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Grothendieck category lemma

Defines: 
proper generator
Keywords: 
Grothendieck category Lemma
Type of Math Object: 
Corollary
Major Section: 
Reference

Mathematics Subject Classification

18-00 no label found18E05 no label found

Comments

The following correction was made in response to pahio's request, accepted.

\subsection{Introduction: proper generator}

Let $\mathcal{C}$ be a category. Let also $U = \left\{U_i\right\}_{i \in I}$ be a family of objects of $\mathbf{C}$. The \emph{family} $U$ is said to be a \emph{family of generators} of the category $\mathbf{C}$ if for any object $A$ of $\mathcal{C}$ and any subobject $B$ of $A$, distinct from $A$, there is at least an index $i \in I$, and a morphism, $u : U_i \to A$, that cannot be factorized through the canonical injection $i : B \to A$. Then, an object $U$ of $\mathbf{C}$ is said to be a \emph{generator} of the category $\mathcal{C}$ provided that the
family $\left\{U_i\right\}_{i \in I}$ is a \emph{family of generators} \cite{NP65} of the category $\mathbf{C}$.

Furthermore, a \emph{generator} of a Grothendieck category $\mathcal{C}$ is called \emph{proper} if $U$ has the property that a monomorphism $i: U' \to U$ induces an isomorphism
$$Hom_{\mathcal{C}}(U,U) \cong Hom_{\mathcal{C}}(U',U)$$ if and only if $i$ is an isomorphism (viz. p. 251 in ref.
\cite{NP65}.

I don't fully understand this sentence:

"...an object $U$ of $\mathcal{C}$ is said to be a generator of the category $\mathcal{C}$ provided that the family $\{U_i\}_{i\in I} is a family of generators of the category $\mathcal{C}$."

Any old object $U$? Shouldn't the object be one of the $U_i$?

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