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Brendan Murphy (May 18 2024 at 17:21):Modules over (associative, unital, not necessarily commutative) rings form abelian categories, and abelian categories are tightly connected to module categories (via Freyd-Mitchell, Gabriel-Popescu, etc). We can tell a similar story for stable infinity categories and modules over E1 ring spectra. What sort of general class do semiring module categories fit into? This is a hazy question, but I hope it makes sense what I'm looking for. Probably this comes down to something like "what exactness properties are true of the category N-Mod of commutative monoids"
Jean-Baptiste Vienney (May 18 2024 at 19:01):Maybe interesting: Proto-exact categories of modules over semirings and hyperrings