Definition 105.12.1. Let $\mathcal{X}$ be an algebraic stack. Let $f : \mathcal{X} \to Y$ be a morphism to an algebraic space $Y$.

We say $f$ is a

*categorical moduli space*if any morphism $\mathcal{X} \to W$ to an algebraic space $W$ factors uniquely through $f$.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.

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 $f$.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)