On the Form of Subobjects in Semi-Abelian and Regular Protomodular Categories

Let Gls denote the category of (possibly large) ordered sets with Galois connections as morphisms between ordered sets. The aim of the present paper is to characterize semi-abelian and regular protomodular categories among all regular categories ℂ, via the form of subobjects of ℂ, i.e. the functor ℂ → Gls which assigns to each object X in ℂ the ordered set Sub(X) of subobjects of X, and carries a morphism f : X → Y to the induced Galois connection Sub(X) → Sub(Y) (where the left adjoint maps a subobject m of X to the regular image of fm, and the right adjoint is given by pulling back a subobject of Y along f). Such functor amounts to a Grothendieck bifibration over ℂ. The conditions which we use to characterize semi-abelian and regular protomodular categories can be stated as self-dual conditions on the bifibration corresponding to the form of subobjects. This development is closely related to the work of Grandis on “categorical foundations of homological and homotopical algebra”. In his work, forms appear as the so-called “transfer functors” which associate to an object the lattice of “normal subobjects” of an object, where “normal” is defined relative to an ideal of null morphism admitting kernels and cokernels.