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Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
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Accepted
Factoring through projective modules is an equivalence relation
This equivalence relation is just a quotient of abelian groups. It will be cleaner to show that the $\mathrm{PHom}_R(M, N)$ is a subgroup of $\mathrm{Hom}_R(M,N).$ Indeed, if $f:M\to P_1 \to N$, $g: …