Toby Bartels wrote:
Then the obvious thing to try is to symmetrise it: An equivalence between A and B is a span A <- X -> B of fully faithful and essentially surjective functors.
It's interesting, I think, that two categories A and B are equivalent if and only if there exists a span A <-- X --> B of functors that are full and faithful and *genuinely* surjective on objects. What's interesting is that "full and faithful and genuinely surjective on objects" is a purely graph-theoretic condition: it doesn't refer to the category structures. This suggests a pain-free way of defining equivalence of n-categories, an idea I learned from Carlos Simpson. For example, if I remember correctly, two bicategories A and B are biequivalent if and only if there exists a span A <-- X --> B of strict functors that are surjective at every level, i.e. locally faithful, locally full, locally surjective on objects, and surjective on objects. Tom -- The University of Edinburgh is a charitable body, registered in Scotland, with registration number SC005336. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]