Lewis Stiller writes: 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. ---------- I am just learning what some of the people on this group invented, so only the enthusiasm of a novice can justify my attempt to explain this. Briefly, this "exchange identity" (or maybe it's called "interchange") is indeed symptomatic of a very general phenomenon in category theory, and the analogy with horizontal and vertical compositions is NOT coincidental. A 2-category is, roughly, a category in which homsets are categories and composition is a (bi)functor. One always has an exchange identity in such a structure. The category of categories is a 2-category, indeed the primordial one. A monoidal category (a category with tensor products satisfying nice axioms) can be viewed as a 2-category in such a way that your exchange identity becomes a special case. Another nice example is the 2-category in which objects are maps from a point into a topological space, morphisms are maps from a unit interval, and morphisms-between-morphisms (so-called 2-morphisms) are maps from a square. Well, here associativity does not hold "on the nose" but only "up to reparametrization" so we are touching upon the great puzzle of "weak n-categories". In any event, this situation provides a very nice geometrical interpretation of the exchange identity. jb ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++