Cofibrant generation of pure monomorphisms in presheaf categories

Abstract

We characterise when the pure monomorphisms in a presheaf category ${\mathbf{Set}}^{\mathcal{C}}$ are cofibrantly generated in terms of the category $\mathcal{C}$. In particular, when $\mathcal{C}$ 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 $a,b\in S$ there is $c\in S$ 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.

26 pages