A for $k$ a field, a vector space over $k$ is module over the ring$k$. Sometimes a vector space over $k$ is called a $k$-linear space. (Compare ‘$k$-linear map’.)

The category of vector spaces is typically denoted Vect, or $Vect_k$ if we wish to make the field $k$ explicit. So

$Vect_k \coloneqq k Mod
\,.$

This category has vector spaces over $k$ as objects, and $k$-linear maps between these as morphisms.