Category of vector spaces and matrices

Question

I was brushing up on linear algebra and the following came to the mind.

Consider a category whose nodes are n-dimensional vector spaces (n>0). Morphisms are matrices (transformations between vector spaces). This construction seems to be forming a category:

• let identities to be identity matrices.
• let composition to be composition of matrices.

Composition is associative and identity rules of category are satisfied, so we have a category.

Questions:

• Any reference studied this category? (any name associated with this category at all in the literature?)
• Is matrix addition associated with any categorical construction in this category?
• What do product, co-product, terminal, and initial objects mean in this category?

Answer 1

The category you describe is well-known. Given a field $k$, the category Vect$_k$ has $k$-vector spaces as its objects and linear maps as its morphisms. Off the top of my head, I would say the trivial vector space is both an initial and terminal object. For more information see here https://ncatlab.org/nlab/show/Vect and here Coproducts and products in the categories of sets, groups and vector spaces.

Answer 2

This is a special case of an abelian category, a category where you can consider things like kernels and cokernels. The category K-Vec that you want to consider can be more generally seen as the category of R-Mod, which is the category with objects as the $R$-modules for an abelian ring $R$ and morphism the homomorphisms between modules, see also here.

Answer 3

Let's look at vector spaces over the field $k$. If you want the morphisms to really be matrices, the vector spaces you're looking at should have ordered bases, and should be finite dimensional.

One way of doing this is to restrict the objects to just $k^n$, for natural numbers $n$ ($n\ge 0$ or $n\ge 1$ depending on whether you think the map when either the target or domain is $0$ is represented by a matrix). That is, of all the (isomorphic) vector spaces of finite dimension $n$, you pick out $k^n$, which comes with a canonical basis. This is exactly the concept of a skeleton category (ncatlab, wikipedia), applied to the category $\mathbf{FinVect}_k$ (ncatlab) of finite dimensional vector spaces over $k$.

Another way is to change the objects from vector spaces to "(finite dimensional) vector spaces with ordered bases". This has a forgetful functor to $\mathbf{FinVect}_k$ by forgetting the basis. I don't know that this category has a standard name, since it is so closely related to $\mathbf{FinVect}_k$ and $\mathbf{Vect}_k$. It also has a skeleton $\{(k^n, (e_1, \dots, e_n))\}$, which is isomorphic to the skeleton of $\mathbf{FinVect}_k$ under the forgetful functor.