The Stacks project

Remark 104.6.1 (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 64.18.2 for notation) as follows. An object of $\mathcal{C}$ will be a system $(U, U', a, i, x', \alpha )$ which forms a commutative diagram

104.6.1.1
\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 ├ętale, $x' : U' \to \mathcal{X}'$ 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

104.6.1.2
\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 (104.6.1.1) with some additional properties and morphisms are commutative diagrams (104.6.1.2) in the category $\textit{Spaces}/\mathcal{X}'$ introduced in Properties of Stacks, Remark 98.3.7. 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.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0CJP. Beware of the difference between the letter 'O' and the digit '0'.