Re: 'Directed Algebraic Topology'

[Martin Escardo <m.escardo@cs.bham.ac.uk>]

"Asymmetric spaces" doesn't give relevant hits in google, but
"asymmetric topology does". People have considered "quasi-uniform
spaces" (losing the symmetry in the same way as metric spaces with
Lavwere's work), and much more. An interesting example, which has shown
up in topos theory via the work of Johnstone, are stably locally compact
spaces/locales, where the patch modification makes (coreflectively) such
spaces symmetric. Some people use two topologies on the same set, or two
subframes of a frame that generate, or variations of this idea.
Symmetric spaces give "positive and negative information", and
asymmetric ones give one of the two only. Some people argue that it is
good to keep the topologies of positive and of negative information
separate. Anyway, I just wanted to say that there is a large body of
work on this. Whether it is relevant for the original question I don't
know, but it is relevant for Barr's subquestion. MHE.

Michael Barr wrote:
> 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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