Dan Piponi wrote:
But I was mainly thinking about how the physicist's definition of tensor needn't be accepted as an irreducible given, but is a consequence of the definition of tensor product through its universal property: http://planetmath.org/encyclopedia/TensorProduct.html
Having said that, I still completely agree with Michael that tensors are a great analogy for monads because I found the concept of a universal property tricky in the same way that I subsequently found monads tricky. BTW I think the concept of a universal property is probably the single most useful idea from category theory that can be used in Haskell programming. I recommend it to everyone :-)
If only my Linear Algebra professor had just uttered the magic words: "For all R-modules M, the functor (-) * M is left-adjoint to the functor Hom(M,-)" I could have just skipped out on the entire semester. Why are teachers always so long-winded? The above is so much clearer! That the above is "the single most useful idea from category theory that can be used in Haskell programming" is so obvious, it belongs in a tutorial titled "You too could have invented Universal Algebra and Category Theory". I nominate Dan Piponi to write it and eagerly await its release! Dan Weston