On injectivity in locally presentable categories

AbstractWe show that some fundamental results about projectivity classes, weakly coreflective subcate-gories and cotorsion theories can be generalized from R -modules to arbitrary locally presentablecategories.  2004 Elsevier Inc. All rights reserved. 1. IntroductionInjectivity in locally presentable categories is well understood(see [2]). The basic resultis that a full subcategory A of a locally presentable category K is a small-injectivity class(i.e., there is a set M of morphisms of K such that A consists of all objects injective w.r.t.each morphism in M ) if and only if A is accessible and closed in K under products and λ -directed colimits for some regular cardinal λ . Accessibility of A can be replaced by A beingalsoclosed under λ -puresubobjects.Here, λ -puresubobjectsare precisely λ -directedcolimits of split subobjects. This result was re-proved for additive locally presentablecategories by H. Krause [13]. Injectivity classes are closely related to weakly reflectivesubcategories. Every small-injectivity class of a locally presentable category

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