Question: Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of short exact sequence. Can we use this to define ext groups which classify extensions? What works and what doesn't work and why?

Answer: 
Jaret Flores's PhD thesis,

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> Jaret Flores, *Homological Algebra for Commutative Monoids*, [arXiv:1503.02309](https://arxiv.org/abs/1503.02309).
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treats precisely this topic.
I might be misunderstanding your question - are you asking about a (co)homology theory for commutative monoids, or about trying to do homological constructions in the category of commutative monoids?

In the former case, I think Grillet has some work on this, see e.g.

Grillet, Pierre-Antoine(1-TULN)
[Commutative semigroup cohomology](http://www.ams.org/mathscinet-getitem?mr=1348250). (English summary)
Comm. Algebra 23 (1995), no. 10, 3573--3587. 

The idea is basically what Mikael describes, although I only know the more primitive version which is cotriple/comonad cohomology as set up by Barr and [Beck](http://www.tac.mta.ca/tac/reprints/articles/2/tr2abs.html). Presumably the problem with trying to apply that set up is ensuring there are enough abelian group objects to use as coefficient modules.