Non-abelian homological algebra and universal higher K-theory

We start with a sketch a fragment of non-abelian homological algebra adopting an intermediate level of generality, which
allows to begin along the lines of Grothendieck's Tohoku lectures replacing abelian categories by right (or left)
exact categories with initial (resp. final) objects. Further analysis leads to the notion of the stable
category of a left exact category and to the notions of quasi-suspended and quasi-triangulated categories.

We define K_0 of a right exact category, introduce a structure of right exact category on the category
of svelte right exact categories and apply the general machinery of non-abelian homological
algebra to obtain a universal higher K-theory. The universal K-theory has all standard (Quillen's)
tools of higher K-theory starting with the long 'exact' sequence (which is almost for free in our
setting) followed by reductions by resolution, additivity of characteristic filtrations and devissage.
Notice that Quillen defined higher K-theory on exact categories (which are a very special case of
right exact categories), but, he established long exact sequence of a localization (which is the hardest
theorem of his remarkable work) and devissage only in the case of abelian categories.
There exists a unique morphism from Quillen's K-theory to the universal K-theory of abelian
categories. According to a recent result of one of my students, Eric Bunch, this morphism is not
an isomorphism, in spite of the fact that both theories give the same answers in all cases
computed via standard tools. If time permits, I will explain why.