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52
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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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.
Quasismooth Derived Manifolds
"... products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the nontransverse intersection of submanifolds is ..."
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products; for example the zeroset of a smooth function on a manifold is not necessarily a manifold, and the nontransverse intersection of submanifolds is
Combinatorics of curvature, and the Bianchi identity, Theory and
 Appl. of Categories
, 1996
"... ABSTRACT. We analyze the Bianchi Identity as an instance of a basic fact of combinatorial groupoid theory, related to the Homotopy Addition Lemma. Here it becomes formulated in terms of 2forms with values in the gauge group bundle of a groupoid, and leads in particular to the (ChernWeil) construct ..."
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Cited by 12 (4 self)
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ABSTRACT. We analyze the Bianchi Identity as an instance of a basic fact of combinatorial groupoid theory, related to the Homotopy Addition Lemma. Here it becomes formulated in terms of 2forms with values in the gauge group bundle of a groupoid, and leads in particular to the (ChernWeil) construction of characteristic classes. The method is that of synthetic di erential geometry, using \the rst neighbourhood of the diagonal " of a manifold as its basic combinatorial structure. We introduce as a tool a new and simple description of wedge ( = exterior) products of di erential forms in this context.
On Differential Structure for Projective Limits of Manifolds
, 1998
"... We investigate the differential calculus defined by Ashtekar and Lewandowski on projective limits of manifolds by means of cylindrical smooth functions and compare it with the C∞ calculus proposed by Fröhlicher and Kriegl in more general context. For products of connected manifolds, a Boman t ..."
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Cited by 10 (2 self)
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We investigate the differential calculus defined by Ashtekar and Lewandowski on projective limits of manifolds by means of cylindrical smooth functions and compare it with the C∞ calculus proposed by Fröhlicher and Kriegl in more general context. For products of connected manifolds, a Boman theorem is proved, showing the equivalence of the two calculi in this particular case. Several examples of projective limits of manifolds are discussed, arising in String Theory and in loop quantization of Gauge Theories.
L∞algebra connections and applications to String and ChernSimons ntransport
, 2008
"... We give a generalization of the notion of a CartanEhresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher Stringlike extensions of Lie algebras. We find (generalized) ChernSimons and BFtheory functionals this way and describe aspect ..."
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Cited by 9 (5 self)
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We give a generalization of the notion of a CartanEhresmann connection from Lie algebras to L∞algebras and use it to study the obstruction theory of lifts through higher Stringlike extensions of Lie algebras. We find (generalized) ChernSimons and BFtheory functionals this way and describe aspects of their parallel transport and quantization. It is known that over a Dbrane the KalbRamond background field of the string restricts to a 2bundle with connection (a gerbe) which can be seen as the obstruction to lifting the P U(H)bundle on the Dbrane to a U(H)bundle. We discuss how this phenomenon generalizes from the ordinary central extension U(1) → U(H) → P U(H) to higher categorical central extensions, like the Stringextension BU(1) → String(G) → G. Here the obstruction to the lift is a 3bundle with connection (a 2gerbe): the ChernSimons 3bundle classified by the first Pontrjagin class. For G = Spin(n) this obstructs the existence of a Stringstructure. We discuss how to describe this obstruction problem in terms of Lie nalgebras and their corresponding categorified CartanEhresmann connections. Generalizations even beyond Stringextensions are then straightforward. For G = Spin(n) the next step is “Fivebrane structures” whose existence is obstructed by certain generalized ChernSimons 7bundles classified by the second Pontrjagin class.
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a pla ..."
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
On the Algebra of Feedback and Systems With Boundary
, 1998
"... this paper. We will, however, consider specific bicategorieswithfeedback and define the notion of a categorywithfeedback, which includes traced symmetric monoidal categories ([21]). In this context, we will consider functors between categorieswithfeedback  functors which model and facilitate ..."
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this paper. We will, however, consider specific bicategorieswithfeedback and define the notion of a categorywithfeedback, which includes traced symmetric monoidal categories ([21]). In this context, we will consider functors between categorieswithfeedback  functors which model and facilitate the interplay between systems and their behaviours which occurs when specifying, building and analysing machines or programs.
Relating firstorder set theories, toposes and categories of classes
 In preparation
, 2006
"... This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a firstorder settheory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions (dssi) on the topos, a forcingst ..."
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Cited by 3 (3 self)
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This paper introduces Basic Intuitionistic Set Theory BIST, and investigates it as a firstorder settheory extending the internal logic of elementary toposes. Given an elementary topos, together with the extra structure of a directed structural system of inclusions (dssi) on the topos, a forcingstyle interpretation of the language of firstorder set theory in the topos is given, which conservatively extends the internal logic of the topos. Since every topos is equivalent to one carrying a dssi, the language of firstorder has a forcing interpretation in every elementary topos. We prove that the set theory BIST+ Coll (where Coll is the strong Collection axiom) is sound and complete relative to forcing interpretations in toposes with natural numbers object (nno). Furthermore, in the case that the structural system of inclusions is superdirected, the full Separation schema is modelled. We show that every cocomplete topos and every realizability topos can be endowed (up to equivalence) with such a superdirected structural system of inclusions. This provides a uniform explanation for why such “realworld ” toposes model Separation. A large part of the paper is devoted to an alternative notion of categorytheoretic model for BIST, which, following the general approach of Joyal and Moerdijk’s Algebraic Set Theory, axiomatizes the structure possessed by categories of classes compatible with ∗Corresponding author. 1Previously, lecturer at HeriotWatt University (2000–2001), and the IT University of
Constructive Nonstandard Representations of Generalized Functions
 Indagationes Mathematicae
, 1998
"... Using techniques of nonstandard analysis Abraham Robinson showed that it is possible to represent each Schwartz distribution T as an integral T (OE) = R f OE, where f is some nonstandard smooth function. We show that the theory based on this representation can be developed within a constructive se ..."
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Using techniques of nonstandard analysis Abraham Robinson showed that it is possible to represent each Schwartz distribution T as an integral T (OE) = R f OE, where f is some nonstandard smooth function. We show that the theory based on this representation can be developed within a constructive setting. Mathematics Subject Classification: 03F60, 03H05, 26E35, 46F10. Keywords: Constructive analysis, nonstandard analysis, generalized functions. 1 Introduction Robinson (1966) demonstrated that Schwartz' theory of distributions could be given a natural formulation using techniques of nonstandard analysis, so that distributions become certain nonstandard smooth functions. In particular, Dirac's deltafunction may then be taken to be the rational function ffi(x) = 1 ß " " 2 + x 2 where " is a positive infinitesimal. As is wellknown, the classical nonstandard analysis is based on strongly nonconstructive assumptions. In this paper we present a constructive version of Robinson's the...
Characterizing algebras of smooth functions on manifolds, to appear
 Comm. Math. Univ. Carolinae (Prague
"... Abstract. Among all C ∞algebras we characterize those which are algebras of smooth functions on smooth separable Hausdorff manifolds. 1. C ∞algebras. An Ralgebra is a commutative ring A with unit together with a ring homomorphism R → A. Then every map p: R n → R m which is given by an mtuple of ..."
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Abstract. Among all C ∞algebras we characterize those which are algebras of smooth functions on smooth separable Hausdorff manifolds. 1. C ∞algebras. An Ralgebra is a commutative ring A with unit together with a ring homomorphism R → A. Then every map p: R n → R m which is given by an mtuple of real polynomials (p1,...,pm) can be interpreted as a mapping A(p) : A n → A m in such a way that projections, composition, and identity are preserved, by just evaluating each polynomial pi on an ntuple (a1,..., an) ∈ A n. A C ∞algebra A is a real algebra in which we can moreover interpret all smooth mappings f: R n → R m. There is a corresponding map A(f) : A n → A m, and again projections, composition, and the identity mapping are preserved.