One defines the $H^n(G,M)$ where $M$ is a $\mathbb{Z}[G]$ module as $Ext^n_{\mathbb{Z}[G]}(\mathbb{Z},M)$ where $\mathbb{Z}$ is viewed as a trivial $\mathbb{Z}[G]$-module.

Is this part of a general pattern for how to define cohomology for non-abelian categories?

In groups we see that we switch to the abelian category of $\mathbb{Z}[G]$-modules. If this idea is indeed extended for general non-abelian categories, what abelian category do we switch to?

  • When you say nonabelian categories, do you mean categories that are not abelian (say, Barr exact) or semiabelian (which include things like Grp, LieAlg)? For the latter see and references there. The people working on semiabelian categories have developed quite a bit of homological algebra in that setting. – David Roberts Oct 30 '10 at 22:31
up vote 6 down vote accepted

Yes, this generalizes to Hochschild cohomology and André-Quillen cohomology.

Given a category $\mathcal{C}$ with finite limits and an object $X$, you can form the category of Beck modules $\operatorname{Ab}(\mathcal{C} / X)$, abelian group objects in the slice category over $X$. When $\mathcal{C}$ is the category of groups, a Beck module over $G$ is precisely a split extension of $G$ with abelian kernel, so Beck modules can be identified with $\mathbb{Z}[G]$-modules.

In nice cases, the category $\operatorname{Ab}(\mathcal{C} / X)$ is abelian, and the forgetful functor $\operatorname{Ab}(\mathcal{C} / X) \to \mathcal{C} / X$ has a left adjoint, called abelianization $\operatorname{Ab}_X$. The Hochschild cohomology of a Beck module $M$ is defined to be $HH^i(X; M) = \operatorname{Ext}^i(\operatorname{Ab}_X X, M)$. In the groups case, $\operatorname{Ab}_G G$ is the augmentation ideal, so Hochschild cohomology is just group cohomology shifted by 1.

Hochschild cohomology itself is only an approximation to André-Quillen cohomology, which is somehow a more homotopically correct notion (I don't know enough about any of this to explain what this means, exactly). In the case of groups, these two cohomology theories coincide. I suggest having a look at Martin Frankland's thesis, which gives a good overview of these things. It works out explicit examples (groups, abelian groups, associative algebras, commutative algebras) very nicely in Appendix A.

  • Thanks a lot! It's very nice to know. Here's a silly question: does this specialize to the regular cohomology if we're working with an a priori abelian category? – Makhalan Duff Oct 30 '10 at 19:08
  • 1
    If by "regular cohomology" you mean "Ext," then I believe the answer is yes: the category of Beck modules over X is the original category, and the cohomology theories become Ext(X, -). At least this is the case in the category of abelian groups according to section 3.2 of Frankland's thesis, and I believe the same argument holds in any abelian category. – Evan Jenkins Oct 31 '10 at 23:47

Your Answer


By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.