PlanetPhysics/Proper Generator in a Grothendieck Category

Introduction: family of generators and generator of a category

Let $𝒞$ be a [[../Cod/|category]]. A family of its [[../TrivialGroupoid/|objects]] ${\left\{U_i\right\}}_{i\in I}$ is said to be a family of [[../Generator/|generators]] of $𝒞$ if for every pair of distinct [[../TrivialGroupoid/|morphisms]] $\alpha ,\beta :A\to B$ there is a morphism $u:U_i\to A$ for some index $i\in I$ such that $\alpha u\ne \beta u$.

One notes that in an [[../DenseSubcategory/|additive category]], ${\left\{U_i\right\}}_{i\in I}$ is a family of generators if and only if for each nonzero morphism $\alpha$ in $𝒞$ there is a morphism $u:U_i\to A$ such that $\alpha u\ne 0$.

An object $U$ in $𝒞$ is called a generator for $𝒞$ if $U\in {\left\{U_i\right\}}_{i\in I}$ with ${\left\{U_i\right\}}_{i\in I}$ being a family of generators for $𝒞$.

Equivalently, (viz. Mitchell) $U$ is a generator for $𝒞$ if and only if the set-valued [[../TrivialGroupoid/|functor]] $H^U$ is an imbedding functor.

Proper generator of a Grothendieck category

A proper generator $U_p$ of a [[../GrothendieckCategory/|Grothendieck category]] $𝒢$ is defined as a generator $U_p$ which has the property that a [[../InjectiveMap/|monomorphism]] $i:U^{\prime }\to U_p$ induces an [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphism]] $\iota$, $Ho{m}_{𝒢}(U_p,U_p)\cong Ho{m}_{𝒢}(U^{\prime },U_p),$ if and only if $i$ is an isomorphism.

\begin{theorem} Any [[../PAdicMeasure/|commutative ring]] is the endomorphism ring of a proper generator in a suitably chosen Grothendieck category. \end{theorem}