## The branching nerve of HDA and the Kan condition (2003)

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Venue: | Theory and Applications of Categories |

Citations: | 7 - 6 self |

### BibTeX

@INPROCEEDINGS{Gaucher03thebranching,

author = {Philippe Gaucher},

title = {The branching nerve of HDA and the Kan condition},

booktitle = {Theory and Applications of Categories},

year = {2003},

pages = {2003}

}

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### Abstract

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.

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Citation Context ... n . In a simplicial set, the face maps are always denoted by # i , the degeneracy maps by # i . Here are the other conventions about simplicial sets and simplicial homology theories (see for example =-=[11]-=- for further information) : 1. Sets : category of sets 2. Sets # op : category of simplicial sets 3. Sets # op + : category of augmented simplicial sets 4. Comp : category of chain complexes of abelia... |

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Citation Context ...mplex, branching, higher dimensional automata, concurrency, homology theory. c # Philippe Gaucher, 2003. Permission to copy for private use granted. 75 76 PHILIPPE GAUCHER of topological spaces as in =-=[2]-=-. The papers [6, 8] demonstrate that the formalism of strict globular #-categories (see Definition 2.1) freely generated by precubical sets (see below) provides a suitable framework for the introducti... |

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Citation Context ...arison with the branching semi-cubical nerve 101 10 Concluding discussion 103 11 Acknowledgments 104 1. Introduction An #-categorical model for higher dimensional automata (HDA) was first proposed in =-=[13]-=-, followed by [9] for a first homological approach using these ideas and cubical models Received by the editors 2001-07-27 and, in revised form, 2003-02-25. Transmitted by Ross Street. Published on 20... |

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Citation Context ... merging semi-globular homology and is denoted by H gl + n+1 (C) := H n (N gl + (C)). 2. Preliminaries The reader who is familiar with papers [6, 7, 8] may want to skip this section. 2.1. Definition. =-=[1, 16, 14]-=- An #-category is a set A endowed with two families of maps (s n = d - n ) n#0 and (t n = d + n ) n#0 from A to A and with a family of partially defined 2-ary operations (# n ) n#0 where for any n # 0... |

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Citation Context ...ponding to the simplicial nerves N gl , N - and N + become? To understand the answer given in Theorem 1.7, we need to recall in an informal way the thin elements conjecture which already showed up in =-=[7]-=-. If C gl (resp. Cs) is the unnormalized chain complex associated to N gl (resp. Ns) and if CR gl (resp. CRs) is the chain complex which is the quotient of C gl (resp. Cs) by the subcomplex generated ... |

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Citation Context ... merging semi-globular homology and is denoted by H gl + n+1 (C) := H n (N gl + (C)). 2. Preliminaries The reader who is familiar with papers [6, 7, 8] may want to skip this section. 2.1. Definition. =-=[1, 16, 14]-=- An #-category is a set A endowed with two families of maps (s n = d - n ) n#0 and (t n = d + n ) n#0 from A to A and with a family of partially defined 2-ary operations (# n ) n#0 where for any n # 0... |

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Citation Context ... non-contracting #-category Before going further in the construction of the negative semi-path #-category of a noncontractings#-category, one needs to recall some well-known facts about globular sets =-=[17]-=-. Let us consider the small category Glob defined as follows : the objects are all natural numbers and the arrows are generated by s and t in Glob(m, m-1) for any m > 0 and by the relations s # s = s ... |

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Citation Context ...s Sets. Let us denote by G # the corresponding category. Let U be the forgetful functor from #Cat to G # . One can prove by standard categorical arguments the existence of a left adjoint F for U (see =-=[12]-=- for an explicit construction of this left adjoint). 3.1. Definition. If G is a globular set, then the #-category FG is called the free #-category generated by the globular set G. Let C be a non-contr... |

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Citation Context ... merging semi-globular homology and is denoted by H gl + n+1 (C) := H n (N gl + (C)). 2. Preliminaries The reader who is familiar with papers [6, 7, 8] may want to skip this section. 2.1. Definition. =-=[1, 16, 14]-=- An #-category is a set A endowed with two families of maps (s n = d - n ) n#0 and (t n = d + n ) n#0 from A to A and with a family of partially defined 2-ary operations (# n ) n#0 where for any n # 0... |

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Citation Context ...K)? The globular nerve becomes Kan. Indeed its simplicial part is nothing else but the simplicial nerve of the path #- category P # F (K) of # F (K) which turns out to be a strict globular #-groupoid =-=[8]-=-. So now we can ask the question : do the branching and merging semi-cubical nerves N - ( # F (K)) and N + ( # F (K)) satisfy the Kan condition as well? The answer is : almost never. A counterexample ... |

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Citation Context ...he 0-morphisms represent the states of the HDA, the 1-morphisms the non-constant execution paths, and the p-morphisms with p # 2 the higher dimensional homotopies between them. R. Cridlig presents in =-=[3]-=- an implementation with CaML of the semantics of a real concurrent language in terms of precubical sets, demonstrating the relevance of this approach. However, in such an #-category C = F (K), if H is... |

11 | A Convenient Category for The Homotopy Theory of Concurrency. arXiv:math.AT/0201252 - Gaucher |

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Citation Context ...= {(0589), (0459)} (the elements in even position are removed). One can also notice that if X is an element of I n or # n , then X = R(X). Both # n and I n are examples of #-complexes in the sense of =-=[15]-=- where the atoms are the elements of the form R({x}) where x is a face of # n (resp. I n ). In such situations, Steiner's paper [15] proves that the calculation rules are very simple and that they can... |

11 |
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Citation Context ... the merging semi-globular homology and is denoted by H gl + n+1(C) := Hn(N gl +(C)). 2. Preliminaries The reader who is familiar with papers [6, 7, 8] may want to skip this section. 2.1. Definition. =-=[1, 16, 14]-=- An !-category is a set A endowed with two families of maps (sn = d-n )n?0 and (tn = d+n )n?0 from A to A and with a family of partially defined 2-ary operations (*n)n?0 where for any n ? 0, *n is a m... |

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Citation Context ...sets Sets. Let us denote by G! the corresponding category. Let U be the forgetful functor from !Cat to G!. One can prove by standard categorical arguments the existence of a left adjoint F for U (see =-=[12]-=- for an explicit construction of this left adjoint). 3.1. Definition. If G is a globular set, then the !-category FG is called the free !-category generated by the globular set G. Let C be a non-contr... |

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