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26
An Invitation to Higher Gauge Theory
, 2010
"... In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2connections on 2bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2group’. We focus on 6 examples. First, every abelian Lie gr ..."
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In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2connections on 2bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2group’, which serves as a gauge 2group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2group’. We also touch upon higher structures such as the ‘gravity 3group’, and the Lie 3superalgebra that governs 11dimensional supergravity. 1
Presentations of OmegaCategories By Directed Complexes
, 1997
"... The theory of directed complexes is extended from free !categories to arbitrary ! categories by defining presentations in which the generators are atoms and the relations are equations between molecules. Our main result relates these presentations to the more standard algebraic presentations; we ..."
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The theory of directed complexes is extended from free !categories to arbitrary ! categories by defining presentations in which the generators are atoms and the relations are equations between molecules. Our main result relates these presentations to the more standard algebraic presentations; we also show that every !category has a presentation by directed complexes. The approach is similar to that used by Crans for pasting presentations. 1991 Mathematics Subject Classification: 18D05. 1 Introduction There are at present three combinatorial structures for constructing !categories: pasting schemes, defined in 1988 by Johnson [8], parity complexes, introduced in 1991 by Street [16, 17] and directed complexes, given by Steiner in 1993 [15]. These structures each consist of cells which have collections of lower dimensional cells as domain and codomain; see for example Definition 2.2 below. They also have `local' conditions on the cells, ensuring that a cell together with its boundin...
Model structures on the category of small double categories, Algebraic and Geometric Topology 8
, 2008
"... Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak ..."
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Abstract. In this paper we obtain several model structures on DblCat, the category of small double categories. Our model structures have three sources. We first transfer across a categorificationnerve adjunction. Secondly, we view double categories as internal categories in Cat and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, DblCat inherits a model structure as a category of algebras over a 2monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, and several nerves
On Braidings, Syllepses, and Symmetries
, 1998
"... this paper is that these are the only differences between (semistrict) braided monoidal 2categories (as defined in [10]) and braided 2D teisi. The interpretation of this is that the main obstacles for proving the conjecture above will be the weakness of functoriality and the weakness of invertibili ..."
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this paper is that these are the only differences between (semistrict) braided monoidal 2categories (as defined in [10]) and braided 2D teisi. The interpretation of this is that the main obstacles for proving the conjecture above will be the weakness of functoriality and the weakness of invertibility. I should mention here that Baez and Neuchl [5, p. 242] (as corrected by me [10, p. 206]) have shown that either one of the functoriality triangles above can be made into an identity, but it is essential to the proof that the other one is not. Defining monoidal 2D teisi as 3D teisi with one object involves a shift of dimension: the arrows, 2arrows and 3arrows of the 3D tas C become the objects, arrows and 2arrows of a 2D tas which will be called the looping of
Homotopy Invariants of Multiple Categories and Concurrency in Computer Science
, 1999
"... this paper is to build several homology theories which are strongly related with the geometric properties of these paths. Concurrent machines can be formalized using cubical complexes (definition (2.1.1)) : the main idea is that a ncube represents the simultaneous execution of n 1transitions (Prat ..."
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this paper is to build several homology theories which are strongly related with the geometric properties of these paths. Concurrent machines can be formalized using cubical complexes (definition (2.1.1)) : the main idea is that a ncube represents the simultaneous execution of n 1transitions (Pratt, 1991). For example, take the following automaton :
Pasting Presentations for OmegaCategories
, 1995
"... The pasting theorem showed that pasting schemes are useful in studying free !categories. It was thought that their inflexibility with respect to composition and identities prohibited wider use. This is not the case: there is a way of dealing with identities which makes it possible to describe !ca ..."
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The pasting theorem showed that pasting schemes are useful in studying free !categories. It was thought that their inflexibility with respect to composition and identities prohibited wider use. This is not the case: there is a way of dealing with identities which makes it possible to describe !categories in terms of generating pasting schemes and relations between generated pastings, i.e., with pasting presentations. In this chapter I develop the necessary machinery for this. The main result, that the !category generated by a pasting presentation is universal with respect to respectable families of realizations, is a generalization of the pasting theorem. Contents 1 Introduction 3 2 Pasting schemes according to Johnson 4 2.1 Graded sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 !categories : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3 Pasting schemes : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.4 The pasting theorem : : ...
Modelling algebraic structures with Prolog (Extended abstract)
"... This paper presents a novel technique of using Prolog with never instantiated variables to manipulate a range of algebraic structures. The paper argues that Prolog is a powerful and underrated tool for use in computational number theory. A detailed example is presented in this extended abstract, and ..."
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This paper presents a novel technique of using Prolog with never instantiated variables to manipulate a range of algebraic structures. The paper argues that Prolog is a powerful and underrated tool for use in computational number theory. A detailed example is presented in this extended abstract, and several in the full paper, showing the advantages of using this technique. The detailed example is an application of higher dimensional category theory which has been used for solving problems in this area. 1 Introduction Among the many problems dealt with in computational algebra two important classes of problems deal with enumerating algebraic structures with certain properties and manipulating or calculating with the elements of such structures. For particular algebraic structures very efficient solutions to these problems are known and such solutions are typically made available as parts of one or more of the large computational packages now available (which include for example Cayley ...
Explicit Choice Higher Dimensional Automata, OmegaMultigraphs, and Process Algebra Operations
"... Explicit choice higher dimensional automata (ECHDA) have been proposed as models for concurrent processes because of their advantages for software engineering. These advantages include a full treatment of true concurrency, the provision of a consistent model throughout the development process, a ..."
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Explicit choice higher dimensional automata (ECHDA) have been proposed as models for concurrent processes because of their advantages for software engineering. These advantages include a full treatment of true concurrency, the provision of a consistent model throughout the development process, a natural partitioning of development processes, and the fact that their one dimensional skeletons correspond to nondeterministic automata and so are already familiar to many software engineers. The provision of ECHDA CASE tools depends upon a rigorous treatment of the connections between process algebra and ECHDA. This paper introduces !multigraphs, shows how they are used to represent ECHDA, and uses them to provide a rigorous treatment of the process algebra operations for ECHDA. In addition we provide a rigorous treatment of a dimension reduction operation which is important for software and specification visualization. Topics and Keywords: Nondeterminism, concurrency, softwar...
Linear Term rewriting systems are higher dimensional string rewriting systems
, 1991
"... This paper illustrates a categorical approach to the theory of rewriting systems along the lines adumbrated by Buchberger [3]. It shows how 2categories provide a simple framework for string rewriting systems and briefly reviews the theory of string rewriting to indicate its simplicity. In contrast ..."
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This paper illustrates a categorical approach to the theory of rewriting systems along the lines adumbrated by Buchberger [3]. It shows how 2categories provide a simple framework for string rewriting systems and briefly reviews the theory of string rewriting to indicate its simplicity. In contrast the theory of term rewriting systems is complicated by the interaction of substitution and rewriting. We outline a 3categorical framework for linear term rewriting systems and show by a detailed example how this separates substitution and rewriting and reduces linear term rewriting to the simplicity of string rewriting, but with one extra dimension. 1 Introduction Rewriting systems have broad applications in Computer Science in areas as diverse as automated theorem proving [12], polynomial ideal theory [2], computational algebra [10], Petri nets, and general models of computation, especially using constructions of free categories with structure [16],[14],[4]. A rewriting rule is a "basic d...
1 OMEGACATEGORIES AND CHAIN COMPLEXES
, 2004
"... There are several ways to construct omegacategories from combinatorial objects such as pasting schemes or parity complexes. We make these constructions into a functor on a category of chain complexes with additional structure, which we call augmented directed complexes. This functor from augmented ..."
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There are several ways to construct omegacategories from combinatorial objects such as pasting schemes or parity complexes. We make these constructions into a functor on a category of chain complexes with additional structure, which we call augmented directed complexes. This functor from augmented directed complexes to omegacategories has a left adjoint, and the adjunction restricts to an equivalence on a category of augmented directed complexes with good bases. The omegacategories equivalent to augmented directed complexes with good bases include the omegacategories associated to globes, simplexes and cubes; thus the morphisms between these omegacategories are determined by morphisms between chain complexes. It follows that the entire theory of omegacategories can be expressed in terms of chain complexes; in particular we describe the biclosed monoidal structure on omegacategories and calculate some internal homomorphism objects. 1.