countably coextensive

A category $\C$ is countably coextensive if it has countable products and, for every countable family of objects $(A_i)_{i \in I}$ , the product functor $\textstyle \prod_{i \in I} (A_i/\C) \to (\coprod_{i \in I} A_i) / \C,$ which maps a family of morphisms $(A_i \to X_i)_{i \in I}$ to their product $\prod_{i \in I} A_i \to \prod_{i \in I} X_i$ , is an equivalence of categories. Equivalently (by using the characterization of countably extensive categories), pushouts along product projections exist, countable products are disjoint, and countable products are stable under pushouts.
The only difference between this property and the property of being infinitary coextensive is that we restrict ourselves to countable families.

Dual property: countably extensive
Related properties: coextensive, countable products, countably codistributive, disjoint products, infinitary coextensive

Relevant implications

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

Examples

There are 4 categories with this property.

Counterexamples

There are 84 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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