Comparison of abelian categories recollements
A triple of abelian categories \({\mathcal A}'\), \({\mathcal A}\), \({\mathcal A}''\) is a recollement situation if there is a collection of functors between them which show that \({\mathcal A}'\) is (equivalent to) a localizing and colocalizing subcategory of \({\mathcal A}\), with quotient category \({\mathcal A}''\). If \({\mathcal A}'\), \({\mathcal B}\), \({\mathcal A}''\) is also a recollement situation, a comparison functor from \({\mathcal A}\) to \({\mathcal B}\) is one that respects the functors implicit in the definitions of the recollement situations. The question then arises as to whether a comparison functor is necessarily an equivalence of categories. For abelian categories in general, an example is given to show that a comparison functor need not be an equivalence; on the other hand, it is known that, in the case of triangulated categories, a comparison functor must be an equivalence. In this paper, the authors give a precise criterion for a comparison functor to be an equivalence, under the condition that \({\mathcal A}\) and \({\mathcal B}\) both have enough projectives and injectives. Given two functors \(F,G: {\mathcal A}''\to{\mathcal A}'\) between abelian categories and a natural transformation \(\xi: F\to G\), MacPherson and Vilonen have constructed a canonical recollement \({\mathcal A}(\xi)\). Given that the categories have enough projectives, the authors determine when a given recollement is equivalent to \({\mathcal A}(\xi)\).
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