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 $(x,y)\leq (x\prime, y\prime)$ if and only if $|y\prime - y| \leq x\prime - x$, so the future cone of any point is a cone symmetric about a horizontal line through the point and with edges at $\pm 45$ degrees to that line.)
The two spaces are
and
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 $p$ to $p^\prime$ 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.
(See also under directed space.)
Foundational work was done by Eric Goubault and his collaborators.
Categorical aspects are looked at in
The fundamental category of a pospace is discussed in
and the possibility of an analogue of covering spaces in
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
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)
Marco Grandis, Directed Algebraic Topology, Models of non-reversible worlds , Cambridge University Press, 2009.
A websearch will find others.
Another approach to model category structures in this area is by Kahl, who uses a Baues type fibration category approach.
Krzysztof Worytkiewicz and Peter Bubenik? have given a model category structure for local pospaces:
Further related references are
L. Fajstrup, Loops, ditopology and deadlocks, Math. Structures Comput. Sci. 10 (4) (2000) 459–480, geometry and concurrency.
L. Fajstrup, M. Raussen, E. Goubault, E. Haucourt, Components of the fundamental category, Appl. Categ. Structures 12 (1) (2004) 81–108, homotopy theory.
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.
U. Fahrenberg, M. Raussen, Reparametrizations of continuous paths, available as preprint R-2006-22 (2006).
E. Goubault, E. Haucourt, Directed algebraic topology and concurrency, (web) (2006).23
E. Goubault, E. Haucourt, Components of the fundamental category, II, technical reports, CEA, Saclay (2006).
M. Raussen, Invariants of directed spaces, available as preprint R-2006-28 (web)(2006).