Aims and applications

Directed algebraic topology is a recent subject which arose in the 1990s, on the one hand in abstract settings for homotopy theory, like [G1], and, on the other hand, in investigations in the theory of concurrent processes, like [FGR1, FGR2]. Its general aim should be stated as ‘modelling non-reversible phenomena’. The subject has a deep relationship with category theory.

The domain of directed algebraic topology should be distinguished from the domain of classical algebraic topology by the principle that directed spaces have privileged directions and directed paths therein need not be reversible. While the classical domain of topology and algebraic topology is a reversible world, where a path in a space can always be travelled backwards, the study of non-reversible phenomena requires broader worlds, where a directed space can have non-reversible paths.

The homotopical tools of directed algebraic topology, corresponding in the classical case to ordinary homotopies, the fundamental group and fundamental n-groupoids, should be similarly ‘non-reversible’: directed homotopies, the fundamental monoid and fundamental n-categories. Similarly, its homological theories will take values in ‘directed’ algebraic structures, like preordered abelian groups or abelian monoids. Homotopy constructions like mapping cone, cone and suspension, occur here in a directed version; this gives rise to new ‘shapes’, like (lower and upper) directed cones and directed spheres, whose elegance is strengthened by the fact that such constructions are determined by universal properties.

Applications will deal with domains where privileged directions appear, such as concurrent processes, rewrite systems, traffic networks, space-time models, biological systems, etc. At the time of writing, the most developed ones are concerned with concurrency: see [FGR1, FGR2, FRGH, Ga1, GG, GH, Go, Ra1, Ra2].