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18
Homotopy Invariants of Higher Dimensional Categories and Concurrency in Computer Science
, 1999
"... The strict globular omegacategories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) omegacategory C three homology theories. The first one is called the globular homology. It contains the oriented loops of C. The two other one ..."
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Cited by 51 (10 self)
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The strict globular omegacategories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) omegacategory 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.
Combinatorics Of Branchings In Higher Dimensional Automata
 Theory Appl. Categ
, 2001
"... We explore the combinatorial properties of the branching areas of execution paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner (or branching) homology of a globular #category and the combinatorics of a new homology theory ca ..."
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Cited by 35 (9 self)
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We explore the combinatorial properties of the branching areas of execution paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner (or branching) homology of a globular #category and the combinatorics of a new homology theory called the reduced branching homology. The latter is the homology of the quotient of the branching complex by the subcomplex generated by its thin elements. Conjecturally it coincides with the non reduced theory for higher dimensional automata, that is #categories freely generated by precubical sets. As application, we calculate the branching homology of some #categories and we give some invariance results for the reduced branching homology. We only treat the branching side. The merging side, that is the case of merging areas of execution paths is similar and can be easily deduced from the branching side.
From Concurrency to Algebraic Topology
, 2000
"... This paper is a survey of the new notions and results scattered in [13], [11] and [12]. Starting from a formalization of higher dimensional automata (HDA) by strict globular !categories, the construction of a diagram of simplicial sets over the threeobject small category gl ! + is exposed. Some of ..."
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Cited by 25 (9 self)
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This paper is a survey of the new notions and results scattered in [13], [11] and [12]. Starting from a formalization of higher dimensional automata (HDA) by strict globular !categories, the construction of a diagram of simplicial sets over the threeobject small category gl ! + is exposed. Some of the properties discovered so far on the corresponding simplicial homology theories are explained, in particular their links with geometric problems coming from concurrency theory in computer science.
Cubical Sets are Generalized Transition Systems
 In LICS'02 (2002). (submitted). References 23
, 2001
"... We show in this article that "labelled" cubical sets (or HigherDimensional Automata) are a natural generalization of transition systems and asynchronous transition systems. This generalizes an older result of [14] which was only holding with precubical sets and subcategories of the cla ..."
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Cited by 5 (2 self)
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We show in this article that "labelled" cubical sets (or HigherDimensional Automata) are a natural generalization of transition systems and asynchronous transition systems. This generalizes an older result of [14] which was only holding with precubical sets and subcategories of the classical (see [29]) categories of transition systems and asynchronous transition systems. This opens up new promises on the actual use of geometric methods (such as [8]) and on comparisons with other methods for verification of concurrent programs.
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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Cited by 4 (4 self)
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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
Homotopy, Concordance and Natural Transformation
, 2007
"... Transformations between (ω)functors are like homotopies or concordances between maps of topological spaces. This statement can be given a precise meaning using the closed structure of ωCat in terms of the extension of the Gray tensor product from 2categories to ωcategories given by Sjoerd Crans. ..."
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Transformations between (ω)functors are like homotopies or concordances between maps of topological spaces. This statement can be given a precise meaning using the closed structure of ωCat in terms of the extension of the Gray tensor product from 2categories to ωcategories given by Sjoerd Crans. The analogous construction is familar in homological algebra from categories of chain complexes. After recalling the basics, we turn to “anafunctors ” (certain spans of functors) and highlight how the general relation between transformations, homotopies and concordances appears in the study of nonabelian ncocycles classifying nbundles.
Nonabelian homotopical cohomology,
"... higher fiber bundles with connection, and their σmodel QFTs ..."
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Transgression of ntransport and nconnections
, 2008
"... After going through some ground work concerning generalized smooth spaces and their differential graded ommutative algebras of forms, I talk about the issue of transgression of transport ωfunctors and of Lie ∞valued connections to smooth mapping spaces. I discuss how what we call transgression of ..."
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After going through some ground work concerning generalized smooth spaces and their differential graded ommutative algebras of forms, I talk about the issue of transgression of transport ωfunctors and of Lie ∞valued connections to smooth mapping spaces. I discuss how what we call transgression of ωfunctors here is really the morphism part of an internal hom, and how that does reproduce the ordinary notion of transgression of differential forms under the relation between ntransport and differential forms.
Nonabelian homotopical cohomology,
"... higher fiber bundles with connection, and their σmodel QFTs ..."
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Nonablian cocycles and their σmodel QFTs
, 2008
"... Nonabelian cohomology can be regarded as a generalization of group cohomology to the case where both the group itself as well as the coefficient object are allowed to be generalized to ∞groupoids or even to general ∞categories. Cocycles in nonabelian cohomology in particular represent higher princ ..."
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Nonabelian cohomology can be regarded as a generalization of group cohomology to the case where both the group itself as well as the coefficient object are allowed to be generalized to ∞groupoids or even to general ∞categories. Cocycles in nonabelian cohomology in particular represent higher principal bundles (gerbes) – possibly equivariant, possibly with connection – as well as the corresponding associated higher vector bundles. We propose, expanding on considerations in [13, 34, 5], a systematic ∞functorial formalization of the σmodel quantum field theory associated with a given nonabelian cocycle regarded as the background field for a brane coupled to it. We define propagation in these σmodel QFTs and recover central aspects of groupoidification [1, 2] of linear algebra. In a series of examples we show how this formalization reproduces familiar structures in σmodels with finite target spaces such as DijkgraafWitten theory and the Yetter model. The generalization to