**algebraic_brain**just posted a link to that old Baez-Shulman paper and got me to reread the early bits again, and I think I understood a little more than the last time I read it.

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.

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Hm, I suspect I have to think more 2-categorically to get it to work out. Man, higher-dimensional categories are confusing.

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Yeah, I think the type &Sigma

**D**.

**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.