Directed Homotopy Theory is a form of homotopy theory which tries to study the properties of directed spaces. Much of the impetus for the theory initially came from work on modelling concurrent process. It can also be seen as a way of studying an ‘evolving’ space. This is discussed in more detail in the entry on motivation for directed homotopy.
The following examples illustrate the sort of problems that arise:
The example uses two directed spaces that are slightly different and use a pospace, i.e. a space with a closed partial order. (Both these use a rectangle with order if and only if , so the future cone of any point is a cone symmetric about a horizontal line through the point and with edges at degrees to that line.)
The two spaces are
In both the space is the rectangle with two smaller rectangles removed. The position of the upper left small rectangle is the same in both, but that of the righthand lower rectangle is shifted slightly to the right in the second picture. The directed homotopy classes of d-paths from to in the two cases are different. The crucial point is that in the second there is such a class that was impossible in the first example, yet the spaces are homeomorphic, so classically would be ‘the same’. The subtlety is in the order.
The first problem is to find a small model of such structures. The fundamental category would be a model, but unlike with the fundamental groupoid in the non-directed case, it is not sufficient to take a ‘base point’ in each connected component. That would ignore the order.
Philippe Gaucher (PPS, Paris) has introduced an interesting related model, namely that of ‘flows’. These are, approximately, topological categories without identity arrows. They are intended as another model of processes. One of his papers on this idea is at Arxiv, published as
Philippe Gaucher, A model category for the homotopy theory of concurrency, Homology Homotopy and Applications, vol. 5 (1):p.549-599, 2003.
Marco Grandis’ work on the area is listed amongst his publications at his (homepage). Such as
Marco Grandis, Directed homotopy theory. I, Cah. Topol. G eom. Di er. Cat eg. 44 (4) (2003) 281–316.
Marco Grandis, Directed homotopy theory. II. Homotopy constructs, Theory Appl. Categ. 10 (2002) No. 14, 369–391 (electronic).
Marco Grandis, The shape of a category up to directed homotopy, Theory Appl. Categ. 15 (2005/06) No. 4, 95–146 (electronic).
Marco Grandis, Modelling fundamental 2-categories for directed homotopy, Homology, Homotopy Appl. 8 (1) (2006) 31–70 (electronic)
L. Fajstrup, Dihomotopy classes of dipaths in the geometric realization of a cubical set: from discrete to continuous and back again, in: R. Kopperman, M. B. Smyth, D. Spreen, J. Webster (Eds.), Spatial Representation: Discrete vs. Continuous Computational Models, no. 04351 in Dagstuhl Seminar Proceedings, IBFI, Schloss Dagstuhl, Germany, 2005.
L. Fajstrup, Dipaths and dihomotopies in a cubical complex, Adv. in Appl. Math. 35 (2) (2005) 188–206.
M. Raussen, Deadlocks and dihomotopy in mutual exclusion models, in: R. Kopperman, M. B. Smyth, D. Spreen, J. Webster (Eds.), Spatial Representation: Discrete vs. Continuous Computational Models, no. 04351 in Dagstuhl Seminar Proceedings, IBFI, Schloss Dagstuhl, Germany, 2005.