Units
November 10, 2015 by Qiaochu Yuan
Yesterday I wrote down a definition of an action of a group 
on a category 
that was slightly incorrect because I neglected to write down any conditions involving units. With the definition I gave it is possible for 
to fail to be an automorphism (it might instead be a nontrivial idempotent endofunctor 
).
The condition we need on units is first that we should have a unit isomorphism
and second that this unit isomorphism should be compatible with the isomorphisms 
in the sense that the composites
and
should both be the identity. If we use the unit isomorphism to replace with 
(which changes the 
), this is just the condition that 
and 
should both be the identity.
Similarly, in our definition of an equivalence 
between two group actions 
, needs to respect these unit isomorphisms in the sense that
Again, if we use the unit isomorphisms to replace 
with on the nose, this is just the condition that 
should be the identity.
Fortunately, in the special case we considered in the previous post, where 
vanishes (and perhaps in general), this produces the same classification of group actions as before, so nothing has gone too badly wrong. Details below the fold.
Concrete details
Recall that in the special case of group actions 
where each 
is the identity , the only information left is in the natural isomorphisms 
. Yesterday we only imposed the usual 2-cocycle condition on these, and saw that equivalences of group actions corresponded to 2-coboundaries. Today we want the additional unit conditions above. This produces a variant of 2-cocycles and 2-coboundaries which we’ll call unital. Explicitly, for 2-cocycles this means 
, and for 2-coboundaries it means we look at 2-coboundaries of the form 
where 
.
We want to show that these give rise to the same cohomology group 
, and we’ll do this very explicitly, by showing that any 2-cocycle is cohomologous to a unital 2-cocycle, and that two unital 2-cocycles are 2-cohomologous iff they are unitally 2-cohomologous. First, using the 2-cocycle condition
and substituting first 
and then 
, we get
and
from which it follows that 
and 
are both constant, and hence must both be equal to 
. Now we just need to modify this 2-cocycle by the 2-coboundary where 
and 
for 
; we get a new 2-cocycle 
satisfying
and we see that 
as desired. Hence every 2-cocycle is cohomologous to a unital 2-cocycle.
Next, suppose 
are two unital 2-cocycles which are cohomologous via a 2-coboundary 
, so that
.
Then setting gives
and hence 
must be a unital 2-coboundary.
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Annoying Precision 
So with that in mind, a group action on C is the same as a monodial functor $G \to EndoFun(C)$, right? (Where $G$ is the monodial category with $G$ as objects, only identities as morphisms and multiplication as tensor product)
Yes, that’s right, sorry for taking so long to reply!
[…] from this data agree. We’ll also want unit isomorphisms satisfying the same compatibility as before. This is just spelling out the definition of a 2-functor from the category of separable extensions […]
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