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Homesection functor

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# section functor

# 1 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:

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

.

###### 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*.

Defines:

localizing subcategory

Keywords:

section functor, adjoint functor, localization

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

18E05*no label found*18-00

*no label found*

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