Restate Galois representation from category theory

Question

Given an arbitrary category $C$, a representation of $G$ that is a category with a single object in $C$ is a functor from $G$ to $C$. A group representation is a representation of $G$ in the category of vector spaces over a field $F$. Then we can restate much theorem of group representation theory: For example maschke's theorem can be states that when the characteristic of $F$ doesn't divide the order of $G$, the category of representations of $G$ over $F$ is semisimple. My question is: Galois representation, p-adic galois representation, p-adic hodge theory and so on can be restate in category theory? I'm sorry if I'm speaking out of turn due to lack of knowledge. Thanks in advance!