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In this paper we provide an axiomatic analysis of the classical diagram chasing method of Mac Lane, that relies on duality and chasing elements of pointed sets, which allows one to generalize this method from abelian categories to non-abelian ones. Among the examples where the generalized method can be used are all modular semiexact categories in the sense of Grandis (which include all Puppe-Mitchell exact categories) and sequentiable categories in the sense of Bourn (which include all semi-abelian categories, and in particular, the categories of group-like structures). The method turns out to be closely related to the essential features of these categories. At the same time, in some sense, it simplifies the usual proofs of some of the standard diagram lemmas in them. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We formulate and prove a “five lemma”, which unifies two independent generalizations of the classical five lemma in an abelian category: the five lemma in a (modular) semi-exact category in the sense of M. Grandis, and the five lemma in a pointed regular protomodular category in the sense of D. Bourn. [ABSTRACT FROM AUTHOR]