Re: 'Directed Algebraic Topology'

["George Janelidze" <janelg@telkomsa.net>]

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?



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