Coproducts and products are the same

Question

I was playing aroudn with finite dimensional vector spaces, and I found that coproducts are the same thing in $\textbf{FinDiVec}$.

What other categories exist where this is the case, and what are some criteria for thist to be true.

Answer 1

I wrote a blog post, Meditation on semiadditive categories, which answers this question in detail. The short answer:

In a category with both finite products and finite coproducts, finite products and coproducts coincide (it is a bit trickier than you might think to say precisely what this means; there's an obvious way of doing it which is wrong, and the correct way of doing it requires handling the 0-ary case separately and first) if and only if the category is enriched over commutative monoids.

Most examples in practice are enriched over abelian groups, and most examples that people care about in practice are abelian categories. This includes examples like vector spaces, modules over a ring, sheaves of abelian groups on topological spaces, quasicoherent sheaves on schemes, etc.