## 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

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Simplicial Objects in Algebraic Topology
- May
- 1967
(Show Context)
Citation Context ...hat kind of identification is better than the other ones. Section 6 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∆... |

97 |
The algebra of oriented simplexes
- Street
- 1987
(Show Context)
Citation Context ...ntification is better than the other ones. Section 6 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
- 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... |

46 |
The Geometry of Concurrency
- Goubault
- 1995
(Show Context)
Citation Context ...ny ways to obtain the required identifications. This problem will be studied in another work because one cannot simply decide what kind of identification is better than the other ones. 3 Some authors =-=[Gou95]-=- [Sok99] use the term cubicalation : this means decomposing a HDA in cubes. 125.3 Deforming HDA The globular and both corner nerves give a satisfactory definition of orthogonal deformations 4 of HDA.... |

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 ... term temporal 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 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) 3�� �� together with an additional set X−1 and an additional map ∂−1 from X0 to X−1 such tha... |

30 |
Voevodsky, Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results), Cah
- Kapranov, A
- 1991
(Show Context)
Citation Context ...)) is obvious. One has H gl p (In(−n,+n)) = Hp−1(P n ) for p ≥ 2 where P n is the ω-category associated to the n-dimensional permutohedron and Hp−1(P n ) the (p − 1)-th simplicial homology group (see =-=[KV91]-=- and [Gau00] for more details). Conjecture 4.7. The globular homology of I n vanishes in strictly positive dimension. 5 Informal discussion 5.1 Examples of calculations The problem of the non-vanishin... |

29 |
The equivalence of ∞-groupoids and crossed complexes. Cahiers Topologie Géom. Différentielle 22
- BROWN, HIGGINS
- 1981
(Show Context)
Citation Context ... of identification is better than the other ones. Section 6 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 ...ion is better than the other ones. Section 6 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... |