Reflective and coreflective subcategories
Manuel Cort\'es-Izurdiaga, Septimiu Crivei, Manuel Saor\'in

TL;DR
This paper characterizes coreflective subcategories in additive categories with split idempotents, pseudokernels, and pseudocokernels, extending classical results to preabelian and pretriangulated categories and exploring applications in various categorical contexts.
Contribution
It provides new characterizations of coreflective subcategories in additive, preabelian, and pretriangulated categories, extending known results for abelian and triangulated categories.
Findings
Characterization of coreflective subcategories in preabelian categories
Extension of classical results to pretriangulated categories
Applications to Grothendieck and module categories
Abstract
Given any additive category with split idempotents, pseudokernels and pseudocokernels, we show that a subcategory is coreflective if, and only if, it is precovering, closed under direct summands and each morphism in has a pseudocokernel in that belongs to . 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 is pretriangulated with split idempotents, then a subcategory 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…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Rings, Modules, and Algebras · Advanced Topics in Algebra
