Remark 106.6.1 (0CJP): Category of lifts

Remark 106.6.1 (Category of lifts). Consider a diagram

\[ \xymatrix{ W \ar[d]_ x \\ \mathcal{X} \ar[r] & \mathcal{X}' } \]

where $\mathcal{X} \subset \mathcal{X}'$ is a thickening of algebraic stacks, $W$ is an algebraic space, and $W \to \mathcal{X}$ is smooth. We will construct a category $\mathcal{C}$ and a functor

\[ p : \mathcal{C} \longrightarrow W_{spaces, {\acute{e}tale}} \]

(see Properties of Spaces, Definition 66.18.2 for notation) as follows. An object of $\mathcal{C}$ will be a system $(U, U', a, i, x', \alpha )$ which forms a commutative diagram

106.6.1.1

\begin{equation} \label{stacks-more-morphisms-equation-object} \vcenter { \xymatrix{ U \ar[d]_ a \ar[r]_ i & U' \ar[dd]^{x'} \\ W \ar[d]_ x & \\ \mathcal{X} \ar[r] & \mathcal{X}' } } \end{equation}

with commutativity witnessed by the $2$-morphism $\alpha : x \circ a \to x' \circ i$ such that $U$ and $U'$ are algebraic spaces, $a : U \to W$ is étale, $x' : U' \to \mathcal{X}'$ is smooth, and such that $U = \mathcal{X} \times _{\mathcal{X}'} U'$. In particular $U \subset U'$ is a thickening. A morphism

\[ (U, U', a, i, x', \alpha ) \to (V, V', b, j, y', \beta ) \]

is given by $(f, f', \gamma )$ where $f : U \to V$ is a morphism over $W$, $f' : U' \to V'$ is a morphism whose restriction to $U$ gives $f$, and $\gamma : x' \circ f' \to y'$ is a $2$-morphism witnessing the commutativity in right triangle of the diagram below

106.6.1.2

\begin{equation} \label{stacks-more-morphisms-equation-morphism} \vcenter { \xymatrix{ & V \ar[ld]_ f \ar[ldd]^ b \ar[rr]_ j & & V' \ar[ld]_{f'} \ar[lddd]^{y'} \\ U \ar[d]_ a \ar[rr]_ i & & U' \ar[dd]_{x'} \\ W \ar[d]_ x & \\ \mathcal{X} \ar[rr] & & \mathcal{X}' } } \end{equation}

Finally, we require that $\gamma $ is compatible with $\alpha $ and $\beta $: in the calculus of $2$-categories of Categories, Sections 4.28 and 4.29 this reads

\[ \beta = (\gamma \star \text{id}_ j) \circ (\alpha \star \text{id}_ f) \]

(more succinctly: $\beta = j^*\gamma \circ f^*\alpha $). Another formulation is that objects are commutative diagrams (106.6.1.1) with some additional properties and morphisms are commutative diagrams (106.6.1.2) in the category $\textit{Spaces}/\mathcal{X}'$ introduced in Properties of Stacks, Remark 100.3.7. This makes it clear that $\mathcal{C}$ is a category and that the rule $p : \mathcal{C} \to W_{spaces, {\acute{e}tale}}$ sending $(U, U', a, i, x', \alpha )$ to $a : U \to W$ is a functor.

The Stacks project

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