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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