## Combinatorics Of Branchings In Higher Dimensional Automata (2001)

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Venue: | Theory Appl. Categ |

Citations: | 37 - 9 self |

### BibTeX

@ARTICLE{Gaucher01combinatoricsof,

author = {Philippe Gaucher},

title = {Combinatorics Of Branchings In Higher Dimensional Automata},

journal = {Theory Appl. Categ},

year = {2001},

volume = {8},

pages = {324--376}

}

### Years of Citing Articles

### OpenURL

### Abstract

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.

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Citation Context ...Sets # op of simplicial sets. The globular #-category # n . Now let us recall the construction of the #-category called by Street the n-th oriental [26]. We use actually the construction appearing in =-=[17]-=-. Let O n be the set of strictly increasing sequences of elements of {0, 1, . . . , n}. A sequence of length p + 1 will be of dimension p. If # = {# 0sp } is a p-cell of O n , then we set # j # = {# 0... |

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Citation Context ...lest of these numbers is called the dimension of x and is denoted by dim(x)). A globular set is a set A endowed with two families of maps (s n ) n#0 and (t n ) n#0 satisfying the same axioms as above =-=[27, 21, 3]-=-. We call s n (x) the n-source of x and t n (x) the n-target of x. Notation. The category of all #-categories (with the obvious morphisms) is denoted by #Cat. The corresponding morphisms are called #-... |

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Citation Context ...unctor 2 This means that for an #-category of length at most 1, H - n+1 (C) # = H + n+1 (C) for any n # 1. In general, this isomorphism is false as shown by 1 The latter point is actually detailed in =-=[13]-=-. 2 Like the branching nerve, the definition of the merging nerve needs to be slightly change, with respect to the definition given in [12]. The correct definition is : an #-functor x from I n to a no... |

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Citation Context ...n and X # Y = X # p Y . Moreover, all elements X of I n satisfy the equality X = R(X). The elements of I n correspond to the loop-free well-formed sub pasting schemes of the pasting scheme cub n [15] =-=[9] or to the-=- molecules of an #-complex in the sense of [25]. The condition "X # n Y exists if and only if X # Y = t n X = s n Y " of [25] is not necessary here because the situation of [25] Figure 2 can... |

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Citation Context ... y because # - 1 A #= # - 1 (x + 2 y). All calculations involving these matrix notations are justified because the Dawson-Pare condition holds in 2-categories due to the existence of connections (see =-=[11]-=- and [7]). The Dawson-Pare condition stands as follows : suppose that a square # has a decomposition of one edge a as a = a 1 + 1 a 2 . Then # has a compatible composition # = # 1 + i # 2 , i.e. such ... |

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Citation Context ...xecution of n 1-transitions. This theoretical idea would be implemented later. Indeed a CaML program translating programs in Concurrent Pascal into a text file coding a precubical set is presented in =-=[10]-=-. At this step, one does not yet consider cubical sets with or without connections since the degenerate elements have no meaning at all from the point of view of computer-scientific modeling (even if ... |

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Citation Context ...ondition which ensures that all "compatible" tilings represent the same object. Let us mention that these special 2-dimensional notations for connections and degeneracies first appeared in [=-=8] and in [23]-=-. Theory and Applications of Categories, Vol. 8, No. 12 337 5. Relation between the simplicial nerve and the branching nerve 5.1. Proposition. [12] Let C be an #-category and # # {-, +}. We set N - n ... |

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Citation Context ...satisfy the equality X = R(X). The elements of I n correspond to the loop-free well-formed sub pasting schemes of the pasting scheme cub n [15] [9] or to the molecules of an #-complex in the sense of =-=[25]. The cond-=-ition "X # n Y exists if and only if X # Y = t n X = s n Y " of [25] is not necessary here because the situation of [25] Figure 2 cannot appear in a composable pasting scheme. The map which ... |

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