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Unifying Constructive and Nonstandard Analysis
 Bull. Symbolic Logic
, 1999
"... This paper is partly a survey of this development. In Section 2 we review the construction of the nonstandard universe N . Section 3 discusses the internal sets and functions of N . Here we present two new results for N : the @ 1 saturation principle and a characterisation of internal functions bet ..."
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Cited by 25 (5 self)
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This paper is partly a survey of this development. In Section 2 we review the construction of the nonstandard universe N . Section 3 discusses the internal sets and functions of N . Here we present two new results for N : the @ 1 saturation principle and a characterisation of internal functions between nonstandard versions of standard sets. We also briefly indicate how to make the Loeb measure construction over hyperfinite sets. Section 4 discusses the relation between nonstandard real numbers and the canonical real numbers of N . In the final section we exemplify the use of the model to prove results in the calculus of several variables, e.g. the Implicit Function Theorem.
Combinatorics of nonabelian gerbes with connection and curvature
, 2008
"... We give a functorial definition of Ggerbes over a simplicial complex when the local symmetry group G is nonAbelian. These combinatorial gerbes are naturally endowed with a connective structure and a curving. This allows us to define a fibered category equipped with a functorial connection over the ..."
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Cited by 24 (2 self)
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We give a functorial definition of Ggerbes over a simplicial complex when the local symmetry group G is nonAbelian. These combinatorial gerbes are naturally endowed with a connective structure and a curving. This allows us to define a fibered category equipped with a functorial connection over the space of edgepaths. By computing the curvature of the latter on the faces of an infinitesimal 4simplex, we recover the cocycle identities satisfied by the curvature of this gerbe. The link with BFtheories suggests that gerbes provide a framework adapted to the geometric formulation of strongly coupled gauge theories.
Rate distortion manifolds as model spaces for cognitive information
 In preparation
, 2007
"... The rate distortion manifold is considered as a carrier for elements of the theory of information proposed by C. E. Shannon combined with the semantic precepts of F. Dretske’s theory of communication. This type of information space was suggested by R. Wallace as a possible geometric–topological desc ..."
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Cited by 15 (9 self)
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The rate distortion manifold is considered as a carrier for elements of the theory of information proposed by C. E. Shannon combined with the semantic precepts of F. Dretske’s theory of communication. This type of information space was suggested by R. Wallace as a possible geometric–topological descriptive model for incorporating a dynamic information based treatment of the Global Workspace theory of B. Baars. We outline a more formal mathematical description for this class of information space and further clarify its structural content and overall interpretation within prospectively a broad range of cognitive situations that apply to individuals, human institutions, distributed cognition and massively parallel intelligent machine design. Povzetek: Predstavljena je formalna definicija prostora za opisovanje kognitivnih procesov. 1
Axioms for Definability and Full Completeness
 in Proof, Language and Interaction: Essays in Honour of Robin
, 2000
"... ion problem for PCF (see [BCL86, Cur93, Ong95] for surveys). The importance of full abstraction for the semantics of programming languages is that it is one of the few quality filters we have. Specifically, it provides a clear criterion for assessing how definitive a semantic analysis of some langu ..."
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Cited by 8 (2 self)
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ion problem for PCF (see [BCL86, Cur93, Ong95] for surveys). The importance of full abstraction for the semantics of programming languages is that it is one of the few quality filters we have. Specifically, it provides a clear criterion for assessing how definitive a semantic analysis of some language is. It must be admitted that to date the quest for fully abstract models has not yielded many obvious applications; but it has generated much of the deepest work in semantics. Perhaps it is early days yet. Recently, game semantics has been used to give the first syntaxindependent constructions of fully abstract models for a number of programming languages, including PCF [AJM96, HO96, Nic94], richer functional languages [AM95, McC96b, McC96a, HY97], and languages with nonfunctional features such as reference types and nonlocal control constructs [AM97c, AM97b, AM97a, Lai97]. A noteworthy feature is that the key definability results for the richer languages are proved by a reduction to...
DIFFERENTIAL RESTRICTION CATEGORIES
"... Abstract. We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of Rn in a way that is completely algebraic. We also give other models for the resulting structure ..."
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Cited by 7 (2 self)
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Abstract. We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of Rn in a way that is completely algebraic. We also give other models for the resulting structure, discuss what it means for a partial map to be additive or linear, and show that differential restriction structure can be lifted through various completion operations.
QUANTUM GAUGE FIELD THEORY IN COHESIVE HOMOTOPY TYPE THEORY
"... Abstract. We implement in the formal language of homotopy type theory a new set of axioms called cohesion. Then we indicate how the resulting cohesive homotopy type theory naturally serves as a formal foundation for central concepts in quantum gauge field theory. This is a brief survey of work by th ..."
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Cited by 6 (4 self)
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Abstract. We implement in the formal language of homotopy type theory a new set of axioms called cohesion. Then we indicate how the resulting cohesive homotopy type theory naturally serves as a formal foundation for central concepts in quantum gauge field theory. This is a brief survey of work by the authors developed in detail elsewhere [48, 45]. Contents
Differential Forms as Infinitesimal Cochains
"... In the context of Synthetic Differential Geometry, we provide, for any manifold, a homotopy equivalence between its deRham complex, and a complex of infinitesimal singular cochains. The equivalence takes wedge product of forms to cup product of singular cochains. The purpose of the present note is ..."
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Cited by 6 (6 self)
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In the context of Synthetic Differential Geometry, we provide, for any manifold, a homotopy equivalence between its deRham complex, and a complex of infinitesimal singular cochains. The equivalence takes wedge product of forms to cup product of singular cochains. The purpose of the present note is to identify the de Rham complex of differential forms on a manifold M with a certain cochain complex related to the singular complex of M . In fact this cochain complex is dual to a certain simplicial subcomplex of the singular complex, consisting of "infinitesimal simplices". The notions make sense in the context of an embedding of the category of smooth manifolds into a suitable topos, more precisely, into a "model for synthetic differential geometry" (SDG). Our comparison is based on some results from [4], and is inspired by Felix and Lavendhomme's [2]. They also provide an identification of the de Rham complex with a complex related to the singular one; they, however, use cubical rather ...
Synthetic Differential Geometry: A Way to Intuitionistic Models of General Relativity in Toposes
, 1996
"... W.Lawvere in [4] suggested a approach to differential geometry and to others mathematical disciplines closed to physics, which allows to give definitions of derivatives, tangent vectors and tangent bundles without passages to the limits. This approach is based on a idea of consideration of all setti ..."
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Cited by 5 (1 self)
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W.Lawvere in [4] suggested a approach to differential geometry and to others mathematical disciplines closed to physics, which allows to give definitions of derivatives, tangent vectors and tangent bundles without passages to the limits. This approach is based on a idea of consideration of all settings not in sets but in some cartesian closed category E , particular in some elementary topos. The synthetic differential geometry (SDG) is the theory developed by A.Kock [1] in a context of Lawvere's ideas. In a base of the theory is an assumption of that a geometric line is not a filed of real numbers, but a some nondegenerate commutative ring R of a line type in E . In this work we shall show that SDG allows to develop a Riemannian geometry and write the Einstein's equations of a field on pseudoRiemannian formal manifold. This give a way for constructing a intuitionistic models of general relativity in suitable toposes. 1 Preliminaries In this paper will be given some metrical notions i...
Algebra of Principal Fibre Bundles and Connections, rejected by TAC 2002; preliminary version at Xiv:math.CT/0005125
"... In this paper, I intend to put together some of the efforts by several people of making aspects of fibre bundle theory into algebra. The initiator of these efforts was Charles Ehresmann, who put the notion of groupoid, and groupoid action in the focus for fibre bundle theory in general, and for conn ..."
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Cited by 3 (2 self)
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In this paper, I intend to put together some of the efforts by several people of making aspects of fibre bundle theory into algebra. The initiator of these efforts was Charles Ehresmann, who put the notion of groupoid, and groupoid action in the focus for fibre bundle theory in general, and for connection theory in particular. In so far as connection theory is concerned, this paper is a sequel to [12], and we presuppose some of the notions presented there: those of Sections 1, 3, 7, 8, and 11, so they will be recalled only sketchily. (The paper may also partly be seen as a rewiting of [7].)