Reflective and coreflective subcategories

Abstract

Given any additive category $\mathcal{C}$ with split idempotents, pseudokernels and pseudocokernels, we show that a subcategory $\mathcal{B}$ is coreflective if, and only if, it is precovering, closed under direct summands and each morphism in $\mathcal{B}$ has a pseudocokernel in $\mathcal{C}$ that belongs to $\mathcal{B}$. We apply this result and its dual to, among others, preabelian and pretriangulated categories. As a consequence, we show that a subcategory of a preabelian category is coreflective if, and only it, it is precovering and closed under taking cokernels. On the other hand, if $\mathcal{C}$ is pretriangulated with split idempotents, then a subcategory $\mathcal{B}$ is coreflective and invariant under the suspension functor if, and only if, it is precovering and closed under taking direct summands and cones. These are extensions of well-known results for AB3 abelian and triangulated categories, respectively. By-side applications of these results allow us: a) To characterize the coreflective subcategories of a given AB3 abelian category which have a set of generators and are themselves abelian, abelian exact or module categories; b) to extend to module categories over arbitrary small preadditive categories a result of Gabriel and De la Pe\~na stating that all fully exact subcategories are bireflective; c) to show that, in any Grothendieck category, the direct limit closure of its subcategory of finitely presented objects is a coreflective subcategory.