State spaces and dipaths up to dihomotopy

Abstract

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.

Invariants of Directed Spaces

“…Naturality issues. As discussed earlier [12,18,7], a general d-map does in general not preserve components. Some coherence with the automorphic flows on the two spaces is needed.…”

“…The fundamental category π 1 (X) [18,7] arises from the trace category T(X) as the category of path components, with the dihomotopy relation. Concatenation on the trace spaces is homotopy invariant and factors over the fundamental category.…”

Algebraic topology and concurrency

“…For definitions and properties, cf. [43,68,20]. The investigation and exploitation of naturality properties for these categories presents still a major challenge.…”

Discovering spaces

“…We already know that the fiberdimension at a point x of a dicoveringX x0 is less than → π 1 (X, x 0 , x), and other connections to the different fundamental categories defined in [8] should be investigated.…”