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 Section 2 of Introduction to extensive and distributive categories by Carboni, Lack, and Walters. 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.

Dual property: countably coextensive
Related properties: countable coproducts, countably distributive, disjoint coproducts, extensive, infinitary extensive

Relevant implications

countably coextensive andself-dual implies countably extensive
countably extensive andfinite products implies countably distributive
countably extensive andself-dual implies countably coextensive
countably extensive implies countable coproducts
countably extensive implies extensive
infinitary extensive implies countably extensive

Examples

There are 23 categories with this property.

Counterexamples

There are 65 categories without this property.

Unknown

There are 0 categories for which the database has no information on whether they satisfy this property.

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