Re: 'Directed Algebraic Topology'

[Marco Grandis <grandis@dima.unige.it>]

Dear Urs,

There are various directed topological structures, which have  
directed homotopies
and fundamental (higher) categories, like:

- preordered topological spaces (simple but poor);
- locally preordered topological spaces (in a suitable sense);
- d-spaces = topological spaces equipped with distinguished paths;
- spaces equipped with distinguished cubes;
- cubical sets (in the combinatorial world);
- generalised metric spaces (in the sense of Lawvere);
- 'inequilogical spaces';
- etc.

I prefer d-spaces, which have also been studied by other authors.  
(Notice that the one-dimensional
information which is added to a topological space has effects in all  
dimension.) However,
directed homology works much better for cubical sets, or spaces with  
distinguished cubes.

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. You could begin by:

- M. Grandis, Directed homotopy theory, I. The fundamental category,  
Cah. Topol. Géom. Différ. Catég. 44 (2003), 281-316.

-, The shape of a category up to directed homotopy, Theory Appl.  
Categ. 15 (2005/06), No. 4, 95-146.


A more complete study can be found in my book.
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.



Marco Grandis

http://www.dima.unige.it/~grandis/


On 21 Sep 2009, at 11:44, Urs Schreiber wrote:

> Marco Grandis wrote:
>
>> My book
>>
>>    'Directed Algebraic Topology'
>>    Models of non-reversible worlds
>>
>> has appeared, at Cambridge University Press.
>
> In that context I am wondering about the following:
>
> it would be nice to have a notion of directed topological space that
> would extend the relation between (nice) topological spaces and
> oo-groupoids to one between (nice) directed topological spaces and
> (oo,1)-categories.
>
> ...
> Has anything like this been considered?


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