k-linear category

Question

In the tasks I have it says

"Let $k$ be a field. Show that the structure of a $k$-linear category on a category $\mathcal{C}$ is equivalent to $\mathcal{C}$ being a module category (see Module category in nLab) over $\mathrm{vect_k}$, the vector spaces over $k$."

I have a few questions to that:

1) I googled and found that "$k$-linear" means that the category is enriched over $\mathrm{vect_k}$. I found the definition of enriched, but that's really abstract, could somebody maybe explain what enriched over $\mathrm{vect_k}$ explicitly means?

2) What exactly does $k$-linear category ON A CATEGORY $\mathcal{C}$ mean? I never found it like that..

3) For the proof he says as a hint "define $v.c$ for $v \in \mathrm{vect_k}$ and $c \in \mathcal{C}$ as the object representing the functor $v \otimes_k \mathrm{Hom}( - , c)$. Why do I have to identify an object like $v.c$ as a functor?

Any further hints on how to proof this are of course always good to see, but I am glad if someone can help me understand the question :D

Thanks in advance!

Answer 1

1. I assume you have found this definition of enriched category and we are talking of enrichment over a monoidal category.

Personally I believe that the best way to think about enriched categories is to consider them as categories whose $\hom$-functor, the compositions mappings, and the identities can be lifted respectively to a $\mathcal V$-valued functor and $\mathcal V$-valued mappings.

I could make this a little more formal but that could be long, so if you need additional details feel free to ask.

A little more concretely, for the case of the monoidal category $\mathbf{Vect}_k$ of $k$-vector spaces (having tensor product $(\otimes)$ as monoidal product, and $k$, the field, i.e. the unique up to isomorphism vector space of dimension 1, as monoidal unit) an enriched category amounts to a category $\mathbf C$ whose $\hom$-sets $\mathbf C[a,b]$ have an additonal (enriching) structure of $k$-linear vector spaces, in which the compositions $\circ_{a,b,c} \colon \mathbf C[b,c] \times \mathbf C[a,b] \to \mathbf C[a,c]$ are $k$-multilinear maps.

2. If you adopt this point of view, and see enriched categories as categories with additional structure (on the $\hom$-sets and on the compositions) then it should be pretty natural to call this additional structure the $k$-linear category structure over the (underlying) category considered (as we talk of the $k$-vector space structure over a set).

3. Since in the formula $v \otimes_k \text{Hom}(-,c)$ you are using the tensor product of $\mathbf{Vect}_k$ I am assuming you are proving that a $k$-linear category is a $\mathbf{Vect}_k$-module category.

Actually the key to understand your hint is in the phrase "the object representing the functor $v \otimes_k \text{Hom}(-,c)$", this implies that $v.c$ is not a functor but it is an object such that you have a natural isomorphism $$\text{Hom}(-,v.c) \cong v \otimes_k \text{Hom}(-,c)$$ of $\mathbf{Vect}_k$-valued functors.

I hope this helps, if not feel free to ask for additional details.

Answer 2

"Let $k$ be a field. Show that the structure of a $k$-linear category on a category $\mathcal{C}$ is equivalent to $\mathcal{C}$ being a module category (see Module category in nLab) over $\mathrm{vect_k}$, the vector spaces over $k$."

I don't think this is true. Consider the category where the objects are 7-dimensional vector spaces over the field $k$ and the morphisms are linear maps. This is a $k$-linear category: that is, it's enriched over $\mathrm{vect_k}$ - or equivalently, in simpler terms, the hom-sets are vector spaces over $k$ and composition is bilinear. But I don't think it's a module category over $\mathrm{vect_k}$, since you can't tensor a 7-dimensional vector space by an arbitrary vector space and get another 7-dimensional vector space in some way that obeys the axioms of a module category.