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section functor
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(Definition)
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Let us consider an Abelian category $\mathcal{C}$ which is locally small and a dense subcategory $\mathcal{A}$ of $\mathcal{C}$ , with $T: \mathcal{C} \to \mathcal{C}/\mathcal{A}$ being the canonical functor. Moreover, let us assume that $T$ has a right adjoint denoted by
$S$ such that one has the following functorial isomorphism, or natural equivalence:
$$Hom_{\mathcal{C}}(X, S(Y)) \cong Hom_{\mathcal{C} / \mathcal{A}}$$ .
Definition 1.1 The right adjoint functor $$S: \mathcal{C}/ \mathcal{A} \to \mathcal{C}$$ of $T$ - which is specified by the essential data above- is called a section functor.
Note: the category $\mathcal{A}$ is defined as a localizing subcategory.
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"section functor" is owned by bci1.
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| Also defines: |
localizing subcategory |
| Keywords: |
section functor, adjoint functor, localization |
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Cross-references: category, natural equivalence, isomorphism, right adjoint, functor, canonical, dense subcategory, locally small, abelian category
There is 1 reference to this entry.
This is version 9 of section functor, born on 2008-10-03, modified 2009-04-05.
Object id is 11130, canonical name is SectionFunctor.
Accessed 823 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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Pending Errata and Addenda
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