The Stacks project

Definition 106.12.1. Let $\mathcal{X}$ be an algebraic stack. Let $f : \mathcal{X} \to Y$ be a morphism to an algebraic space $Y$.

  1. We say $f$ is a categorical moduli space if any morphism $\mathcal{X} \to W$ to an algebraic space $W$ factors uniquely through $f$.

  2. We say $f$ is a uniform categorical moduli space if for any flat morphism $Y' \to Y$ of algebraic spaces the base change $f' : Y' \times _ Y \mathcal{X} \to Y'$ is a categorical moduli space.

Let $\mathcal{C}$ be a full subcategory of the category of algebraic spaces.

  1. We say $f$ is a categorical moduli space in $\mathcal{C}$ if $Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and any morphism $\mathcal{X} \to W$ with $W \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ factors uniquely through $f$.

  2. We say is a uniform categorical moduli space in $\mathcal{C}$ if $Y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and for every flat morphism $Y' \to Y$ in $\mathcal{C}$ the base change $f' : Y' \times _ Y \mathcal{X} \to Y'$ is a categorical moduli space in $\mathcal{C}$.

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Your email address will not be published.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



View Definition 106.12.1 as pdf