The Stacks project

Definition 99.39.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Consider a $2$-commutative solid diagram

99.39.1.1
\begin{equation} \label{stacks-morphisms-equation-diagram} \vcenter { \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r]_-x \ar[d]_ j & \mathcal{X} \ar[d]^ f \\ \mathop{\mathrm{Spec}}(A) \ar[r]^-y \ar@{..>}[ru] & \mathcal{Y} } } \end{equation}

where $A$ is a valuation ring with field of fractions $K$. Let

\[ \gamma : y \circ j \longrightarrow f \circ x \]

be a $2$-morphism witnessing the $2$-commutativity of the diagram. (Notation as in Categories, Sections 4.28 and 4.29.) Given (99.39.1.1) and $\gamma $ a dotted arrow is a triple $(a, \alpha , \beta )$ consisting of a morphism $a : \mathop{\mathrm{Spec}}(A) \to \mathcal{X}$ and $2$-arrows $\alpha : a \circ j \to x$, $\beta : y \to f \circ a$ such that $\gamma = (\text{id}_ f \star \alpha ) \circ (\beta \star \text{id}_ j)$, in other words such that

\[ \xymatrix{ & f \circ a \circ j \ar[rd]^{\text{id}_ f \star \alpha } \\ y \circ j \ar[ru]^{\beta \star \text{id}_ j} \ar[rr]^\gamma & & f \circ x } \]

is commutative. A morphism of dotted arrows $(a, \alpha , \beta ) \to (a', \alpha ', \beta ')$ is a $2$-arrow $\theta : a \to a'$ such that $\alpha = \alpha ' \circ (\theta \star \text{id}_ j)$ and $\beta ' = (\text{id}_ f \star \theta ) \circ \beta $.


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