Pure monomorphisms in Set^C are cofibrantly generated iff C satisfies a combinatorial condition equivalent to the existence of a stable independence relation, with the monoid case reducing to existence of c linking any a and b.
Pr¨ ufer, Untersuchungen ¨ uber die Zerlegbarkeit der abz¨ ahlbaren p rim¨ aren Abelschen Gruppen, Mathematische Zeitschrift 17 (1923), 35–61
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Cofibrant generation of pure monomorphisms in presheaf categories
Pure monomorphisms in Set^C are cofibrantly generated iff C satisfies a combinatorial condition equivalent to the existence of a stable independence relation, with the monoid case reducing to existence of c linking any a and b.