Group action on a category Motivating example: 
We get a functor from the category of real vector spaces to the category of complex vector spaces by complexifying (i.e. tensoring over $\mathbb{R}$ with $\mathbb{C}$). Let $\sigma$ denote complex conjugation. This induces a functor, $\sigma^*$, from the category of complex vector spaces into itself, also by tensoring with $\mathbb{C}$, but over $\mathbb{C}$ and via $\sigma$. The image of this can then be identified with the pairs $(V,f)$ of complex vector spaces $V$ together with a morphism $\sigma^*V\rightarrow V$, s.t. $f\circ \sigma^*f=\operatorname{id}_V$ where we secretely identify $(\sigma^*)^2V$ with $V$.
Question: 
What would be the good notion of a group acting on a category to say in the above example that the fixed points of that action are precisely the real vector spaces? 
Remarks:


*

*In particular this notion should also work for an arbitrary galois
field extension $L|K$ and the group $Gal(L|K)$ acting on the category
of $L$ vector spaces (Or a similar category which can be constructed
from it)

*The question came up, when I was reading about some abstract aspects of Hodge theory, where one sometimes considers (the category of) "complex Hodge structures", has $Gal(\mathbb{C}|\mathbb{R})$ and wants to identify the real Hodge structures as fixed points under a $Gal(\mathbb{C}|\mathbb{R})$. This can be done similarly as in the motivating example but I was wondering if there was a general framework for doing this.
 A: There are two possibilities. One is the "naïve" or "strict" notion. A strict $G$-action on a category $\mathcal{C}$ is a group homomorphism $G \to \mathrm{Aut} (\mathcal{C})$, where $\mathrm{Aut} (\mathcal{C})$ is the group of automorphisms of $\mathcal{C}$. The fixed points of a strict $G$-action on $\mathcal{C}$ are defined in the obvious way and form a subcategory. 
The better notion is this. A pseudo $G$-action on a category $\mathcal{C}$ consists of the following data:


*

*For each element $g \in G$, a functor $g_* : \mathcal{C} \to \mathcal{C}$.

*For the unit element $e$ of $G$, an isomorphism $\eta : \mathrm{id}_\mathcal{C} \Rightarrow e_*$.

*For each pair $(g, h)$ of elements of $G$, an isomorphism $\mu_{g, h} : g_* \circ h_* \Rightarrow (g h)_*$.


These isomorphisms are subject to coherence conditions:
\begin{align}
\mu_{g, e} \bullet (\mathrm{id}_{g_*} \circ \eta) & = \mathrm{id}_{g_*} \\
\mu_{e, g} \bullet (\eta \circ \mathrm{id}_{g_*}) & = \mathrm{id}_{g_*} \\
\mu_{g, h k} \bullet (\mathrm{id}_{g_*} \circ \mu_{h, k}) & = \mu_{g h, k} \bullet (\mu_{g, h} \circ \mathrm{id}_{k_*})
\end{align}
Given a pseudo $G$-action on $\mathcal{C}$, a fixed point is an object $X$ in $\mathcal{C}$ together with an isomorphism $\lambda_g : X \to g_* X$ for every element $g \in G$, such that $g_* (\lambda_h) \circ \lambda_h = \lambda_{g h}$ for every pair $(g, h)$ of elements of $G$. There is an obvious notion of morphism of fixed points, and we get a category of fixed points. 
Given a finite Galois field extension $K \hookrightarrow L$, the category of $L$-vector spaces has an obvious strict $\mathrm{Gal} (L | K)$-action. Unfortunately, the fixed points of this strict action are the zero-dimensional $L$-vector spaces. On the other hand, any strict action is also a pseudo action, and you can show that the category of fixed points of the pseudo action is equivalent to the category of $K$-vector spaces in a canonical way. 
Incidentally, the above is an example of faithfully flat descent.
