since $\Delta _ f$ is representable by algebraic spaces (hence the diagonal of $\Delta _ f$ is separated), we see that the category of dotted arrows in the first commutative diagram of the lemma is a setoid by Lemma 101.39.2. All the other statements of the lemma are consequences of $2$-diagramatic computations which we omit. $\square$

The Stacks project

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