We introduce a new simplicial nerve of higher dimensional automata whose simplicial homology groups shifted by one yield a new definition of the globular homology. With this new definition, the drawbacks noticed with the construction of [Gau00c] disappear. Moreover the important morphisms which associate to every globe its corresponding branching area and merging area of execution paths become morphisms of simplicial sets. Contents

Abstract. The primary purpose of this work is to characterise strict ω-categories as simplicial sets with structure. We prove the Street-Roberts conjecture which states that they are exactly the “complicial sets ” defined and named by John Roberts in his handwritten notes of that title [26].2 VERITY

We introduce a new simplicial nerve of higher dimensional automata whose simplicial homology groups shifted by one yield a new definition of the globular homology. With this new definition, the drawbacks noticed with the construction of [Gau00c] disappear. Moreover the important morphisms which associate to every globe its corresponding branching area and merging area of execution paths become morphisms of simplicial sets. Contents

We explore the combinatorial properties of the branching areas of paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner homology of a globular ω-category and the combinatorics of a new homology theory called the reduced negative corner homology. This latter is the homology of the quotient of the corner complex by the sub-complex generated by its thin elements. As application, we calculate the corner homology of some ω-categories and we give some invariance results for the reduced corner homology. We only treat the negative side. The positive side, that is to say the case of merging areas of paths is similar and can be easily deduced from the negative side.

The strict globular ω-categories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) ω-category 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 to reinterprete some geometric problems coming from computer science.

We explore the combinatorial properties of the branching areas of paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner homology of a globular ω-category and the combinatorics of a new homology theory called the reduced negative corner homology. This latter is the homology of the quotient of the corner complex by the sub-complex generated by its thin elements. As application, we calculate the corner homology of some ω-categories and we give some invariance results for the reduced corner homology. We only treat the negative side. The positive side, that is to say the case of merging areas of paths is similar and can be easily deduced from the negative side.

The strict globular ω-categories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) ω-category 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.

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 explore the combinatorial properties of the branching areas of paths in higher dimensional automata. Mathematically, this means that we investigate the combinatorics of the negative corner homology of a globular ω-category and the combinatorics of a new homology theory called the reduced negative corner homology. This latter is the homology of the quotient of the corner complex by the sub-complex generated by its thin elements. It conjecturally coincides with the non reduced theory for higher dimensional automata. As application, we calculate the corner homology of some ω-categories and we give some invariance results for the reduced corner homology. We only treat the negative side. The positive side, that is to say the case of merging areas of paths is similar and can be easily deduced from the negative side. 1

The strict globular ω-categories formalize the execution paths of a parallel automaton and the homotopies between them. One associates to such (and any) ω-category 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.