$H$-objects of a category $\mathcal{A}$ as $H$-fixed points of a $G$-category

Question

This question is pretty self contained in the title. Let $G$ be a group regarded as a category with a single object and consider a functor $F:G \to Cat$. This exactly the data of a $G$-category that I will denote by $\mathcal{C}$. This has an obvious notion of $H$-fixed points category, $\mathcal{C}^H$ for $H \leq G$.

I would like to know if given a category $\mathcal{A}$ one could find a $G$-category $\mathcal{C}$ such that $\mathcal{C}^H=Fun(H,\mathcal{A})$ for every subgroup $H \leq G$.

Answer 1

Yes, take $C = A$ with the trivial action of $G$. I will write $BG$ for the one-object category with automorphisms $G$; this is really a different object from $G$ and really should be indicated with different notation. Then we have

$$BG \cong \text{pt}/G$$

meaning that $G$ is the homotopy quotient of a point by the trivial action of $G$, and hence

$$[BH, A] \cong [\text{pt}/H, A] \cong [\text{pt}, A]^H \cong A^H \cong C^H$$

by the universal property of homotopy quotients. More explicitly, you can just verify in detail that if the action of $H$ is trivial then a homotopy fixed point for the action of $H$ is precisely an object together with an action of $H$.