The heart of the Banach spaces

Consider an exact category in the sense of Quillen. Assume that in this category every morphism has a kernel and that every kernel is an inflation. In their seminal 1982 paper, Beĭlinson, Bernstein and Deligne consider in this setting a t-structure on the derived category and remark that its heart can be described as a category of formal quotients. They further point out that the category of Banach spaces is an example, and that here a similar category of formal quotients was studied by Waelbroeck already in 1962. In the current article, we give a direct and rigorous construction of the latter category by considering first the monomorphism category. Then we localize with respect to a multiplicative system. Our approach gives rise to a heart-like category not only for the Banach spaces. In particular, the main results apply to categories in which the set of all kernel–cokernel pairs does not form an exact structure. Such categories arise frequently in functional analysis.