The Stacks project

4.44 Categories of dotted arrows

We discuss certain “categories of dotted arrows” in $(2,1)$-categories. These will appear when formulating various lifting criteria for algebraic stacks, see for example Morphisms of Stacks, Section 100.39 and More on Morphisms of Stacks, Section 105.8.

Definition 4.44.1. Let $\mathcal{C}$ be a $(2,1)$-category. Consider a $2$-commutative solid diagram
\begin{equation} \label{categories-equation-dotted-arrows} \vcenter { \xymatrix{ S \ar[r]_-x \ar[d]_ j & X \ar[d]^ f \\ T \ar[r]^-y \ar@{..>}[ru] & Y } } \end{equation}

in $\mathcal{C}$. Fix a $2$-isomorphism

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

witnessing the $2$-commutativity of the diagram. Given ( and $\gamma $, a dotted arrow is a triple $(a, \alpha , \beta )$ consisting of a morphism $a \colon T \to X$ and and $2$-isomorphisms $\alpha : a \circ j \to x$, $\beta : y \to f \circ a$ 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 $.

In the situation of Definition 4.44.1, there is an associated category of dotted arrows. This category is a groupoid. It may depend on $\gamma $ in general. The next two lemmas say that categories of dotted arrows are well-behaved with respect to base change and composition for $f$.

Lemma 4.44.2. Let $\mathcal{C}$ be a $(2,1)$-category. Assume given a $2$-commutative diagram

\[ \xymatrix{ S \ar[r]_-{x'} \ar[d]_ j & X' \ar[d]^ p \ar[r]_ q & X \ar[d]^ f \\ T \ar[r]^-{y'} & Y' \ar[r]^ g & Y } \]

in $\mathcal{C}$, where the right square is $2$-cartesian with respect to a $2$-isomorphism $\phi \colon g \circ p \to f \circ q$. Choose a $2$-arrow $\gamma ' : y' \circ j \to p \circ x'$. Set $x = q \circ x'$, $y = g \circ y'$ and let $\gamma : y \circ j \to f \circ x$ be the $2$-isomorphism $\gamma = (\phi \star \text{id}_{x'}) \circ (\text{id}_ g \star \gamma ')$. Then the category $\mathcal{D}'$ of dotted arrows for the left square and $\gamma '$ is equivalent to the category $\mathcal{D}$ of dotted arrows for the outer rectangle and $\gamma $.

Proof. There is a functor $\mathcal{D}' \to \mathcal{D}$ which is $(a, \alpha , \beta ) \mapsto (q \circ a, \text{id}_ q \star \alpha , (\phi \star \text{id}_ a) \circ (\text{id}_ g \star \beta ))$ on objects and $\theta \mapsto \text{id}_ q \star \theta $ on arrows. Checking that this functor $\mathcal{D}' \to \mathcal{D}$ is an equivalence follows formally from the universal property for $2$-fibre products as in Section 4.31. Details omitted. $\square$

Lemma 4.44.3. Let $\mathcal{C}$ be a $(2,1)$-category. Assume given a solid $2$-commutative diagram

\[ \xymatrix{ S \ar[r]_-x \ar[dd]_ j & X \ar[d]^ f \\ & Y \ar[d]^ g \\ T \ar[r]^-z \ar@{..>}[ruu] & Z } \]

in $\mathcal{C}$. Choose a $2$-isomorphism $\gamma \colon z \circ j \to g \circ f \circ x$. Let $\mathcal{D}$ be the category of dotted arrows for the outer rectangle and $\gamma $. Let $\mathcal{D}'$ be the category of dotted arrows for the solid square

\[ \xymatrix{ S \ar[r]_-{f \circ x} \ar[d]_ j & Y \ar[d]^ g \\ T \ar[r]^-z \ar@{..>}[ru] & Z } \]

and $\gamma $. Then $\mathcal{D}$ is equivalent to a category $\mathcal{D}''$ which has the following property: there is a functor $\mathcal{D}'' \to \mathcal{D}'$ which turns $\mathcal{D}''$ into a category fibred in groupoids over $\mathcal{D}'$ and whose fibre categories are isomorphic to categories of dotted arrows for certain solid squares of the form

\[ \xymatrix{ S \ar[r]_-x \ar[d]_ j & X \ar[d]^ f \\ T \ar[r]^-y \ar@{..>}[ru] & Y } \]

and some choices of $2$-isomorphism $y \circ j \to f \circ x$.

Proof. Construct the category $\mathcal{D}''$ whose objects are tuples $(a,\alpha ,\beta ,b,\eta )$ where $(a,\alpha ,\beta )$ is an object of $\mathcal{D}$ and $b \colon T \rightarrow Y$ is a $1$-morphism and $\eta \colon b \rightarrow f \circ a$ is a $2$-isomorphism. Morphisms $(a,\alpha ,\beta ,b,\eta ) \rightarrow (a',\alpha ',\beta ',b',\eta ')$ in $\mathcal{D}''$ are pairs $(\theta _1,\theta _2)$, where $\theta _1 \colon a \rightarrow a'$ defines an arrow $(a, \alpha , \beta ) \rightarrow (a', \alpha ', \beta ')$ in $\mathcal{D}$ and $\theta _2 \colon b \rightarrow b'$ is a $2$-isomorphism with the compatibility condition $\eta ' \circ \theta _2 = (\text{id}_ f \star \theta _1) \circ \eta $.

There is a functor $\mathcal{D}'' \rightarrow \mathcal{D}'$ which is $(a, \alpha , \beta , b, \eta ) \mapsto (b, (\text{id}_ f \star \alpha ) \circ (\eta \star \text{id}_ j), (\text{id}_ g \star \eta ^{-1}) \circ \beta )$ on objects and $(\theta _1,\theta _2) \mapsto \theta _2$ on arrows. Then $\mathcal{D}'' \rightarrow \mathcal{D}'$ is fibred in groupoids.

If $(y, \delta , \epsilon )$ is an object of $\mathcal{D}'$, write $\mathcal{D}_{y,\delta }$ for the category of dotted arrows for the last displayed diagram with $y \circ j \rightarrow f \circ x$ given by $\delta $. There is a functor $\mathcal{D}_{y,\delta } \rightarrow \mathcal{D}''$ given by $(a, \alpha , \eta ) \mapsto (a, \alpha , (\text{id}_ g \star \eta ) \circ \epsilon , y, \eta )$ on objects and $\theta \mapsto (\theta , \text{id}_ y)$ on arrows. This exhibits an isomorphism from $\mathcal{D}_{y,\delta }$ to the fibre category of $\mathcal{D}'' \rightarrow \mathcal{D}'$ over $(y,\delta ,\epsilon )$.

There is also a functor $\mathcal{D} \rightarrow \mathcal{D}''$ which is $(a,\alpha ,\beta ) \mapsto (a,\alpha ,\beta ,f \circ a, \text{id}_{f \circ a})$ on objects and $\theta \mapsto (\theta , \text{id}_ f \star \theta )$ on arrows. This functor is fully faithful and essentially surjective, hence an equivalence. Details omitted. $\square$

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