A representability theorem for some huge abelian categories

TL;DR

This paper introduces quasi--locally presentable categories, proves a representability theorem for certain functors, and applies it to abelianization of triangulated categories, providing new insights and proofs in category theory.

Contribution

It defines quasi--locally presentable categories and establishes a new representability theorem, extending Brown's theorem and analyzing functor representability.

Findings

Abelianization of well generated triangulated categories is quasi--locally presentable

A new proof of Brown's representability theorem is provided

Examples of non-representable functors are discussed

Abstract

We define quasi--locally presentable categories as big unions of coreflective subcategories which 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 Brown representability theorem. Examples of functors which are not representable are also given.