## About the globular homology of higher dimensional automata (2000)

### BibTeX

@MISC{Gaucher00aboutthe,

author = {Philippe Gaucher},

title = {About the globular homology of higher dimensional automata},

year = {2000}

}

### OpenURL

### Abstract

We introduce a new simplicial nerve of higher dimensional automata whose homology groups yield a new definition of the globular homology. With this new definition, the drawbacks noticed with the construction of [Gau99] 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.

### Citations

298 |
Simplicial objects in algebraic topology
- May
- 1967
(Show Context)
Citation Context ..., and how to construct a lot of other invariants for HDA. Section 7 deals with the proofs omitted from Section 4. 2 Conventions and notations Here are the conventions of notations for the sequel (see =-=[May67]-=- [BH81] [Str87] [Ste91] [Gau99] [Gau00]). 1. ωCat : category of globular ω-categories 2. ωCat1 : category of globular ω-categories with non-1-contracting ω-functors 3. Sets : category of sets 4. Sets∆... |

99 |
The algebra of oriented simplexes
- Street
- 1987
(Show Context)
Citation Context ...nstruct a lot of other invariants for HDA. Section 7 deals with the proofs omitted from Section 4. 2 Conventions and notations Here are the conventions of notations for the sequel (see [May67] [BH81] =-=[Str87]-=- [Ste91] [Gau99] [Gau00]). 1. ωCat : category of globular ω-categories 2. ωCat1 : category of globular ω-categories with non-1-contracting ω-functors 3. Sets : category of sets 4. Sets∆op : category o... |

48 | Homotopy invariants of higher dimensional categories and concurrency in computer science, in: Geometry and concurrency
- Gaucher
(Show Context)
Citation Context ...e a new simplicial nerve of higher dimensional automata whose homology groups yield a new definition of the globular homology. With this new definition, the drawbacks noticed with the construction of =-=[Gau99]-=- 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 1 Introdu... |

47 | The Geometry of Concurrency - Goubault - 1995 |

43 | Algebraic topology and concurrency
- Fajstrup, Goubault, et al.
(Show Context)
Citation Context ...er authors are developing an “algebraic topology” of HDA using pospaces. Roughly speaking, a pospace is a topological space endowed with a local partial order for every point of the space. I refer to =-=[FGR98]-=- for their terminology. The three simplicial nerves can be probably constructed for pospaces. And due to the non-dependence on the choice of a cubification of a HDA, they will give rise to interesting... |

41 |
Théorie des ensembles
- Bourbaki
- 1958
(Show Context)
Citation Context ...mutes with all face and degeneracy maps. We denote by Sets∆op + the category of augmented simplicial sets. Every simplicial set X can be made an augmented simplicial set by setting X−1 = ∅. Following =-=[Bou82]-=-, there is exactly one map ∂−1 from X0 to the empty set X−1 : ∂−1 = ∅. We obtain a canonical inclusion functor from Sets∆op to Sets∆op + . The functor H∗ from Sets∆op to Ab can be extended to a functo... |

41 |
Simplicial methods and the interpretation of triple cohomology
- Duskin
- 1970
(Show Context)
Citation Context ...mporal graph of C means the 1-category τ1C. The cubical homology of C (i.e. the homology of the total cubical complex of C) is used nowhere in this work. 3�� �� 3 Cut of ω-categories Definition 3.1. =-=[Dus75]-=- An augmented simplicial set is a simplicial set ((Xn)n≥0,(∂i : Xn+1 −→ Xn)0≤i≤n+1,(ǫi : Xn −→ Xn+1)0≤i≤n) together with an additional set X−1 and an additional map ∂−1 from X0 to X−1 such that ∂−1∂0 ... |

37 | Combinatorics of branchings in higher dimensional automata
- Gaucher
(Show Context)
Citation Context ...n We will see in Section 5 the reason of the terminology “globular”. In the case of an ω-category C of length at most one, that is if x∗0 y ∈ C implies that x or y is 0-dimensional, then we proved in =-=[Gau00]-=- that the three simplicial cuts N ± (C) and N gl (C) have the same homology groups in dimension greater or equal than 2. The following theorem uses the notion of cut to characterize the morphisms h − ... |

29 |
The equivalence of ∞-groupoids and crossed complexes. Cahiers Topologie Géom. Différentielle 22
- BROWN, HIGGINS
- 1981
(Show Context)
Citation Context ...w to construct a lot of other invariants for HDA. Section 7 deals with the proofs omitted from Section 4. 2 Conventions and notations Here are the conventions of notations for the sequel (see [May67] =-=[BH81]-=- [Str87] [Ste91] [Gau99] [Gau00]). 1. ωCat : category of globular ω-categories 2. ωCat1 : category of globular ω-categories with non-1-contracting ω-functors 3. Sets : category of sets 4. Sets∆op : ca... |

25 |
Tensor products of infinity-categories
- Steiner
- 1991
(Show Context)
Citation Context ...a lot of other invariants for HDA. Section 7 deals with the proofs omitted from Section 4. 2 Conventions and notations Here are the conventions of notations for the sequel (see [May67] [BH81] [Str87] =-=[Ste91]-=- [Gau99] [Gau00]). 1. ωCat : category of globular ω-categories 2. ωCat1 : category of globular ω-categories with non-1-contracting ω-functors 3. Sets : category of sets 4. Sets∆op : category of simpli... |

6 | Classifying holes of arbitrary dimensions in partially ordered cubes
- Sokolowski
- 1999
(Show Context)
Citation Context ... to (G,ev) is a natural transformation of functors φ from F to G which makes the following diagram commutative for any n ≥ 0 : evn Fn � �� φn �� ��evn �� Gn The terminology of “cuts” is borrowed from =-=[Sok99]-=-. There is no ambiguity to denote all evn by the same notation ev in the sequel. The map ev of N-graded sets is called the evaluation map and a cut (F,ev) will be always denoted by F. Denote by Cut th... |