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The extensive completion of a distributive category

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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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