Combinatorics of branchings in higher dimensional automata

Details Combinatorics of branchings in higher dimensional automata Combinatorics of branchings in higher dimensional automata

1999-12-08

arxiv.org/abs/math/9912059v4

Theory and Applications of Categories, Vol. 8, 2001, No. 12, pp 324-376

Mathematics

Category Theory

Final version, see http://www.tac.mta.ca/tac/volumes/8/n12/abstract.html

Scientific paper

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 $\omega$-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 $\omega$-categories freely generated by precubical sets. As application, we calculate the branching homology of some $\omega$-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.

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