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Chetan Vuppulury (Dec 01 2022 at 13:19):I was wondering if the category of sheaves of commutative monoids over any site could be embedded fully faithfully and exactly into the category of modules over some commutative rig?
Jean-Baptiste Vienney (Dec 01 2022 at 13:50):I don't know a lot about sheaves and sites but Chris Heunen is speaking about generalizing the Freyd-Mitchell theorem by replacing rings by rigs in this paper: Semimodule enrichment
I quote the first phrase of the introduction:
"Mitchell’s celebrated theorem states that every Abelian category can be embedded in the category of modules over a ring [18]. This article is a first part of a generalisation from rings to semiring, which could hopefully give a representation theorem for semantic models of linear (quantum) computation."
I'm not sure if it's useful in your context but I let you see.
Jean-Baptiste Vienney (Dec 01 2022 at 13:54):Maybe we can try this: @Chris Heunen
Jean-Baptiste Vienney (Dec 01 2022 at 15:50):But I guess that no theorem like this has yet been proved. The maths without negatives are today far from the state of development of the maths with negatives.
Fernando Yamauti (Dec 01 2022 at 18:42):@Chetan Vuppulury I'm not familiar with rigs, but for every bounded topoi, the category of internal suplattices is equivalent to the category of modules over the quantale of relations on the respective bound.