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Towards a unified treatment of induction, I: the general recursion theorem, unfinished draft manuscript
, 1996
"... The recursive construction of a function f: A → Θ consists, paradigmatically, of finding a functor T and maps α: A → TA and θ: TΘ → Θ such that f = α; Tf; θ. The role of the functor T is to marshall the recursive subarguments, and apply the function f to them in parallel. This equation is called pa ..."
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The recursive construction of a function f: A → Θ consists, paradigmatically, of finding a functor T and maps α: A → TA and θ: TΘ → Θ such that f = α; Tf; θ. The role of the functor T is to marshall the recursive subarguments, and apply the function f to them in parallel. This equation is called partial correctness of the recursive program, because we have also to show that it terminates, i.e. that the recursion (coded by α) is well founded. This may be done by finding another map g: A → N, called a loop variant, where N is some standard well founded srtucture such as the natural numbers or ordinals. In set theory the functor T is the covariant powerset; in the study of the free algebra for a free theory Ω (such as in proof theory) it is the polynomial Σr∈Ω(−)ar(r), and it is often something very crude. We identify the properties of the category of sets needed to prove the general recursion theorem, that these data suffice to define f uniquely. For any pullbackpreserving functor T, a structure similar to the von Neumann hierarchy is developed which analyses the free Talgebra if it exists, or deputises for it otherwise. There is considerable latitude in the choice of ambient category, the functor T and the class of predicates admissible in the induction scheme. Free algebras, set theory, the familiar ordinals and novel forms of them which have arisen in theoretical computer science are treated in a uniform fashion. The central idea in the paper is a categorical definition of well founded coalgebra α: A. TA, namely that any pullback diagram of the form
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.
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 ..."
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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...
On Some Properties Of Pure Morphisms Of Commutative Rings
"... We prove that pure morphisms of commutative rings are e#ective A descent morphisms where A is a (COMMUTATIVE RINGS) indexed category given by (i) finitely generated modules, or (ii) flat modules, or (iii) finitely generated flat modules, or (iv) finitely generated projective modules. 1. ..."
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We prove that pure morphisms of commutative rings are e#ective A descent morphisms where A is a (COMMUTATIVE RINGS) indexed category given by (i) finitely generated modules, or (ii) flat modules, or (iii) finitely generated flat modules, or (iv) finitely generated projective 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. ..."
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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.
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 ..."
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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.
DUALITY AND TRACES FOR INDEXED MONOIDAL CATEGORIES
"... Abstract. By the Lefschetz fixed point theorem, if an endomorphism of a topological space is fixedpointfree, then its Lefschetz number vanishes. This necessary condition is not usually sufficient, however; for that we need a refinement of the Lefschetz number called the Reidemeister trace. Abstrac ..."
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Abstract. By the Lefschetz fixed point theorem, if an endomorphism of a topological space is fixedpointfree, then its Lefschetz number vanishes. This necessary condition is not usually sufficient, however; for that we need a refinement of the Lefschetz number called the Reidemeister trace. Abstractly, the Lefschetz number is a trace in a symmetric monoidal category, while the Reidemeister trace is a trace in a bicategory; in this paper we relate these contexts using indexed symmetric monoidal categories. In particular, we will show that for any symmetric monoidal category with an associated indexed symmetric monoidal category, there is an associated bicategory which produces refinements of trace analogous to the Reidemeister trace. This bicategory also produces a new notion of trace for parametrized spaces with dualizable fibers, which refines the obvious “fiberwise ” traces by incorporating the action of the fundamental group of the base space. We also advance the basic theory of indexed monoidal categories, including introducing a string diagram calculus which makes calculations much more tractable. This abstract framework lays the foundation for generalizations of these ideas to other