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# Grothendieck category lemma

# 0.1 Introduction: proper generator

###### Definition 0.1.

Let us recall that a *generator* of a Grothendieck category $\mathcal{G}$ is called *proper* if $U$ has the property that a monomorphism $i:U^{{\prime}}\to U$ induces an isomorphism

$Hom_{{\mathcal{G}}}(U,U)\cong Hom_{{\mathcal{G}}}(U^{{\prime}},U)$ |

if and only if $i$ is an isomorphism (viz. p. 251 in ref. [1]).

# 0.2 Grothendieck category lemma

###### Lemma 0.1.

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

# References

- AG4
Alexander Grothendieck et al.
*Séminaires en Géometrie Algèbrique- 4*, Tome 1, Exposé 1 (or the Appendix to Exposée 1, by ‘N. Bourbaki’ for more detail and a large number of results.), AG4 is freely available in French; also available here is an extensive Abstract in English. - 1 Nicolae Popescu. Abelian Categories with Applications to Rings and Modules., Academic Press: New York and London, 1973 and 1976 edns., (English translation by I. C. Baianu.)
- 2 Leila Schneps. 1994. The Grothendieck Theory of Dessins d’Enfants. (London Mathematical Society Lecture Note Series), Cambridge University Press, 376 pp.
- 3
David Harbater and Leila Schneps. 2000.
Fundamental groups of moduli and the Grothendieck-Teichmüller group,
*Trans. Amer. Math. Soc*. 352 (2000), 3117-3148. MSC: Primary 11R32, 14E20, 14H10; Secondary 20F29, 20F34, 32G15.

Defines:

proper generator

Keywords:

Grothendieck category Lemma

Related:

GrothendieckCategory

Type of Math Object:

Corollary

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Reference

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## Mathematics Subject Classification

18-00*no label found*18E05

*no label found*

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## Comments

## correction: re: proper generator

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}.

## Re: correction: re: proper generator

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 $U_i$?