The Stacks project

Definition 19.2.4. Let $\mathcal{C}$ be a category, let $I \subset \text{Arrows}(\mathcal{C})$, and let $\alpha $ be an ordinal. An object $A$ of $\mathcal{C}$ is said to be $\alpha $-small with respect to $I$ if whenever $\{ B_\beta \} $ is a system over $\alpha $ with transition maps in $I$, then the map (19.2.0.1) is an isomorphism.

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View Definition 19.2.4 as pdf