Characterisations of purity in a locally finitely presented additive category: A short functorial proof

Abstract

In this expository article, we will give an efficient functorial proof of the equivalence of various characterisations of purity in a finitely accessible additive category $\mathcal{C}$. The complications of the proofs for specific choices of $\mathcal{C}$ are contained in the description of fp-injective and injective objects in $(\mathrm{fp}\mathcal{C},\mathrm{Ab})$, the category of additive functors $\mathcal{C}\to \mathrm{Ab}$. For example, the equivalence of many characterisations of purity in a module category $\mathcal{A}-\mathrm{Mod}$ is a simple corollary of what we will prove here, since we know which objects are fp-injective, and which objects are injective, in $(\mathcal{A}-\mathrm{mod},\mathrm{Ab})$.