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 $k$ (the ground field) explicit. So

$Vect_k \coloneqq k Mod
\,.$

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

Multisorted notion

Alternatively, one sometimes defines “vector space” as a two-sorted notion; taking the field $k$ as one of the sorts and a module over $k$ as the other. More generally, the notion of “module” can also be considered as two-sorted, involving a ring and a module over that ring.

This is occasionally convenient; for example, one may define the notion of topological vector space or topological module as an internalization in $Top$ of the multisorted notion. This procedure is entirely straightforward for topological modules, as the notion of module can be given by a two-sorted Lawvere theory $T$, whence a topological module (for instance) is just a product-preserving functor $T \to Top$. One may then define a topological vector space as a topological module whose underlying (discretized) ring sort is a field.