Duality in non-abelian algebra II. From Isbell bicategories to Grandis exact categories

This paper continues the study of self-dual axioms on forms, i.e. faithful amnestic functors, motivated by properties of subobject forms in non-abelian algebra (which in many cases are Grothendieck bifibrations), and in particular, by properties of forms of substructures of group-like structures. In this paper we explore axiomatic origins of this kind, of a hierarchy of contexts introduced by M. Grandis for his projective approach to non-abelian homological algebra. This reveals new links between those contexts and the theory of factorization systems. Among other things, we show that a Grandis exact category is the same as an Isbell bicategory whose form (fibration) of projections is isomorphic to the form (opfibration) of injections.