The Category of Matrices III
Now here’s the really important thing: There’s a functor that assigns the finite-dimensional vector space of -tuples of elements of to each object of . Such a vector space of -tuples comes with the basis , where the vector has a in the th place and a elsewhere. In matrix notation:
and so on. We can write (remember the summation convention), so the vector components of the basis vectors are given by the Kronecker delta. We will think of other vectors as column vectors.
Given a matrix we clearly see a linear transformation from to . Given a column vector with components (where the index satisfies ), we construct the column vector (here ). But we’ve already established that matrix multiplication represents composition of linear transformations. Further, it’s straightforward to see that the linear transformation corresponding to a matrix is the identity on (depending on the range of the indices on the Kronecker delta). This establishes that we really have defined a functor.
But wait, there’s more! The functor is linear over , so it’s a functor enriched over . The Kronecker product of matrices corresponds to the monoidal product of linear transformations, so the functor is monoidal, too. Following the definitions, we can even find that our functor preserves duals.
So we’ve got a functor from our category of matrices to the category of finite-dimensional vector spaces, and it preserves all of the relevant structure.