Le mercredi 30 Novembre 2005 23:24, vous avez écrit :
Perhaps it has not been sufficiently emphasized that semi-categories and the like are not really "generalizations" of categories (though formally they may appear so).
Indeed, at least in my case, a flow must not be viewed as a generalization of the notion of small categories. Let me explain a little bit what I am doing with these objects. I was not very explicit in my previous post. And so the terminology I use is not in "competition". I want to model HDA, at least those coming from precubical sets. I use a set of states X^0 and between each state A and B of the HDA, there is a topological space P_{A,B}X whose elements represent the non-constant execution paths from A to B. The topology of this space models the concurrency of the situation between A and B. And execution paths can be composed with a strictly asssociative law. There does not necessarily exist a loop from a given state A to itself : so P_{A,A}X can be empty. This fact is one reason among several other ones why I remove the identity maps. Inside this model, I am able to define what is a dihomotopy equivalence. The main problem to define dihomotopy is that some contractible parts of "the directed spaces of execution paths" must not be contracted. Otherwise in the categorical localization, the relevant geometric information is lost. In particular, initial and final states must be unchanged by a dihomotopy equivalence. A very simple example : take two execution paths going from one initial state 0 to one final state 1. If contractions in the direction of time are allowed, one finds in the same equivalence class a loop. Some examples of unwanted final states are deadlocks of concurrent systems : a deadlock is nothing else but a final state from a geometric viewpoint. Flows allow to propose a solution of this problem : in fact I introduced this notion of flows on purpose, to make the following solution work. The first kind of dihomotopy equivalence is a morphism f:X->Y such that f^0:X^0->Y^0 is a bijection and such that Pf:PX->PY is a weak homotopy equivalence. It turns out that there is a model structure on Flows whose weak equivalences are exactly the preceding kind of morphisms. By imposing the condition f^0:X^0->Y^0 bijective, we do not take any risk : nothing is contracted in the direction of time. So no geometric information is lost. But this kind of identification is too rigide ! The second kind of dihomotopy equivalence is generated by taking a representative set of inclusions of posets P1\subset P2, where P1 and P2 are finite bounded posets and where the inclusions preserve the bottom element and the top element (which are different by hypothesis in a bounded poset). For example, the inclusion of posets {0<1}\subset{0<A<1} represents the directed segment (going from the initial state 0 to the final state 1) identified with the composition of two directed segments. This second kind of dihomotopy equivalence models "refinement of observation". Of course, initial and final states are still preserved. The category of flows up to dihomotopy equivalences is between the homotopy category of the model structure associated to the first kind of dihomotopy equivalence and the homotopy category of the Bousfield localization of the same model structures with respect to the set of Q(P_1\subset P_2) where Q is the cofibrant replacement functor. I call the weak equivalences of the Bousfield localization "quasi-dihomotopy". Morally speaking, quasi-dihomotopy is like dihomotopy except in non-observable areas of the time flow where the topological configuration can changed. pg.