A categorical and homological framework for Banach space theory. Adjoint functors, limits and colimits, comma categories and Kan extensions are developed inside the category Ban of Banach spaces and bounded operators, then used to build the homotopy category KOM of chain complexes and the derived category obtained by formally inverting quasi-isomorphisms. Derived functors and Ext^n are defined in that setting, and the obstruction that Ext cannot itself be a functor into Ban motivates enlarging Ban to an Abelian or Quillen-exact 'heart' - Waelbroeck's quasi-Banach quotients, Wegner's Mon[D], Noel's QESP and the Clausen-Scholze condensed approach - so that the standard machinery of homological algebra becomes available to functional analysts.
Part II of a survey (to appear in Extracta Mathematicae) on what category theory offers Banach space theory, continuing Part I and the book Homological Methods in Banach Space Theory. It builds the m…