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21
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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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
Maps II: Chasing Diagrams in Categorical Proof Theory
, 1996
"... In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, ..."
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Cited by 7 (4 self)
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In categorical proof theory, propositions and proofs are presented as objects and arrows in a category. It thus embodies the strong constructivist paradigms of propositionsastypes and proofsasconstructions, which lie in the foundation of computational logic. Moreover, in the categorical setting, a third paradigm arises, not available elsewhere: logicaloperationsasadjunctions. It offers an answer to the notorious question of the equality of proofs. So we chase diagrams in algebra of proofs. On the basis of these ideas, the present paper investigates proof theory of regular logic: the f; 9gfragment of the first order logic with equality. The corresponding categorical structure is regular fibration. The examples include stable factorisations, sites, triposes. Regular logic is exactly what is needed to talk about maps, as total and singlevalued relations. However, when enriched with proofsasarrows, this familiar concept must be supplied with an additional conversion rule, conn...
Type Theory via Exact Categories (Extended Abstract)
 In Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science LICS '98
, 1998
"... Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why ..."
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Partial equivalence relations (and categories of these) are a standard tool in semantics of type theories and programming languages, since they often provide a cartesian closed category with extended definability. Using the theory of exact categories, we give a categorytheoretic explanation of why the construction of a category of partial equivalence relations often produces a cartesian closed category. We show how several familiar examples of categories of partial equivalence relations fit into the general framework. 1 Introduction Partial equivalence relations (and categories of these) are a standard tool in semantics of programming languages, see e.g. [2, 5, 7, 9, 15, 17, 20, 22, 35] and [6, 29] for extensive surveys. They are usefully applied to give proofs of correctness and adequacy since they often provide a cartesian closed category with additional properties. Take for instance a partial equivalence relation on the set of natural numbers: a binary relation R ` N\ThetaN on th...
A Characterization Of The Left Exact Categories Whose Exact Completions Are Toposes
, 1999
"... We characterize the categories with finite limits whose exact completions are toposes. We review the examples in the literature and also find new examples and counterexamples. ..."
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Cited by 6 (2 self)
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We characterize the categories with finite limits whose exact completions are toposes. We review the examples in the literature and also find new examples and counterexamples.
A note on the free regular and exact completions and their infinitary generalizations
 THEORY AND APPLICATIONS OF CATEGORIES
, 1996
"... Free regular and exact completions of categories with various ranks of weak limits are presented as subcategories of presheaf categories. Their universal properties can then be derived with standard techniques as used in duality theory. ..."
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Free regular and exact completions of categories with various ranks of weak limits are presented as subcategories of presheaf categories. Their universal properties can then be derived with standard techniques as used in duality theory.
A unified approach to algebraic set theory
 the proceedings of the Logic Colloquium 2006, arXiv:0710.3066
, 2007
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A Note On The Exact Completion Of A Regular Category, And Its Infinitary Generalizations
, 1999
"... A new description of the exact completion C ex/reg of a regular category C is given, using a certain topos Shv(C) of sheaves on C; the exact completion is then constructed as the closure of C in Shv(C) under finite limits and coequalizers of equivalence relations. An infinitary generalization is ..."
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A new description of the exact completion C ex/reg of a regular category C is given, using a certain topos Shv(C) of sheaves on C; the exact completion is then constructed as the closure of C in Shv(C) under finite limits and coequalizers of equivalence relations. An infinitary generalization is proved, and the classical description of the exact completion is derived.
Fibrations and Calculi of Fractions
 Journal of pure and applied algebra
, 1994
"... Given a fibration E ! B and a class \Sigma of arrows of B, one can construct the free fibration (on E over B such that all reindexing functors over elements of \Sigma are equivalences. ..."
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Given a fibration E ! B and a class \Sigma of arrows of B, one can construct the free fibration (on E over B such that all reindexing functors over elements of \Sigma are equivalences.
A minimalist twolevel foundation for constructive mathematics
, 811
"... We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other lev ..."
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We present a twolevel theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the settheoretic version introduced in [MS05] with collections. The other level is given by an extensional set theory that is interpreted in the first one by means of a quotient model. This twolevel theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the “proofsasprograms ” paradigm and acts as a programming language.