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

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 $\mathcal{C}$.

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