TITLE: Homological algebra for commutative monoids?
QUESTION [26 upvotes]: 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?

REPLY [5 votes]: Jaret Flores's PhD thesis,

Jaret Flores, Homological Algebra for Commutative Monoids, arXiv:1503.02309.

treats precisely this topic.