I'd like to modify it. I got too clever, forgetting that commuting with sums doesn't need any additional naturality.
Date: Mon, 29 Nov 93 09:52:49 EST From: stiller@blaze.cs.jhu.edu
Is there a categorical analogue of the following well-known matrix identity: If A, B, C, D are matrices over a field and juxtaposition denotes matrix multiplication and \otimes denotes Kronecker product, then
(A\otimes B)(C\otimes D)=(AC)\otimes (BD)
whenever the dimensions are consistent?
Note: I am a category theory novice, but the proof of this identity uses so few of the field axioms, and is so useful, (and curiously looks like the interchange of horizontal and vertical compositions of natural transformations which I presume is coincidental) that I was just curious if an analogue of this identity is true in more general categories than matrices over rings. -- Lewis Stiller. Dept. of Computer Science. The Johns Hopkins University. stiller@cs.jhu.edu. "Tertan I am, but what is Tertan? Of this time, of that place, of some parentage, what does it matter?"
The stated identity is obviously true in any additive category with a monoidal structure, provided the monoidal structure commutes with the finite sums. This is guaranteed if the monoidal structure is part of a closed monoidal structure. Just a consequence of the functoriality of the finite sums. Michael