# The extensive completion of a distributive category

## J.R.B. Cockett and Stephen Lack

A category with finite products and finite coproducts is said to be distributive if the canonical map $A \times B + A \times C \to A \times (B + C)$ is invertible for all objects $A$, $B$, and $C$. Given a distributive category $\cal D$, we describe a universal functor $\cal D \to \cal D_{ex}$ preserving finite products and finite coproducts, for which $\cal D_{ex}$ is extensive; that is, for all objects $A$ and $B$ the functor $\cal D_{ex}/A \times \cal D_{ex}/B \to \cal D_{ex}/(A + B)$ is an equivalence of categories.

As an application, we show that a distributive category $\cal D$ has a full distributive embedding into the product of an extensive category with products and a distributive preorder.

Keywords: distributive category, extensive category, free construction.

2000 MSC: 18D99, 18A40, 18B15.

Theory and Applications of Categories, Vol. 8, 2001, No. 22, pp 541-554.

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