The Stacks project

Definition 106.3.1. Thickenings.

  1. We say an algebraic stack $\mathcal{X}'$ is a thickening of an algebraic stack $\mathcal{X}$ if $\mathcal{X}$ is a closed substack of $\mathcal{X}'$ and the associated topological spaces are equal.

  2. Given two thickenings $\mathcal{X} \subset \mathcal{X}'$ and $\mathcal{Y} \subset \mathcal{Y}'$ a morphism of thickenings is a morphism $f' : \mathcal{X}' \to \mathcal{Y}'$ of algebraic stacks such that $f'|_\mathcal {X}$ factors through the closed substack $\mathcal{Y}$. In this situation we set $f = f'|_\mathcal {X} : \mathcal{X} \to \mathcal{Y}$ and we say that $(f, f') : (\mathcal{X} \subset \mathcal{X}') \to (\mathcal{Y} \subset \mathcal{Y}')$ is a morphism of thickenings.

  3. Let $\mathcal{Z}$ be an algebraic stack. We similarly define thickenings over $\mathcal{Z}$ and morphisms of thickenings over $\mathcal{Z}$. This means that the algebraic stacks $\mathcal{X}'$ and $\mathcal{Y}'$ are endowed with a structure morphism to $\mathcal{Z}$ and that $f'$ fits into a suitable $2$-commutative diagram of algebraic stacks.

Comments (4)

Comment #2051 by Matthieu Romagny on

In (3), second sentence, change into two times.

Comment #5964 by Dario WeiƟmann on

typo in (3): are algebraic stack -> are algebraic stacks ( or just delete it and say "are endowed")

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Your email address will not be published.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 0BPP. Beware of the difference between the letter 'O' and the digit '0'.