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31
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
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.
Homotopy invariants of higher dimensional categories and concurrency in computer science
, 1999
"... The strict globular ωcategories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) ωcategory C three homology theories. The first one is called the globular homology. It contains the oriented loops of C. The two other ones are ca ..."
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The strict globular ωcategories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) ωcategory C three homology theories. The first one is called the globular homology. It contains the oriented loops of C. The two other ones are called the negative (resp. positive) corner homology. They contain in a certain manner the branching areas of execution paths or negative corners (resp. the merging areas of execution paths or positive corners) of C. Two natural linear maps called the negative (resp. the positive) Hurewicz morphism from the globular homology to the negative (resp. positive) corner homology are constructed. We explain the reason why these constructions allow the reinterpretation of some geometric problems coming from computer science.
Homotopy invariants of higher dimensional categories and concurrency in computer science
, 2008
"... The strict globular ωcategories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) ωcategory C three homology theories. The first one is called the globular homology. It contains the oriented loops of C. The two other ones are ca ..."
Abstract
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The strict globular ωcategories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) ωcategory C three homology theories. The first one is called the globular homology. It contains the oriented loops of C. The two other ones are called the negative (resp. positive) corner homology. They contain in a certain manner the branching areas of execution paths or negative corners (resp. the merging areas of execution paths or positive corners) of C. Two natural linear maps called the negative (resp. the positive) Hurewicz morphism from the globular homology to the negative (resp. positive) corner homology are constructed. We explain the reason why these constructions allow to reinterprete some geometric problems coming from computer science.
Homotopy invariants of higher dimensional categories and concurrency in computer science
, 1999
"... The strict globular ωcategories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) ωcategory C three homology theories. The first one is called the globular homology. It contains the oriented loops of C. The two other ones are ca ..."
Abstract
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The strict globular ωcategories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) ωcategory C three homology theories. The first one is called the globular homology. It contains the oriented loops of C. The two other ones are called the negative (resp. positive) corner homology. They contain in a certain manner the branching areas of execution paths or negative corners (resp. the merging areas of execution paths or positive corners) of C. Two natural linear maps called the negative (resp. the positive) Hurewicz morphism from the globular homology to the negative (resp. positive) corner homology are constructed. We explain the reason why these constructions allow the reinterpretation of some geometric problems coming from computer science.
About the globular homology of higher dimensional automata
, 2001
"... We introduce a new simplicial nerve of higher dimensional automata whose simplicial homology groups shifted by one yield a new definition of the globular homology. With this new definition, the drawbacks noticed with the construction of [Gau00c] disappear. Moreover the important morphisms which asso ..."
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We introduce a new simplicial nerve of higher dimensional automata whose simplicial homology groups shifted by one yield a new definition of the globular homology. With this new definition, the drawbacks noticed with the construction of [Gau00c] disappear. Moreover the important morphisms which associate to every globe its corresponding branching area and merging area of execution paths become morphisms of simplicial sets.