My understanding is that a field $(k, +, \times)$ is a set $k$ with abelian group structures $(k, 0, +)$ and $(k \setminus \{0\}, 1, \times)$ such that multiplication distributes over addition.
Can this be generalized to the categorical setting? For example, if we adjoin to some category $\mathcal{C}$ a coproduct and product structures $(\mathcal{C}, \mathbb{0}, +)$ and $(\mathcal{C}, \mathbb{1}, \times)$ such that products distribute over coproducts, is this an analogue of a field?
A problem I see is that multiplication with 0 doesn't have an inverse, which I'm not sure how to talk about in a categorical setting.
TLDR; How can I think of fields in terms of category theory?