PlanetPhysics/Section Functor

Essential data

Let us consider an [[../AbelianCategory2/|abelian category]] $𝒞$ which is locally small and a [[../DenseSubcategory/|dense subcategory]] $𝒜$ of $𝒞$, with $T:𝒞\to 𝒞/𝒜$ being the canonical [[../TrivialGroupoid/|functor]]. Moreover, let us assume that $T$ has a right adjoint denoted by $S$ such that one has the following functorial [[../IsomorphicObjectsUnderAnIsomorphism/|isomorphism]], or [[../IsomorphismClass/|natural equivalence]]:

$Ho{m}_{𝒞}(X,S(Y))\cong Ho{m}_{𝒞/𝒜}$.

The right adjoint functor $S:𝒞/𝒜\to 𝒞$ of $T$-- which is specified by the essential data above-- is called a section functor .

Note: the [[../Cod/|category]] $𝒜$ is defined as a localizing subcategory .

Reference cited.