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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.
@article{TAC_2001_8_a21,
author = {J.R.B. Cockett and Stephen Lack},
title = {The extensive completion of a distributive category},
journal = {Theory and applications of categories},
pages = {541--554},
publisher = {mathdoc},
volume = {8},
year = {2001},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2001_8_a21/}
}
J.R.B. Cockett; Stephen Lack. The extensive completion of a distributive category. Theory and applications of categories, Tome 8 (2001), pp. 541-554. http://geodesic.mathdoc.fr/item/TAC_2001_8_a21/