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)
There is a (skeletal) category Mat whose objects are finite ordinals, whose arrows n --> m are mxn matrices, and whose composition is matrix multiplication. Mat becomes a monoidal category with Kronecker product as its tensor product. Your equation expresses the functoriality of this tensor product Mat x Mat ---> Mat which is also strictly associative. The category Vect of finite dimensional vector spaces is equivalent to Mat and ordinary tensor product of vector spaces. Every monoidal category is equivalent to a strictly associative one (coherence theorem), and Mat is a concrete way of doing this for Vect. Category theorists call your equation "the middle-four-interchange-law" since it involves interchanging the middle two of a string of four terms. It is the axiom which makes 2-categories interesting. I made much use of all this, including Kronecker product, in my Myhill Lectures at Buffalo, 20-23 April 1993 [available as Macquarie University Math Report 93-130 (June 1993), submitted for publication]. Sincerely, Ross ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++