Essential data

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

has a right adjoint denoted by

such that one has the following functorial isomorphism, or natural equivalence:

$\displaystyle Hom_{\mathcal{C}}(X, S(Y)) \cong Hom_{\mathcal{C} / \mathcal{A}}$

Definition 1.1 The right adjoint functor

$\displaystyle S: \mathcal{C}/ \mathcal{A} \to \mathcal{C}$

- which is specified by the essential data above- is called a section functor.

Note: the category $\mathcal{A}$ is defined as a localizing subcategory.
Reference cited.

"section functor" is owned by bci1.

(view preamble | get metadata)

See Also: adjoint functor, natural equivalence

Also defines:

localizing subcategory

Keywords:

section functor, adjoint functor, localization

Log in to rate this entry.
(view current ratings)

Cross-references: reference, category, natural equivalence, isomorphism, right adjoint, functor, canonical, dense subcategory, locally small, abelian category
There is 1 reference to this entry.

This is version 6 of section functor, born on 2008-10-03, modified 2008-10-18.
Object id is 11130, canonical name is SectionFunctor.
Accessed 248 times total.

Classification:

AMS MSC:	18-00 (Category theory; homological algebra :: General reference works )
	18E05 (Category theory; homological algebra :: Abelian categories :: Preadditive, additive categories)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact