countably extensive

A category $\C$ is countably extensive if it has countable coproducts and, for every countable family of objects $(A_i)_{i \in I}$, the coproduct functor $$\textstyle \prod_{i \in I} (\C/A_i) \to \C/(\coprod_{i \in I} A_i),$$ which maps a family of morphisms $(X_i \to A_i)_{i \in I}$ to their coproduct $\coprod_{i \in I} X_i \to \coprod_{i \in I} A_i$, is an equivalence of categories. Equivalently, pullbacks along coproduct inclusions exist, countable coproducts are disjoint, and countable coproducts are stable under pullbacks. For a proof of this equivalent characterization, see Carboni–Lack–Walters, Section 2. This covers the finite case, but the countable case is similar.The only difference between this property and the property of being infinitary extensive is that we restrict ourselves to countable families. A typical example of a countably extensive category which is not infinitary extensive is the category of measurable spaces.