coextensive

Coextensive is a term used in category theory to describe a dual notion to extensive categories. It concerns how products and coproducts interact in a way that mirrors the familiar behavior of coproducts in extensive categories, but with the roles of products and coproducts reversed.

Definition and intuition

- A category C that has finite products and finite coproducts is called coextensive if, for every pair of objects A and B, the canonical relation between arrows into a product and pairs of arrows into the factors holds in a precise sense. Concretely, this is often expressed by a condition on slice categories: arrows into the product A×B correspond to compatible pairs of arrows into A and into B, in a way that makes C/(A×B) equivalent to C/A × C/B (up to appropriate coherence). This dualizes the standard extensive condition, where arrows into a coproduct correspond to pairs of arrows into the summands.

Relation to extensivity

- Coextensivity is the dual notion of extensivity. A category is extensive if coproducts behave well with respect to pullbacks, and there is a canonical equivalence involving the slice categories over a coproduct. Dually, a coextensive category has the corresponding property with products and slice categories over a product. Consequently, a category C is coextensive exactly when its opposite category C^op is extensive.

Examples and context

- The concept arises in the study of how categorical structures distribute over products and coproducts, with applications in topos theory and related areas. Specific examples are discussed in the literature on coextensive categories, including work by researchers such as Carboni, Lack, and Walters, among others.

See also

- Extensive category

- Slice category

- Pullback and product-coproduct interactions

- Opposite category and duality

Note: This article provides a concise overview of the concept and its dual relationship to extensivity. For formal definitions, proofs, and a catalog of examples, refer to specialized texts on coextensive categories.