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The branching nerve of HDA and the Kan condition. (English) Zbl 1029.55014
The formalism of strict globular \(\omega\)-categories freely generated by precubical sets provides a suitable framework for the introduction of new algebraic tools devoted to the study of deformations of higher dimensional automata (HDA) (see the author’s paper [Math. Struct. Comput. Sci. 10, No.4, 481-524 (2000; Zbl 0956.68097)] and [Theory Appl. Categ. 8, 324-376, (2001; Zbl 0977.55012)]). In general, one can associate to any strict globular \(\omega\)-category three augmented simplicial nerves called the globular nerve, the branching and the merging semi-cubical nerve, but if the strict globular \(\omega\)-category is freely generated by a precubical set, then the corresponding homology theories contain some information about the geometry of the HDA modeled by the cubical set. If one adds 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, the homology theories do not change and the globular nerve satisfies the Kan condition. However, both branching and the merging semi-cubical nerves never satisfy it, except in some uninteresting cases.
In this framework, the author introduces two new nerves (branching and merging semi-globular nerves) satisfying the Kan condition. He also conjetures that these nerves have the same simplicial homology as the branching and merging semi-cubical nerves, respectively.

MSC:
55U10 Simplicial sets and complexes in algebraic topology
68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
18G35 Chain complexes (category-theoretic aspects), dg categories
68Q70 Algebraic theory of languages and automata
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