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[parent] Grothendieck category lemma (Corollary)

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 $i: U' \to U$ 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. [1]).

Grothendieck category lemma

Lemma 0.1   Any Grothendieck category $\mathcal{G}$ has a proper generator.

Bibliography

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.



"Grothendieck category lemma" is owned by bci1.
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See Also: Grothendieck category

Also defines:  proper generator
Keywords:  Grothendieck category Lemma

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Cross-references: viz, isomorphism, induces, monomorphism, property, Grothendieck category, generator
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This is version 23 of Grothendieck category lemma, born on 2008-10-04, modified 2009-01-29.
Object id is 11134, canonical name is GrothendieckCategoryLemma.
Accessed 644 times total.

Classification:
AMS MSC: 18-00 (Category theory; homological algebra :: General reference works )
 18E05 (Category theory; homological algebra :: Abelian categories :: Preadditive, additive categories)
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correction: re: proper generator by bci1 on 2009-01-28 18:50:02
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}.


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