Jason (jcreed) wrote,
Jason
jcreed

Have been going back and reading about ditopologies again. The idea is that you slap a poset structure an a topological space in a local way, so that transitivity need not hold "globally". It's extremely similar to the way you define a manifold without referring to an ambient space: you take an open cover, and require a poset structure on each open set in it such that they are compatible where the overlap, like a manifold's atlas of charts.

Anyway, these things got trendy a few years back because homotopies in them --- well, dihomotopies --- let you talk about concurrency and deadlock and saftey in a cute, diagrammatic way. But they're another obvious place to start thinking about higher-order algebra. Homotopies in ordinary topological spaces make up omega-groupouids, and "directing" the topology seems to get rid of commutativity in the right way. I seems pretty obvious that a ditopology ought to give rise to a weak omega-category --- I figure this all got worked out by someone, but I haven't really combed the literature very aggressively.

Except for the obvious fact that still nobody's exactly sure what weak omega-categories are. So... I looked at Tom Leinster's survey again, and I think there is maybe still a niche for some sort of ditopological definition.
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