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24
A Fibrational Theory of Geometric Morphisms
, 1998
"... Introduction Category theory can be viewed as an elementary, i.e. essentially first order, theory independent from set theory. In an elementary topos, i.e. a category satisfying a number of elementary axioms, one can perform all constructions that one performes with sets in everyday mathematics. Ne ..."
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Introduction Category theory can be viewed as an elementary, i.e. essentially first order, theory independent from set theory. In an elementary topos, i.e. a category satisfying a number of elementary axioms, one can perform all constructions that one performes with sets in everyday mathematics. Nevertheless, the language of category theory is not expressive enough to capture those categorical notions that make reference to set theory. Amongst those are: (co)completeness, (local) smallness, existence of a small set of generators and wellpoweredness. If we want to replace the category of sets by a category B whose objects are regarded as index sets we need an abstract theory of families. Such a theory is the theory of fibred categories. We can choose B as a topos but for most purposes it suffices that B has pullbacks. A category fibred over B is a functor P : E ! B
Sheaf Representation for Topoi
, 1997
"... It is shown that every (small) topos is equivalent to the category of global sections of a sheaf of socalled hyperlocal topoi, improving on a result of Lambek & Moerdijk. It follows that every boolean topos is equivalent to the global sections of a sheaf of wellpointed topoi. Completeness theo ..."
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It is shown that every (small) topos is equivalent to the category of global sections of a sheaf of socalled hyperlocal topoi, improving on a result of Lambek & Moerdijk. It follows that every boolean topos is equivalent to the global sections of a sheaf of wellpointed topoi. Completeness theorems for higherorder logic result as corollaries. The main result of this paper is the following. Theorem (Sheaf representation for topoi). For any small topos E, there is a sheaf of categories e E on a topological space, such that: (i) E is equivalent to the category of global sections of e E, (ii) every stalk of e E is a hyperlocal topos. Moreover, E is boolean just if every stalk of e E is wellpointed. Before defining the term "hyperlocal," we indicate some of the background of the theorem. The original and most familiar sheaf representations are for commutative rings (see [12, ch. 5] for a survey); e.g. a wellknown theorem due to Grothendieck [9] asserts that every commutative r...
Proofsearch in typetheoretic languages: an introduction
 Theoretical Computer Science
, 2000
"... We introduce the main concepts and problems in the theory of proofsearch in typetheoretic languages and survey some specific, connected topics. We do not claim to cover all of the theoretical and implementation issues in the study of proofsearch in typetheoretic languages; rather, we present som ..."
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We introduce the main concepts and problems in the theory of proofsearch in typetheoretic languages and survey some specific, connected topics. We do not claim to cover all of the theoretical and implementation issues in the study of proofsearch in typetheoretic languages; rather, we present some key ideas and problems, starting from wellmotivated points of departure such as a definition of a typetheoretic language or the relationship between languages and proofobjects. The strong connections between different proofsearch methods in logics, type theories and logical frameworks, together with their impact on programming and implementation issues, are central in this context.
Distribution Algebras and Duality
, 2000
"... INTRODUCTION By an Svalued distribution on a topos E bounded over a base topos S it is meant here a cocontinuous Sindexed functor : E ! S. Since introduced by F. W. Lawvere in 1983, considerable progress has been made in the study of distributions on toposes from a variety of viewpoints [19, 15, ..."
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INTRODUCTION By an Svalued distribution on a topos E bounded over a base topos S it is meant here a cocontinuous Sindexed functor : E ! S. Since introduced by F. W. Lawvere in 1983, considerable progress has been made in the study of distributions on toposes from a variety of viewpoints [19, 15, 24, 5, 6, 12, 7, 8, 9]. However, much work still remains to be done in this area. The purpose of this paper is to deepen our understanding of topos distributions by exploring a (dual) latticetheoretic notion of distribution algebra. We characterize the distribution algebras in E relative to S as the Sbicomplete Satomic Heyting algebras in E . As an illustration, we employ distribution algebras explicitly in order to give an alternative description of the display locale (complete spread) of a distribution [10, 12, 7].
THE COMPREHENSIVE FACTORIZATION AND TORSORS
, 2010
"... This is an expanded, revised and corrected version of the first author's preprint [1]. The discussion of onedimensional cohomology H1 in a fairly general category E involves passing to the (2)category Cat(E) of categories in E. In particular, the coe cient object is a category B in E and the tors ..."
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This is an expanded, revised and corrected version of the first author's preprint [1]. The discussion of onedimensional cohomology H1 in a fairly general category E involves passing to the (2)category Cat(E) of categories in E. In particular, the coe cient object is a category B in E and the torsors that H1 classifies are particular functors in E. We only impose conditions on E that are satisfied also by Cat(E) and argue that H1 for Cat(E) is a kind of H2 for E, and so on recursively. For us, it is too much to ask E to be a topos (or even internally complete) since, even if E is, Cat(E) is not. With this motivation, we are led to examine morphisms in E which act as internal families and to internalize the comprehensive factorization of functors into a nal functor followed by a discrete bration. We de ne Btorsors for a category B in E and prove clutching and classification theorems. The former theorem clutches ƒech cocycles to construct torsors while the latter constructs a coefficient category to classify structures locally isomorphic to members of a given internal family of structures. We conclude with applications to examples.
DESCENT THEORY FOR SCHEMES
"... Abstract. We give a complete characterization of the class of quasicompact morphisms of schemes that are stable effective descent morphisms for the SCHEMESindexed category given by quasicoherent sheaves of modules. 1. ..."
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Abstract. We give a complete characterization of the class of quasicompact morphisms of schemes that are stable effective descent morphisms for the SCHEMESindexed category given by quasicoherent sheaves of modules. 1.
Relational Limits in General Polymorphism
, 1993
"... Parametric models of polymorphic lambda calculus have the structure of enriched categories with cotensors and ends in some generalized sense, and thus have many categorical data types induced by them. The !order minimum model is a parametric model. 1 Introduction Higher order quantifier of polymor ..."
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Parametric models of polymorphic lambda calculus have the structure of enriched categories with cotensors and ends in some generalized sense, and thus have many categorical data types induced by them. The !order minimum model is a parametric model. 1 Introduction Higher order quantifier of polymorphic lambda calculus has several meanings. Two inventors of the calculus use different symbols. When Girard wrote V X:F (X) [10] (\PiX:F (X) in [12]), it corresponded to a higher order quantified formula 8X:F (X) via CurryHoward isomorphism. When Reynolds wrote \DeltaX:F (X) [33], it was the type of polymorphism, especially of parametric polymorphism [34]. The third interpretation leaded by categorical semantics is that the quantified type, we write 8X:F (X), is a kind of limits. The notation \PiX:F (X) suggests that it might be regarded as a product of all F (X) where X ranges over all types. That is to say, \PiX:F (X) is the collection of all sections (a section is a function sending a...
A Presentation Of The Initial LiftAlgebra
 Journal of Pure and Applied Algebra
, 1997
"... The object of study of the present paper may be considered as a model, in an elementary topos with a natural numbers object, of a nonclassical variation of the Peano arithmetic. The new feature consists in admitting, in addition to the constant (zero) s0 2 N and the unary operation (the success ..."
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The object of study of the present paper may be considered as a model, in an elementary topos with a natural numbers object, of a nonclassical variation of the Peano arithmetic. The new feature consists in admitting, in addition to the constant (zero) s0 2 N and the unary operation (the successor map) s1 : N ! N, arbitrary operations su : N u ! N of arities u `between 0 and 1'. That is, u is allowed to range over subsets of a singleton set.
FUNDAMENTAL PUSHOUT TOPOSES
"... Abstract. The author [2, 5] introduced and employed certain ‘fundamental pushout toposes ’ in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spir ..."
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Abstract. The author [2, 5] introduced and employed certain ‘fundamental pushout toposes ’ in the construction of the coverings fundamental groupoid of a locally connected topos. Our main purpose in this paper is to generalize this construction without the local connectedness assumption. In the spirit of [16, 10, 8] we replace connected components by constructively complemented, or definable, monomorphisms [1]. Unlike the locally connected case, where the fundamental groupoid is localic prodiscrete and its classifying topos is a Galois topos, in the general case our version of the fundamental groupoid is a locally discrete progroupoid and there is no intrinsic Galois theory in the sense of [19]. We also discuss covering projections, locally trivial, and branched coverings without local connectedness by analogy with, but also necessarily departing from, the locally connected case [13, 11, 7]. Throughout, we work abstractly in a setting given axiomatically by a category V of locally discrete locales that has as examples the categories D of discrete locales, and Z of zerodimensional locales [9]. In this fashion we are led to give unified and often simpler proofs of old theorems in the locally connected case, as well as new ones without that assumption.
Predicate Logic for Functors and Monads
, 2010
"... Abstract. This paper starts from the elementary observation that what is usually called a predicate lifting in coalgebraic modal logic is in fact an endomap of indexed categories. This leads to a systematic review of basic results in predicate logic for functors and monads, involving induction and c ..."
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Abstract. This paper starts from the elementary observation that what is usually called a predicate lifting in coalgebraic modal logic is in fact an endomap of indexed categories. This leads to a systematic review of basic results in predicate logic for functors and monads, involving induction and coinduction principles for functors and compositional modal operators for monads. 1