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.
Why do this?
There are several ways to justify this definition. The basic justification is that categories themselves naturally live in a 2-category: accordingly, their endomorphisms form monoidal categories, and we’ve written down above is precisely a (strong) monoidal functor from , regarded as a discrete monoidal category, to . Said another way, the automorphisms form not a group but a 2-group, and what we’ve written above is the definition of a morphism of 2-groups.
If we’d gone with the more naive definition, we would’ve ended up with a notion of groups acting on categories which wouldn’t have transported across equivalence of categories.
Where do 2-cocycles come into this?
Suppose that every is the identity functor . (Abstractly, this is a natural thing to do if every automorphism is equivalent to the identity; then we can replace each with the identity while changing the appropriately.) Then each is a natural isomorphism , or equivalently an element of the group of units of the center , and the compatibility condition becomes
Hence, in this special case, the precisely describe a 2-cocycle with coefficients in .
What about cohomology?
The point of 2-cocycles is that they can be used to describe second group cohomology; where does that come into the picture?
Abstractly, second group cohomology can be described as homotopy classes of maps , where is the classifying space of and is the classifying space of the classifying space of , or equivalently the Eilenberg-MacLane space . These are the same as homotopy classes of maps of 2-groups (by the homotopy hypothesis), and if the 2-group is connected in the sense that (which, as above, is precisely the condition that every automorphism is equivalent to the identity), then
as 2-groups, and hence in this special case actions of on are classified by group cohomology with coefficient in the group of units of the center of . In general, maps of 2-groups can be thought of as “nonabelian group cohomology,” with coefficients in a 2-group.
Concretely, let’s write down what it ought to mean for two group actions to be equivalent. For starters, we should probably ask for a family of natural isomorphisms
and next we should ask for some compatibility between these isomorphisms and the “2-cocycles” . The natural compatibility to ask for is that the composites
are equal. In the special case where all of the are equal to the identity , the isomorphisms again take values in , and the compatibility above becomes
which, with a little rearranging (using the fact that is abelian), becomes
This means precisely that the 2-cocycles and differ by a 2-coboundary in the usual sense.
Okay, but do you have any interesting examples?
Let be the category of vector spaces over a field . Every automorphism of as a -linear category turns out to be equivalent to the identity. (Without the -linearity condition, any automorphism of induces an automorphism of .) The center is itself, so its group of units is . Hence the set of isomorphism classes of actions of a group on can be identified with the cohomology group
Puzzle 1: This cohomology group also appears when classifying projective representations of over . Is this a coincidence?
Puzzle 2: In terms of the description above as the set of isomorphism classes of actions, where does the group structure on this set come from, and why is it abelian?