The Stacks project

Remark 8.4.9. (Cutting down a “big” stack to get a stack.) Let $\mathcal{C}$ be a site. Suppose that $p : \mathcal{S} \to \mathcal{C}$ is functor from a “big” category to $\mathcal{C}$, i.e., suppose that the collection of objects of $\mathcal{S}$ forms a proper class. Finally, suppose that $p : \mathcal{S} \to \mathcal{C}$ satisfies conditions (1), (2), (3) of Definition 8.4.1. In general there is no way to replace $p : \mathcal{S} \to \mathcal{C}$ by a equivalent category such that we obtain a stack. The reason is that it can happen that a fibre categories $\mathcal{S}_ U$ may have a proper class of isomorphism classes of objects. On the other hand, suppose that

  1. for every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ there exists a set $S_ U \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$ such that every object of $\mathcal{S}_ U$ is isomorphic in $\mathcal{S}_ U$ to an element of $S_ U$.

In this case we can find a full subcategory $\mathcal{S}_{small}$ of $\mathcal{S}$ such that, setting $p_{small} = p|_{\mathcal{S}_{small}}$, we have

  1. the functor $p_{small} : \mathcal{S}_{small} \to \mathcal{C}$ defines a stack, and

  2. the inclusion $\mathcal{S}_{small} \to \mathcal{S}$ is fully faithful and essentially surjective.

(Hint: For every $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ let $\alpha (U)$ denote the smallest ordinal such that $\mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U) \cap V_{\alpha (U)}$ surjects onto the set of isomorphism classes of $\mathcal{S}_ U$, and set $\alpha = \sup _{U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})} \alpha (U)$. Then take $\mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_{small}) = \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}) \cap V_\alpha $. For notation used see Sets, Section 3.5.)

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