Categories enriched over a field

Question

Let $K$ be a field, regarded as a monoidal category in the following way:

1. Objects are elements $x\in K$;
2. Morphisms between $x\to y$ are only identities;
3. The (strictly symmetric) monoidal structure $x\otimes y$ is given by the product $x\cdot y $ (it is not closed because of $0$)

Is there a special name for categories enriched over $K$?

Whatever these are, they have to fulfill quite strange properties: let $V\in K\text{-Cat}$, then $\hom(v,w)\otimes \hom(w,z) = \hom(v,w)$, which taking into account the fact that each $\hom(v,w)$ is a scalar in $K$, implies that $\hom(v,w)\hom(w,z) = \hom(v,z)$, i.e. $\hom(v,v)=1$ and $\hom(v,w) = \hom(w,v)^{-1}$.

I started wondering if a $K$-vector space could be regarded as a $K$-category, but this seems to be false. I thought that $K$ was a $K$-category, but it seems that if it is, then $\hom(x,y)=x^{-1}y$, impossible if $x=0$.

Answer 1

You make no use of the additive structure, so let's instead talk about categories enriched over monoids $M$. As you observe, in such an enrichment only the invertible elements of $M$ arise, so let's instead talk about categories enriched over groups $G$. Then my claim is that

a category enriched over $G$ is the same thing as a set on which $G$ acts freely and transitively, or in other words a $G$-torsor.

The point is that an equivalent definition of a $G$-torsor is that it is a $G$-set $S$ in which, for every $x, y \in S$, there is a unique $g \in G$ such that $gx = y$. Given this unique $g$ there is a unique enrichment of $S$ over $G$ which sets $\text{Hom}(x, y) = g$. Conversely, given a $G$-enriched category, if $\text{Hom}(x, y) = g$ then define $gx = y$. This definition is independent of the choice of $y$ precisely because composition is compatible with the group operation in $G$.

So, unfortunately, categories enriched over monoids are not too interesting. It's much more interesting to enrich categories over monoids equipped with poset structures; these give you generalizations of (Lawvere) metric spaces.

In general, $V$-enriched categories may be thought of as at least analogous to $V$-module categories. You can try to pass from one to the other if certain adjunctions exist.