The left heart and exact hull of an additive regular category

Abstract

Quasi-abelian categories are abundant in functional analysis and repre-sentation theory. It is known that a quasi-abelian category 8 is a cotilting torsionfree class of an abelian category. In fact, this property characterizes quasi-abelian cate-gories. This ambient abelian category is derived equivalent to the category 8, and can be constructed as the heart ZJe (8) of a t-structure on the bounded derived cate-gory Db(8) or as the localization of the category of monomorphisms in 8. However, there are natural examples of categories in functional analysis which are not quasi-abelian, but merely one-sided quasi-abelian or even weaker. Examples are the category of LB-spaces or the category of complete Hausdorff locally convex spaces. In this paper, we consider additive regular categories as a generalization of quasi-abelian categories that covers the aforementioned examples. Additive regular categories can be characterized as those subcategories of abelian categories which are closed under subobjects. As for quasi-abelian categories, we show that such an ambient abelian category of an additive regular category 8 can be found as the heart of a t-structure on the bounded derived category Db(8), or as the localization of the category of monomor-phisms of 8. In our proof of this last construction, we formulate and prove a version of Auslander's formula for additive regular categories. Whereas a quasi-abelian category is an exact category in a natural way, an addi-tive regular category has a natural one-sided exact structure. Such a one-sided exact category can be 2-universally embedded into its exact hull. We show that the exact hull of an additive regular category is again an additive regular category.