## The Extensive Completion Of A Distributive Category (2001)

Venue: | Theory Appl. Categ |

Citations: | 6 - 1 self |

### BibTeX

@ARTICLE{Cockett01theextensive,

author = {J.r.b. Cockett and Stephen Lack and D Ex /b},

title = {The Extensive Completion Of A Distributive Category},

journal = {Theory Appl. Categ},

year = {2001},

volume = {8},

pages = {8--541}

}

### OpenURL

### Abstract

A category with finite products and finite coproducts is said to be distributive if the canonical map AB+AC # A (B +C) is invertible for all objects A, B, and C. Given a distributive category D , we describe a universal functor D # D ex preserving finite products and finite coproducts, for which D ex is extensive; that is, for all objects A and B the functor D ex /A D ex /B # D ex /(A + B) is an equivalence of categories. As an application, we show that a distributive category D has a full distributive embedding into the product of an extensive category with products and a distributive preorder. 1.

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Citation Context ...ory C , forms the category of relations in C , then splits certain idempotents in this category; the resulting category is then the category of relations in the exact completion C ex/reg : see [4] or =-=[12]-=- and the references therein. Notation. The identity morphism on an object A is denoted by A. Our notation for coproducts is: (f g) : A+B # C for the morphism induced by f : A # C and g : B # C, and i ... |

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Citation Context ... B. These categories are particularly important in geometry -- see for instance [13, 15] --- but also in proof theory [1], categorical Galois theory [2], and descent morphisms for internal structures =-=[14]-=-. All of the above examples of distributive categories are also extensive except for P(X). More generally, any distributive lattice, viewed as a preorder, is a distributive category which is not exten... |

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NSW 2006, Australia Email: robin@cpsc.ucalgary.ca and stevel@maths.usyd.edu.au This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anonymous ftp at ftp://ftp.tac.mta.ca/pub/tac/html/volumes/8/n22/n22.{dvi,ps} THEORY AND APPLICATIONS O
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(Show Context)
Citation Context ...py classes of maps, the poset P(X) of subsets of a set X, or the opposite of the category of commutative rings. The papers [3, 6] contain an introduction to distributive categories; see also the book =-=[16]-=-. A category E with finite coproducts is said to be extensive if the functors E /AE /B # E /(A + B) sending a pair (f : X # A, g : Y # B) to f + g : X + Y # A + B are equivalences of categories for al... |