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?



Homological algebra for commutative monoids?
Your question can be understood as how to do Homological Algebra over the Field with one Element.


Deitmar, in <http://arxiv.org/abs/math/0608179> , section 6, gives an example of what can go wrong if you try to do sheaf cohomology directly via resolutions...


You might also want to look at his <http://arxiv.org/abs/math/0605429> ; in order to construct K-theory of monoids he sets up an analogue of the Q-construction. The Hom-sets in the resulting category are sort of Exts, maybe something to start with...


Durov, in <http://arxiv.org/abs/0704.2030> , follows the simplicial approach for commutative monads, of which commutative monoids are a special case


