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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!

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  • $\begingroup$ Btw: it could be useful if you could provide the references you are using. $\endgroup$ – Giorgio Mossa Feb 11 at 18:42
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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 $(0)$, the zero-vector space, 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.

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  • $\begingroup$ Hey, thanks for the answer! I understood 1) and 2) - thanks for that! in 3) I understood what you wrote, its kind of that there is such a $c' \in \mathcal{C}$ s.t.h $(\mathrm{Hom}(-, c') \cong v \otimes \mathrm{Hom}(-, c)$. and then I can define $v.c$ to be $c'$. Right? I have given that a lot of thought the last hours, but unfortunately I have absolutely no idea, how to find such a $c'$. Do you have any idea there? $\endgroup$ – P. Schulze Feb 12 at 12:57
  • $\begingroup$ Unfortunately no. It is not clear to me why such a functor should be representable. Is there any additional hypothesis you forgot in the question? Alternatively could you provide the references you have on the subject? (where does the question come from? from which book or article are you getting the definitions?) $\endgroup$ – Giorgio Mossa Feb 12 at 16:22
  • $\begingroup$ I can start the same way, you did: Unfortunately no. Or at least not all of them. The questions are from a handwritten-collection of exercise I got from my professor. Some of the definitions in there are taken from arxiv.org/abs/math/0111139, "k-linear" and "representable" aren't explained at all, it seams like part of the exercises was to do some research on those terms. The only thing I can add is the next question (as they sometimes correlate) where it says: $\endgroup$ – P. Schulze Feb 12 at 17:15
  • $\begingroup$ Show: k-linear functor $F: \mathcal{C} \rightarrow \mathcal{C'}$ with $\mathcal{C}, \mathcal{C'}$ $k$-linear amounts t a $\mathbb{C}$-module functor $\mathcal{C} \rightarrow \mathcal{C'}$ ... but I don't think this I really helpful $\endgroup$ – P. Schulze Feb 12 at 17:15
  • $\begingroup$ A possible way to proceed could be to find a universal element for the $\mathcal V$-functor $v \otimes_k \text{Hom}(-,c)$ but I am not sure what such an element could be. $\endgroup$ – Giorgio Mossa Feb 13 at 10:53

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