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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 ..."
Abstract

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 ..."
Abstract

Cited by 24 (8 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.
The branching nerve of HDA and the Kan condition
 Theory and Applications of Categories
, 2003
"... One can associate to any strict globular omegacategory three augmented simplicial nerves called the globular nerve, the branching and the merging semicubical nerves. If this strict globular omegacategory is freely generated by a precubical set, then the corresponding homology theories contain dif ..."
Abstract

Cited by 7 (6 self)
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One can associate to any strict globular omegacategory three augmented simplicial nerves called the globular nerve, the branching and the merging semicubical nerves. If this strict globular omegacategory is freely generated by a precubical set, then the corresponding homology theories contain different informations about the geometry of the higher dimensional automaton modeled by the precubical set. Adding inverses in this omegacategory to any morphism of dimension greater than 2 and with respect to any composition laws of dimension greater than 1 does not change these homology theories. In such a framework, the globular nerve always satisfies the Kan condition. On the other hand, both branching and merging nerves never satisfy it, except in some very particular and uninteresting situations. In this paper, we introduce two new nerves (the branching and merging semiglobular nerves) satisfying the Kan condition and having conjecturally the same simplicial homology as the branching and merging semicubical nerves respectively in such framework. The latter conjecture is related to the thin elements conjecture already introduced in our previous papers.
The branching nerve of HDA and the Kan condition
, 2001
"... We have already seen that in reasonable situations, i.e. when the path ωcategory associated to a higher dimensional automata (HDA) is an ωgroupoid, only the globular nerve satisfies the Kan condition. Indeed for the branching and merging nerves, this condition generally does not hold. This drawbac ..."
Abstract
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We have already seen that in reasonable situations, i.e. when the path ωcategory associated to a higher dimensional automata (HDA) is an ωgroupoid, only the globular nerve satisfies the Kan condition. Indeed for the branching and merging nerves, this condition generally does not hold. This drawback is overcome here by introducing two new nerves (the left and right globular nerves) satisfying the Kan condition in any reasonable situation and having conjecturally the same simplicial homology as the branching and merging nerves respectively for any ωcategory freely generated by a cubical set.
The branching nerve of HDA and the Kan condition
, 2001
"... We have already seen that in reasonable situations, i.e. when the path ωcategory associated to a higher dimensional automata (HDA) is an ωgroupoid, only the globular nerve satisfies the Kan condition. Indeed for the branching and merging (semicubical) nerves, this condition generally does not hol ..."
Abstract
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We have already seen that in reasonable situations, i.e. when the path ωcategory associated to a higher dimensional automata (HDA) is an ωgroupoid, only the globular nerve satisfies the Kan condition. Indeed for the branching and merging (semicubical) nerves, this condition generally does not hold. This drawback is overcome here by introducing two new nerves (the branching and merging semiglobular nerves) satisfying the Kan condition in any reasonable situation and having conjecturally the same simplicial homology as the branching and merging semicubical nerves respectively for any ωcategory freely generated by a precubical set.
Combinatorics of branchings in higher dimensional automata
, 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 ωcategory and the combinatorics of a new homology theory ca ..."
Abstract
 Add to MetaCart
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 ωcategory 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 ωcategories freely generated by precubical sets. As application, we calculate the branching homology of some ωcategories 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. 1 1
and
, 2000
"... We show the equivalence of two kinds of strict multiple category, namely the wellknown globular ocategories, and the cubical ocategories with connections. # 2002 ..."
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We show the equivalence of two kinds of strict multiple category, namely the wellknown globular ocategories, and the cubical ocategories with connections. # 2002