A representability theorem for some huge abelian categories

We define quasi-locally presentable categories as big unions of a chain of coreflective subcategories that are locally presentable. Under appropriate hypotheses we prove a representability theorem for exact contravariant functors defined on a quasi-locally presentable category taking values in abelian groups. We show that the abelianization of a well generated triangulated category is quasi-locally presentable, and we obtain a new proof of the Brown representability theorem. Examples of functors that are not representable are also given.

Keywords

representable functor, quasi-locally presentable category, abelianization, triangulated category with coproducts

2010 Mathematics Subject Classification

18A40, 18E30

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Published 4 December 2012