Dear Michael, The asymmetry you are mentioning is very different. One of your old theorems, expressed in a language recently used by Maria Manuel Clementino, Dirk Hofmann, Walter Tholen and others, says that the category of topological spaces can be identified with the category of (T,V)-categories (=lax (T,V)-algebras) for V = 2 = {0,1} considered as a symmetric monoidal category, and T being the ultrafilter monad on the category of sets. If T is the identity monad on the category of sets, a (T,V)-category becomes just a V-category (=a category enriched in V), and - at least when V is a quantale - it makes sense to call a V-category X "symmetric" if X(a,b) = X(b,a) for all objects a and b in X (where X(a,b) denotes the internal hom object). Example 1 (the old observation of Bill Lawvere - as you know of course): When V = R+ (suitable monoidal category of nonnegative real numbers), a V-category is exactly an asymmetric metric space, while a symmetric V-category is an ordinary metric space (well, ignoring the axiom d(x,y) = 0 => x = y of course). Example 2 (obvious): When V = 2, a V-category is a preorder (=reflexive and transitive binary relation on a set), while a symmetric V-category is an equivalence relation (=symmetric preorder). But when T is not an identity monad as in your old theorem, the symmetry does not even make sense - simply because, say, a relation between T(X) and X cannot be symmetric. That is, the ordinary topological spaces are already "much more asymmetric" than asymmetric metric spaces! And what I said about (asymmetric) metric spaces can be repeated for (quasi)uniform spaces with "pro" involved: see [M. M. Clementino, D. Hofmann, W. Tholen, One setting for all: Metric, Topology, Uniformity, Approach Structure, Applied Categorical Structures 12, 2004, 127-154]. So again, "the ordinary topological spaces are much more asymmetric than asymmetric uniform spaces (=quasiuniform spaces)". But yes, quasiuniform spaces have been and are studied seriously - particularly categorically and particularly by Guillaume Brummer and Hans-Peter Kunzi in Cape Town. George ----- Original Message ----- From: "Michael Barr" <barr@math.mcgill.ca> To: "Urs Schreiber" <urs.schreiber@googlemail.com>; <categories@mta.ca> Sent: Monday, September 21, 2009 5:56 PM Subject: categories: Re: 'Directed Algebraic Topology' One obvious thing that comes to mind are asymmetric spaces--a metric without the symmetry axiom. This can obviously be extended to uniform spaces, although I am not aware anyone has. As for topological spaces, I know of nothing there. Michael On Mon, 21 Sep 2009, 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.
More generally, it would be nice to have a notion of "r-directed topological space" for r in N that would extend the relation between (nice) topological spaces and oo-groupoids to one of (nice) "r-directed spaces" and (oo,r)-cateories.
(Probably such a notion of directed spaces can't be supporrted by plain topological spaces with direction information, but requires filtered directed spaces or the like. )
Has anything like this been considered?
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]