section functor

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.



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