2025/06/25 by Cox, Sean, Feigert, Jonathan, Kamsma, Mark +2 · 1 citation
#03C48 #03C60 #18C05 #18C35 #20M30 (Secondary) #20M50 (Primary) #Category Theory (math.CT) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2506.20278
We characterise when the pure monomorphisms in a presheaf category SetC are cofibrantly generated in terms of the category C. In particular, when C is a monoid S this characterises cofibrant generation of pure monomorphisms between sets with an S-action in terms of S: this happens if and only if for all a, b ∈ S there is c ∈ S such that a = cb or ca = b. We give a model-theoretic proof: we prove that our characterisation is equivalent to having a stable independence relation, which in turn is equivalent to cofibrant generation. As a corollary, we show that pure monomorphisms in acts over the multiplicative monoid of natural numbers are not cofibrantly generated.