Re: 'Directed Algebraic Topology'

[Urs Schreiber <urs.schreiber@googlemail.com>]

On Tue, Sep 22, 2009 at 10:37 AM, Marco Grandis <grandis@dima.unige.it> wrote:

> In my web page you can find references to many papers of mine on this
> domain, and such
> papers have many references to other authors.

[...]

> The latter does not cover higher fundamental categories, which - in
> dimension 2 - can be found
> in:
>
> -, Modelling fundamental 2-categories for directed homotopy, Homology
> Homotopy Appl. 8 (2006), 31-70.
>
>
> -, Lax 2-categories and directed homotopy, Cah. Topol. Géom. Différ. Catég.
> 47 (2006), 107-128.
>
> -, Absolute lax 2-categories, Appl. Categ. Struct. 14 (2006), 191-214.


Thanks for these references. While I haven't read all of them in
detail, I am aware of many of them, I think. In fact, the question I
asked arose in discussion of nLab entries on directed space

 http://ncatlab.org/nlab/show/directed+space

and directed homotopy theory

 http://ncatlab.org/nlab/show/directed+homotopy+theory

(which still are greatly in need of improvement)

that list some of these.

My question revolves around the issue whether and to which degree
forming the fundamental category or 2-category or ... or
(oo,n)-catgory of a directed space -- for instance a d-space --
establishes an equivalence, in a suitable sense, between directed
spaces and these categorical structures that is analogous to the
(Quillen) equivalence between (nice) topological spaces and
oo-groupoids (modeled as Kan complexes) that is given by forming the
fundamental oo-groupoid Pi(X) = S(X) given by the singular simplicial
complex.

It would seem that in order to have the formation of the "fundamental
(oo,1)-category" (if any) of a directed space be a suitable
equivalence of sorts, one would need something like filtered or
stratified directed spaces.

Do you know if this has been considered?

Meanwhile probably Peter Bubenik's message to the mailing list will
have appeared, where he says that with David Spivak he is in the
process of investigating the connection between directed topological
spaces and (oo,1)-categories. I am wondering what model of directed
spaces they are using and to which extent they find an equivalence.


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