Cofibrant generation of pure monomorphisms in presheaf categories

TL;DR

This paper characterizes when pure monomorphisms in presheaf categories are cofibrantly generated, linking this property to the structure of the underlying category and providing a model-theoretic proof.

Contribution

It provides a new characterization of cofibrant generation of pure monomorphisms in presheaf categories based on the structure of the category and model-theoretic methods.

Findings

Pure monomorphisms in presheaf categories are cofibrantly generated under specific conditions.

For monoids, cofibrant generation relates to the existence of certain elements satisfying algebraic conditions.

Pure monomorphisms in acts over natural numbers' multiplicative monoid are not cofibrantly generated.

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.