Jason (jcreed) wrote,
Jason
jcreed

I don't have access to the 1980 paper that allegedly proves that there are only two symmetric monoidal closed structures on Cat, so I tried a baby version of the problem, reconstructing the proof on Grph. It's tougher than I thought to get all the way there, but I see a few inroads. It's impressive how strong the assumption is latent in the word "closed". To assume, as is a consequence of assuming monoidal-closedness, that A ⊗ — preserves colimits gives you a lot!

For every object in Grph can be gotten by taking some suitable colimit of the one-vertex graph 1 and the one-edge graph 2. This means that I really only need to understand
11, 12, 22
(and perhaps a few graph homomorphisms between them) to understand the action of ⊗ on the whole category.

Also, I know ⊗ has an identity object, I. Can I be the empty graph? No, because of preservation of colimits, G ⊗ 0 = 0, so that invalidates I being the identity if ever G is not zero. So now consider what I could possibly be. Whatever it is, it's the colimit of a bunch of copies of 1 and 2. And observe that 1 ⊗ I = 1 doesn't have any arrows in it. But again since ⊗ preserves colimits, 1 ⊗ I has to be the colimit of a bunch of copies of 11 and 12. Weeelll, maybe there are no copies of 12 in there, since we haven't shown that I has any arrows, but at least there's a copy of 11. Since colimits never get rid of arrows, only identify them with other arrows, we conclude that
a) 11 has no arrows in it.
b) Either 12 has no arrows, or I has no arrows (which would cause copies of 12 to show up in the colimit that's equal to 1 ⊗ I = 1)

Consider the case where I has no arrows. It's a collection of n vertices. But 1 = I ⊗ 1 = n copies of 1, so n = 1, and 1 = I. Therefore also we know the value of 12: it's 2. Down this road is the non-cartesian tensor product that I described yesterday. All that remains is to somehow prove that 22 is a four-arrow square.

The case where 12 has no arrows ought to lead inexorably to the cartesian product, but I haven't made any further progress on that front.
Tags: categories, math, monoidal
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