Anybody know if there's a standard fancy name for the generalization of a binary relation over sets X and Y where it's not just an assignment of a truth value to every element of X × Y, but an assignment of a set to each element of X × Y? I looked around at "multirelations" on google, but this doesn't seem to be the right thing.
Calling them "polyrelations", there is another punchline of the Baez-Shulman intro that I want to be true (but the paper goes off in a different direction) namely that a "not necessarily covering" covering of C is equivalent to a functor from C to the category PolyRel, whose objects are sets, and whose morphisms are polyrelations. (A "not necessarily covering" covering of C is just a choice of D and a functor from D to C, avoiding any path lifting condition) The problem is I can't really seem to define polyrelation composition in any way that works.
Hm, I suspect I have to think more 2-categorically to get it to work out. Man, higher-dimensional categories are confusing.
Yeah, I think the type &SigmaD.D→C (that is: choose a category D, and then provide a functor from D to C) is morally equivalent to the lax functors from C (regarded as a bicategory) into the bicategory PolyRel, whose objects are sets, whose morphisms X → Y are "polyrelations" on X and Y, that is, a choice of a set for each element of X × Y, and whose 2-cells R → S : X → Y are collections of maps R(x,y) → S(x,y), one for each x∈X, y∈Y. To compose two polyrelations, you take the set of two-step paths, viewing each of the polyrelations as a bipartite directed graph.