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section functor (Definition)

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 $T$ has a right adjoint denoted by $S$ 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

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

Reference cited.



"section functor" is owned by bci1.

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Other names:  right adjoint functor
Keywords:  adjoint functor, natural transformations, secyion functor

Cross-references: category, isomorphism, functor, dense subcategory, abelian category
There are 3 references to this object.

This is version 2 of section functor, born on 2009-04-05, modified 2009-04-05.
Object id is 618, canonical name is SectionFunctor.
Accessed 510 times total.

Classification:
Physics Classification: 00. (GENERAL)
 02. (Mathematical methods in physics)
 03. (Quantum mechanics, field theories, and special relativity )
Pending Errata and Addenda
None.
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