What is classified by cohomology?

What is classified by cohomology?

[Hasse Riemann]

 

Hi all categorists
 
Here is another questions i think about and need your help with.
 
3>
What does the cohomology H^n(X;coefficients) classify, for X a more general object then a group and especially when X is a category?
I know that the case X=group gives n-torsors.

And how come that the classification is independant of the coefficients?
 
Best regards
Rafael Borowiecki

Re: What is classified by cohomology?

[Fernando Muro]

Dear Hasse,

> What does the cohomology H^n(X;coefficients) classify, for X a more general
> object then a group and especially when X is a category?
> I know that the case X=group gives n-torsors.

Group cohomology also classifies crossed extensions:

Huebschmann, Johannes
Crossed $n$-fold extensions of groups and cohomology.
Comment. Math. Helv. 55 (1980), no. 2, 302--313. 

There is a similar approach for cohomology of categories in:

Baues, Hans-Joachim(D-MPI); Minian, Elias Gabriel(D-MPI)
Track extensions of categories and cohomology. (English summary)
$K$-Theory 23 (2001), no. 1, 1--13. 

The cases n = 0, 1, 2, 3 were known before, see references therein.

Best,

Fernando

-- 
Fernando Muro
Universitat de Barcelona, Departament d'Àlgebra i Geometria
http://atlas.mat.ub.es/personals/muro/