PlanetPhysics/Grothendieck Category Lemma

Introduction: proper generator

Let us recall that a generator of a Grothendieck category ${\mathcal {G}}$ is called proper if $U$ has the property that a monomorphism $i:U'\to U$ induces an isomorphism $Hom_{\mathcal {G}}(U,U)\cong Hom_{\mathcal {G}}(U',U)$ if and only if $i$ is an isomorphism (viz. p. 251 in ref. ).

Grothendieck category lemma

\begin{lemma} Any Grothendieck category ${\mathcal {G}}$ has a proper generator. \end{lemma}