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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]