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Exact Completions and Toposes
 University of Edinburgh
, 2000
"... Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and ..."
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Cited by 14 (4 self)
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Toposes and quasitoposes have been shown to be useful in mathematics, logic and computer science. Because of this, it is important to understand the di#erent ways in which they can be constructed. Realizability toposes and presheaf toposes are two important classes of toposes. All of the former and many of the latter arise by adding &quot;good &quot; quotients of equivalence relations to a simple category with finite limits. This construction is called the exact completion of the original category. Exact completions are not always toposes and it was not known, not even in the realizability and presheaf cases, when or why toposes arise in this way. Exact completions can be obtained as the composition of two related constructions. The first one assigns to a category with finite limits, the &quot;best &quot; regular category (called its regular completion) that embeds it. The second assigns to
The Extensive Completion Of A Distributive Category
 Theory Appl. Categ
, 2001
"... 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 wh ..."
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Cited by 6 (1 self)
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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.
THE EXTENSIVE COMPLETION OF A DISTRIBUTIVE CATEGORY J.R.B. COCKETT AND STEPHEN LACK
"... ABSTRACT. A category with finite products and finite coproducts is said to be distributive ifthe canonical map A × B + A × C → A × (B + C) is invertible for all objects A, B, andC. Given a distributive category D, we describe a universal functor D → Dex preserving finite products and finite coproduc ..."
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ABSTRACT. A category with finite products and finite coproducts is said to be distributive ifthe canonical map A × B + A × C → A × (B + C) is invertible for all objects A, B, andC. Given a distributive category D, we describe a universal functor D → Dex preserving finite products and finite coproducts, for which Dex is extensive; thatis,for all objects A and B the functor Dex/A × Dex/B → Dex/(A + B) is an equivalence of categories.
Exact completions and small sheaves
, 2012
"... We prove a general theorem which includes most notions of “exact completion” as special cases. The theorem is that “κary exact categories” are a reflective sub2category of “κary sites”, for any regular cardinal κ. A κary exact category is an exact category with disjoint and universal κsmall ..."
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We prove a general theorem which includes most notions of “exact completion” as special cases. The theorem is that “κary exact categories” are a reflective sub2category of “κary sites”, for any regular cardinal κ. A κary exact category is an exact category with disjoint and universal κsmall coproducts, and a κary site is a site whose covering sieves are generated by κsmall families and which satisfies a solutionset condition for finite limits relative to κ. In the unary