Although partially ordered spaces (pospaces) and directed spaces originally arose in Analysis and General Topology, the recent interest in them has resulted from attempts to model state spaces of evolving processes, particularly those that arise in modelling concurrent programming. They also occur in models of space-time and in modelling certain modal logics.
The basic idea is that of a topological space in which points can be compared using a partial order corresponding to ‘precedes’.
A partially ordered space or pospace, $X$, is a topological space with a (globally defined) closed partial order, $\leq$, so considering $\leq$ as a subset of $X \times X$, it is a closed subset. (It follows that $X$ must be a Hausdorff space.)
A dimap $f: X \to Y$ between two pospaces, $X$ and $Y$, is a continuous map that respects the partial order:
The standard directed interval, $I_d = ([0,1], \leq)$, is a pospace that gives rise to the directed space of the same name.
Any pospace $X$ gives rise to a directed space by taking the directed paths to be, well, directed paths: dimaps from $I_d$ to $X$. (It can be helpful to use directed paths of variable length based on $\stackrel{\to}{[0,r]}$ the directed version of the interval of length $r$.)
There are several useful notions of ‘homotopy’ between dimaps. Choosing between them depends on the intended use and/or the situation being modelled.
where $X \times I_d$ is given the product partial order. (The pospace $I_d$ is defined below amongst the examples.)
We say $a$ and $b$ are dihomotopic, the relation being called directed homotopy.
Eric: I can imagine that “directed” stuff might begin to garner more attention here, so could we settle on a notation? For example, I’ve used $\uparr$ to indicate that a space is directed so that the directed interval is denoted $\uparr I$. Above, it is given as $I_d$. I am slightly partial (but not firmly set) to using $\uparr$ to denote that a space is directed or partially ordered (or even preordered).
Second, I have a “continuum” allergy and never like to see a definition that explicitly evokes the continuum to define important objects. Is there a more combinatoric or abstract way to define pospaces, directed spaces, and directed homotopy that does not evoke the continuum immediately?
How/if is the standard directed interval related to interval object?
Tim: I am also alergic to too much ‘continuum’ use. The problem is to get a substitute that allows and controls subdivision. Ideas anyone? Gabriel Minian (paper in K-theory used a variant of Baues’s cofibration category notion to handle various categories including that of simplicial complexes. He used an Ind-object instead of an interval object. His theory is not a directed one, of course. Marco Grandis also looked at the problem of subdivision in some papers and going right back there were notes by George Hoff and Marcel Evrard trying to do the homotopy theory of categories using concatentated intervals. I can try to find references if it should proved useful.
As to combinatorial definitions, pospace is not infected with ‘continuumitis’, but the directed interval is. Perhaps what you want is that the directed interval is both a directed object and an interval object, but that may lead to a clash of definitions, so better an amalgam of the two definitions should be sought.
One can try to use a domain theoretic definition of things a bit like the papers such as A Domain of spacetime intervals for General relativity by Keye Martin and Prakash Panangaden. (see on Prakash’s page http://www.cs.mcgill.ca/~prakash/accepted.html).
Toby: As Tim says, there's nothing about the continuum in the basic notions of pospace; that comes in only when you introduce dihomotopy, and already ordinary homotopy involves the continuum. Really we should talk about what interval objects there may be in the category of pospaces and what homotopies with respect to them are like; the continuum is simple our first example of that.
As for notation, notice that we've actually got three: $I_d$, $\uparrow I$, and $\vec{I}$ (applied to $[0,r]$ above). Personally, I'd just use $I$; let the context tell us that it's directed. (Nobody needs special notation to distinguish the set $R$ of real numbers from the poset $R$, after all; why should we need it when we topologise?)