Directed Homotopy Theory is a form of homotopy theory which tries to study the properties of directed spaces. Initially it used pospaces as the objects, but with the introduction of d-spaces by Grandis, these have become, perhaps, the more common objects of study.
Much of the impetus for the theory has come from work on modelling concurrent process. It can also be seen as a way of studying an ‘evolving’ space.
(See also under directed space.)
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)
David Roberts: Marco Grandis has come out with a book 'Directed Algebraic Topology', published by CUP. Haven’t checked details on it, but presumably it contains material from the above papers
Zoran: I have not held the book in my hands yet but some info from the web is incorporated into new book entry Directed Algebraic Topology.
Urs Schreiber: hey, you forgot to reference the book here. I did it for you below:
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
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).
Tim: A related topic is that of higher dimensional automata. Perhaps someone who knows a little more than I do about this could comment on the links with other aspects of the nLab.