What's the meaning of defining a functor in a natural way?

Question

I have a question: Given a group $\mathbf{G}$, there is homomorphism $\rho$ $\colon$ $\mathbf{G}$ $\to$ $\mathbf{GL(V)}$. BTW, $\rho$ is a representation of a group $\mathbf{G}$ on a vector space.

Now the task is to define a functor $F$ $\colon$ $\mathbf{G}$ $\to$ $\mathbf{Vec}_K$ in a natural way.

The second part is a converse way, that is, given a functor $F$ $\colon$ $\mathbf{G}$ $\to$ $\mathbf{Vec}_K$, define a representation of $\mathbf{G}$ on a vector space in a natural way.

Can anyone give me ideas?

Answer 1

It seems that what you are after is the correspondence between representations of $G$ and functors from the groupoid G to the category of vector spaces over $k$. (By "the groupoid $G$" I mean the category with one object "$\bullet$", and $\mathrm{Hom}(\bullet,\bullet) = G$.)

To define a representation $\rho:G\to \mathrm{GL}_k(V)$, you need the following data:

1. a vector space $V$;
2. for each $g\in G$, a linear map $\rho(g):V\to V$ (subject to the relevant compatibility conditions making $\rho$ a group homomorphism).

To define a functor $F:G\to\mathrm{Vec}(k)$, you need:

1. a vector space $F(\bullet)$;
2. for each $g\in\mathrm{Hom}(\bullet,\bullet) = G$, a linear map $F(g): F(\bullet)\to F(\bullet)$ (subject to the relevant compatibility conditions making $F$ a functor).

What would be the natural thing to do?