  Quasi-abelian categories are abundant in functional analysis and
representation theory. It is known that a quasi-abelian category $\mathcal{E}$
is a cotilting torsionfree class of an abelian category. In fact, this property
characterizes quasi-abelian categories. This ambient abelian category is
derived equivalent to the category $\mathcal{E}$, and can be constructed as the
heart $\mathcal{LH}(\mathcal{E})$ of a $\operatorname{t}$-structure on the
bounded derived category $\operatorname{D^b}(\mathcal{E})$ or as the
localization of the category of monomorphisms in $\mathcal{E}.$
  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 $\operatorname{LB}$-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 $\mathcal{E}$ can be found as the
heart of a $\operatorname{t}$-structure on the bounded derived category
$\operatorname{D^b}(\mathcal{E})$, or as the localization of the category of
monomorphisms of $\mathcal{E}$. 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
additive 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.
