Finite coproducts in variable categories

Ross Street – 14 May 1997

When we move away from ordinary category theory to enriched or variable category theory, it is no longer obvious what is the most useful replacement for "finite coproducts". In the enriched case, where the base category V is monoidally locally finitely presentable, I proposed that tensors x#a with finitely presented objects x in V should be included along with the more obvious finite coproducts a_1 + . . . + a_n. My reason for going back to such basics was to try to understand how Michael Batanin's trees (or forests) could be regarded as natural numbers in the context of globular sets. I tried to convince the 17 members (including myself) of the Seminar that suspension of globular sets is a "finite copower" (as you might expect from our experience with chain complexes). I explained why the globular category of higher spans was the globular category of sets. Why is the bicategory structure on ordinary spans in Set intrinsic to the category of graphs? If we could understand this, we could try to intrinsically understand the monoidal structure (in Batanin's sense) on the globular category of higher spans. A graphical (new?) view of distributors (profunctors) was presented: but why is composition intrinsic?