Abstract not found

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 sub-complex 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.

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 three-object 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.

One can associate to any strict globular omega-category three augmented simplicial nerves called the globular nerve, the branching and the merging semi-cubical nerves. If this strict globular omega-category 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 omega-category 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 semi-globular nerves) satisfying the Kan condition and having conjecturally the same simplicial homology as the branching and merging semi-cubical nerves respectively in such framework. The latter conjecture is related to the thin elements conjecture already introduced in our previous papers.

We have already seen that in reasonable situations, i.e. when the path ω-category associated to a higher dimensional automata (HDA) is an ω-groupoid, 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 ω-category freely generated by a cubical set.

We have already seen that in reasonable situations, i.e. when the path ω-category associated to a higher dimensional automata (HDA) is an ω-groupoid, only the globular nerve satisfies the Kan condition. Indeed for the branching and merging (semi-cubical) nerves, this condition generally does not hold. This drawback is overcome here by introducing two new nerves (the branching and merging semi-globular nerves) satisfying the Kan condition in any reasonable situation and having conjecturally the same simplicial homology as the branching and merging semi-cubical nerves respectively for any ω-category freely generated by a precubical set.

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 ω-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 sub-complex generated by its thin elements. Conjecturally it coincides with the non reduced theory for higher dimensional automata, that is ω-categories freely generated by precubical sets. As application, we calculate the branching homology of some ω-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. 1 1

We show the equivalence of two kinds of strict multiple category, namely the wellknown globular o-categories, and the cubical o-categories with connections. # 2002