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 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

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?"

There is of course also the pointwise product of matrices of the same size. (In the case of complex square matrices I believe this is known as the Schur or Schur-Hadamard product.) This product and the usual product of matrices do not satisfy the middle four interchange law, in the case of matrices over a commutative ring. However, there are examples in the study of bicategories where they do _ in a sense. Consider the bicategory of categories, profunctors and transformations, PRO_. For categories X,A the hom category PRO_(X,A) is given by SET_^(A^op x X) and note that it has sums which, with respect to PRO_, we will call "local" sums. PRO_ has sums in the usual sense which are as in CAT_. Call them global sums. Global sum injections have right adjoints and the process of taking right adjoints carries global sum diagrams to (global) product diagrams. So an arrow in PRO_ from a global sum to a global sum is a matrix of arrows (just as in any category in which sums and products coincide). A composite of such matrices is given by matrix multiplication where the component multiplications are composites in PRO_ and the component additions are provided by local sums. These considerations apply to transformations in PRO_ too so that horizontal composition of transformations is also given by matrix multiplication when the objects under consideration are global sums. But vertical composition of transformations is just pointwise vertical composition. So here there is a middle four interchange law between pointwise and ordinary matrix multiplication _ but note that the component multiplications are not the same. The component multiplications of ordinary matrix product are provided by horizontal composition _ which middle four interchanges with vertical composition. The Kronecker product can also be considered in this example. Global binary products for CAT_ extend to a tensor product for PRO_. This tensor distributes over global sum and the remarks of Ross on this question apply to this example. At least this is the story modulo a discussion of the coherent isomorphisms which are easily spotted. This example admits a number of generalizations and specializations. RJ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++