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48
Twisted differential nonabelian cohomology Twisted (n−1)brane nbundles and their ChernSimons (n+1)bundles with characteristic (n + 2)classes
, 2008
"... We introduce nonabelian differential cohomology classifying ∞bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shif ..."
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We introduce nonabelian differential cohomology classifying ∞bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shifted abelian ngroup B n−1 U(1). Notable examples are String 2bundles [9] and Fivebrane 6bundles [133]. The obstructions to lifting ordinary principal bundles to these, hence in particular the obstructions to lifting Spinstructures to Stringstructures [13] and further to Fivebranestructures [133, 52], are abelian ChernSimons 3 and 7bundles with characteristic class the first and second fractional Pontryagin class, whose abelian cocycles have been constructed explicitly by Brylinski and McLaughlin [35, 36]. We realize their construction as an abelian component of obstruction theory in nonabelian cohomology by ∞Lieintegrating the L∞algebraic data in [132]. As a result, even if the lift fails, we obtain twisted String 2 and twisted Fivebrane 6bundles classified in twisted nonabelian (differential) cohomology and generalizing the twisted bundles appearing in twisted Ktheory. We explain the GreenSchwarz mechanism in heterotic string theory in terms of twisted String 2bundles and its magnetic dual version – according to [133] – in terms of twisted Fivebrane 6bundles. We close by transgressing differential cocycles to mapping
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
Presheaves as Configured Specifications
"... The paper addresses a notion of configuring systems, constructing them from specified component parts with specified sharing. This notion is independent of any underlying specification language and has been abstractly identified with the taking of colimits in category theory. Mathematically it is kn ..."
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The paper addresses a notion of configuring systems, constructing them from specified component parts with specified sharing. This notion is independent of any underlying specification language and has been abstractly identified with the taking of colimits in category theory. Mathematically it is known that these can be expressed by presheaves and the present paper applies this idea to configuration. We interpret the category theory informally as follows. Suppose C is a category whose objects are interpreted as specifications, and for which each morphism u : X ! Y is interpreted as contravariant "instance reduction", reducing instances of specification Y to instances of X . Then a presheaf P : Set C op represents a collection of instances that is closed under reduction. We develop an algebraic account of presheaves in which we present configurations by generators (for components) and relations (for shared reducts), and we outline a proposed configuration language based on the techniques. Oriat uses diagrams to express colimits of specifications, and we show that Oriat's category Diag(C) of finite diagrams is equivalent to the category of finitely presented presheaves over C.
A Machine Assisted Proof of the HahnBanach Theorem
, 1997
"... We describe an implementation of a pointfree proof of the Alaoglu and the HahnBanach theorems in Type Theory. The proofs described here are formalisations of the proofs presented in "The HahnBanach Theorem in Type Theory" [4]. The implementation was partially developed simultaneously with [4] and i ..."
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We describe an implementation of a pointfree proof of the Alaoglu and the HahnBanach theorems in Type Theory. The proofs described here are formalisations of the proofs presented in "The HahnBanach Theorem in Type Theory" [4]. The implementation was partially developed simultaneously with [4] and it was a help in the development of the informal proofs. 1 Introduction We present a machine assisted formalisation of pointfree topology in MartinLof's type theory. The continuum and the basic definitions needed in a pointfree approach to functional analysis are given and in this setting we describe implementations of localic formulations of the Alaoglu and the HahnBanach theorems. The classical HahnBanach theorem says that, if M is a subspace of a normed linear space A and f is a bounded linear functional on M , then f can be extended to a linear functional F on A so that kFk = kfk. (In our proof we use the equivalent formulation: if kfk 1 then f can be extended to F so that kFk 1.) A...
Crystallographic Topology 2: Overview And Work In Progress
, 1999
"... This overview describes an application of contemporary geometric topology and stochastic process concepts to structural crystallography. In this application, crystallographic groups become orbifolds, crystal structures become Morse functions on orbifolds, and vibrating atoms in a crystal become vect ..."
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This overview describes an application of contemporary geometric topology and stochastic process concepts to structural crystallography. In this application, crystallographic groups become orbifolds, crystal structures become Morse functions on orbifolds, and vibrating atoms in a crystal become vector valued Gaussian measures with the RadonNikodym property. Intended crystallographic benefits include new methods for visualization of space groups and crystal structures, analysis of the thermal motion patterns seen in ORTEP drawings, and a classification scheme for crystal structures based on their Heegaard splitting properties. 1 Introduction Geometric topology and structural crystallography concepts are combined to define a research area we call Structural Crystallographic Topology, or just Crystallographic Topology. The first paper in the series[30] describes basic crystallography concepts (crystallographic groups, lattice complexes, and crystal structures) and their replacement topo...
Abstract Syntax with Variable Binding
, 1999
"... The permutation model of set theory with atoms (FMsets), devised by Fraenkel and Mostowski in the 1930s, supports notions of `nameabstraction' and `fresh name' that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variablebinding operations. In ..."
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The permutation model of set theory with atoms (FMsets), devised by Fraenkel and Mostowski in the 1930s, supports notions of `nameabstraction' and `fresh name' that provide a new way to represent, compute with, and reason about the syntax of formal systems involving variablebinding operations. Inductively defined FMsets involving the nameabstraction set former (together with cartesian product and disjoint union) can correctly encode syntax modulo renaming of bound variables. In this way, the standard theory of algebraic data types can be extended to encompass signatures involving binding operators. In particular, there is an associated notion of structural recursion for defining syntaxmanipulating functions (such as capture avoiding substitution, set of free var...
Duality For Simple omegaCategories And Disks
"... A. Joyal [J] has introduced the category D of the socalled finite disks, and used it to define the concept of #category, a notion of weak #category. We introduce the notion of an #graph being composable (meaning roughly that 'it has a unique composite'), and call an #category simple if it is fr ..."
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A. Joyal [J] has introduced the category D of the socalled finite disks, and used it to define the concept of #category, a notion of weak #category. We introduce the notion of an #graph being composable (meaning roughly that 'it has a unique composite'), and call an #category simple if it is freely generated by a composable #graph. The category S of simple #categories is a full subcategory of the category, with strict #functors as morphisms, of all #categories. The category S is a key ingredient in another concept of weak #category, called protocategory [MM1], [MZ]. We prove that D and S are contravariantly equivalent, by a duality induced by a suitable schizophrenic object living in both categories. In [MZ], this result is one of the tools used to show that the concept of #category and that of protocategory are equivalent in a suitable sense. We also prove that composable #graphs coincide with the #graphs of the form T # considered by M.Batanin [B], which were characterized by R. Street (as announced in [S]) and called `globular cardinals'. Batanin's construction, using globular cardinals, of the free #category on a globular set plays an important role in our paper. We give a selfcontained presentation of Batanin's construction that suits our purposes.
CHARACTERIZING ARTIN STACKS
"... Abstract. We study properties of morphisms of stacks in the context of the homotopy theory of presheaves of groupoids on a small site C. There is a natural method for extending a property P of morphisms of sheaves on C to a property P of morphisms of presheaves of groupoids. We prove that the proper ..."
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Abstract. We study properties of morphisms of stacks in the context of the homotopy theory of presheaves of groupoids on a small site C. There is a natural method for extending a property P of morphisms of sheaves on C to a property P of morphisms of presheaves of groupoids. We prove that the property P is homotopy invariant in the local model structure on P (C, Grpd)L when P is stable under pullback and local on the target. Using the homotopy invariance of the properties of being a representable morphism, representable in algebraic spaces, and of being a cover, we obtain homotopy theoretic characterizations of algebraic and Artin stacks as those which are equivalent to simplicial objects in C satisfying certain analogues of the Kan conditions. The definition of Artin stack can naturally be placed within a hierarchy which roughly measures how far a stack is from being representable. We call the higher analogues of Artin stacks nalgebraic stacks, and provide a characterization of these in terms of simplicial objects. A consequence of this characterization is that, for presheaves of groupoids, nalgebraic is the same as 3algebraic for all n ≥ 3. As an application of these results we show that a stack is nalgebraic if and only if the homotopy orbits of a group action on it is. 1.
JOYAL’S ARITHMETIC UNIVERSE AS LISTARITHMETIC PRETOPOS
"... Abstract. We explain in detail why the notion of listarithmetic pretopos should be taken as the general categorical definition for the construction of arithmetic universes introduced by André Joyal to give a categorical proof of Gödel’s incompleteness results. We motivate this definition for three ..."
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Abstract. We explain in detail why the notion of listarithmetic pretopos should be taken as the general categorical definition for the construction of arithmetic universes introduced by André Joyal to give a categorical proof of Gödel’s incompleteness results. We motivate this definition for three reasons: first, Joyal’s arithmetic universes are listarithmetic pretopoi; second, the initial arithmetic universe among Joyal’s constructions is equivalent to the initial listarithmetic pretopos; third, any listarithmetic pretopos enjoys the existence of free internal categories and diagrams as required to prove Gödel’s incompleteness. In doing our proofs we make an extensive use of the internal type theory of the categorical structures involved in Joyal’s constructions. The definition of listarithmetic pretopos is equivalent to the general one that I came to know in a recent talk by André Joyal. 1.