Cool stuff! I had previously seen that Vect has a biproduct structure, but never even considered that matrices could be appropriately generalized using it.
One thought I had was about handling potentially infinite-dimensional vector spaces. In that case, if V has a basis {vi:i∈I}, we still get a decomposition into a direct sum of copies of R:
V≅⨁i∈IR,
However, when the indexing set I is infinite, the coproducts and products in Vect differ, so the argument from the post doesn't go through verbatim. But it still... (read more)
Cool stuff! I had previously seen that Vect has a biproduct structure, but never even considered that matrices could be appropriately generalized using it.
One thought I had was about handling potentially infinite-dimensional vector spaces. In that case, if V has a basis {vi:i∈I}, we still get a decomposition into a direct sum of copies of R:
V≅⨁i∈IR,However, when the indexing set I is infinite, the coproducts and products in Vect differ, so the argument from the post doesn't go through verbatim. But it still... (read more)