Topological deformation of higher dimensional automata

Directed Algebraic Topology is a recent field, deeply linked with ordinary and higher dimensional Category Theory. A 'directed space', e.g. an ordered topological space, has directed homotopies (which are generally non-reversible) and fundamental n-categories (replacing the fundamental n-groupoids of the classical case). Finding a simple model of the latter is a nontrivial problem, whose solution gives relevant information on the given 'space'; a problem which is also of interest in general Category Theory, as it requires equivalence relations which are more general than categorical equivalence. Taking on a previous work on 'The shape of a category up to directed homotopy', we study now the fundamental 2-category of a directed space. All the notions of 2-category theory used here are explicitly reviewed.

“…We study the fundamental 2-category of the 'hollow cube' X ⊂ ↑ [0,1] 3 (see (8)), starting with the construction of a future 2-equivalent model.…”

Section: Modelling a Fundamental 2-categorymentioning

confidence: 99%

“…Its present applications deal mostly with the analysis of concurrent processes (see [5,6,7,8,9]), but its natural range covers non-reversible phenomena, in any domain. Other references for Directed Algebraic Topology and its applications can be found in [14].…”

Section: Introductionmentioning

confidence: 99%

Modelling fundamental 2-categories for directed homotopy

Grandis

2006

“…It is only quite recently that these have come into full bloom, see [21] for a start, where many algebraic invariants are also introduced. The "unification" of these approaches has lead us to the concept of a globular CW-complex [23] which I will briefly describe in Section 8.…”

Section: • S Is a Set Of States With Initial State I • L Is A Set Of mentioning

confidence: 99%

“…In general, we restrict ourself to topological spaces which have the homotopy type of a CW-complex [41]. P. Gaucher and myself have introduced in [23] a particular kind of CW-complex, which we called globular CW-complex, and which is essentially a CW-complex whose ncells are all directed. A n-cell is homeomorphic to I × I n−1 quotiented by relations…”

Section: Cubical Setsmentioning

confidence: 99%

Some geometric perspectives in concurrency theory

Goubault

2003

Self Cite

Concurrency, i.e., the domain in computer science which deals with parallel (asynchronous) computations, has very strong links with algebraic topology; this is what we are developing in this paper, giving a survey of "geometric" models for concurrency. We show that the properties we want to prove on concurrent systems are stable under some form of deformation, which is almost homotopy. In fact, as the "direction" of time matters, we have to allow deformation only as long as we do not reverse the direction of time. This calls for a new homotopy theory: "directed" or di-homotopy. We develop some of the geometric intuition behind this theory and give some hints about the algebraic objects one can associate with it (in particular homology groups). For some historic as well as for some deeper reasons, the theory is at a stage where there is a nice blend between cubical, ω-categorical and topological techniques.

“…This is the po-space introduced in [15] -for different purposes -under the term Glob(X). All dipaths from the minimal to the maximal point have the form α x : I → ΣX, t → [(x, t)] for a fixed x ∈ X -or are monotone reparametrizations of those.…”

Section: Dihomotopymentioning

confidence: 99%

State spaces and dipaths up to dihomotopy

Raussen

2003

Geometric models have been used by several authors to describe the behaviour of concurrent sytems in computer science. A concurrent computation corresponds to an oriented path (dipath) in a (locally) partially ordered state space, and di homotopic dipaths correspond to equivalent computations. This paper studies several invariants of the state space in the spirit of those of algebraic topology, but taking partial orders into account as an important part of the structure. We use several categories of fractions of the fundamental category of the state space and define and investigate the related quotient categories of "components". For concurrency applications, the resulting categories can be interpreted as a dramatic reduction of the size of the state space to be considered.