The aim of this series of papers is to develop a self-dual categorical approach to some topics in non-abelian algebra, which is based on replacing the framework of a category with that of a category equipped with a functor to it. The present paper gives some preliminary steps in this direction, where several known structures on a category, which arise in the categorical treatment of these topics, are viewed as such functors; as a result, we obtain some new conceptual links between these structures.
Keywords: biform, cover relation, ex2-category, factorization system, faithful amnestic functor, form, Grothendieck fibration, ideal of null morphisms, subobject, universalizer
2010 MSC: 18D30, 18A32, 18A20, 18E10, 18G50, 06A15, 06A75, 06F99
Theory and Applications of Categories, Vol. 29, 2014, No. 11, pp 315-331.