Yesterday we decided that it might be interesting to describe various categories as “fixed points” of Galois actions on various other categories, whatever that means: for example, perhaps real Lie algebras are the “fixed points” of a Galois action on complex Lie algebras. To formalize this we need a notion of group actions on categories and fixed points of such group actions.

So let be a group and be a category. For starters, we should probably ask for a functor for each . Next, we might naively ask for an equality of functors

but this is too strict: functors themselves live in a category (of functors and natural transformations), and so we should instead ask for natural isomorphisms

.

These natural isomorphisms should further satisfy the following compatibility condition: there are two ways to use them to write down an isomorphism , and these should agree. More explicitly, the composite

should be equal to the composite

.

(There’s also some stuff going on with units which I believe we can ignore here. I think we can just require that on the nose and nothing will go too horribly wrong.)

These natural isomorphisms can be regarded as a natural generalization of 2-cocycles, and the condition above as a natural generalization of a cocycle condition. Below the fold we’ll describe this and other aspects of this definition in more detail, and we’ll end with two puzzles about the relationship between this story and group cohomology.