# The Stacks Project

## Tag 0CLA

Definition 91.38.1. Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks. Consider a $2$-commutative solid diagram $$\tag{91.38.1.1} \vcenter{ \xymatrix{ \mathop{\rm Spec}(K) \ar[r]_-x \ar[d]_j & \mathcal{X} \ar[d]^f \\ \mathop{\rm Spec}(A) \ar[r]^-y \ar@{..>}[ru] & \mathcal{Y} } }$$ 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 4.27 and 4.28.) Given (91.38.1.1) and $\gamma$ a dotted arrow is a triple $(a, \alpha, \beta)$ consisting of a morphism $a : \mathop{\rm Spec}(A) \to \mathcal{X}$ 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 $\beta' = (\text{id}_f \star \theta) \circ \beta$.

The code snippet corresponding to this tag is a part of the file stacks-morphisms.tex and is located in lines 8585–8624 (see updates for more information).

\begin{definition}
\label{definition-fill-in-diagram}
Let $f : \mathcal{X} \to \mathcal{Y}$ be a morphism of algebraic stacks.
Consider a $2$-commutative solid diagram

\label{equation-diagram}
\vcenter{
\xymatrix{
\Spec(K) \ar[r]_-x \ar[d]_j & \mathcal{X} \ar[d]^f \\
\Spec(A) \ar[r]^-y \ar@{..>}[ru] & \mathcal{Y}
}
}

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 \ref{categories-section-formal-cat-cat}
and \ref{categories-section-2-categories}.)
Given (\ref{equation-diagram}) and $\gamma$
a {\it dotted arrow} is a triple $(a, \alpha, \beta)$ consisting of a
morphism $a : \Spec(A) \to \mathcal{X}$ 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 {\it 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
$\beta' = (\text{id}_f \star \theta) \circ \beta$.
\end{definition}

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