Subject: categories: Re: semi-categories

From: Philippe Gaucher <[email-masked]>
Date: Fri, 2 Dec 2005 13:25:46 +0100

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.