Algebra Universalis Duality in non-abelian algebra III. Normal categories and 0-regular varieties Zurab Janelidze and Thomas Weighill Abstract. Normal categories are pointed categorical counterparts of 0-regular vari- eties, i.e., varieties where each congruence is uniquely determined by the equivalence class of a fixed constant 0. In this paper, we give a new axiomatic approach to nor- mal categories, which uses self-dual axioms on a functor defined using subobjects of objects in the category. We also show that a similar approach can be developed for 0-regular varieties, if we replace subobjects with subsets of algebras containing 0. 1. Introduction Axiomatic duality in categories goes back to [35]. In that paper Mac Lane proposes to use dual axioms on a category (equipped perhaps with an ad- ditional structure) for generalizing and organizing results, such as the iso- morphism theorems, from the concrete context of groups. He then lists such axioms suitable for the context of abelian groups, and arrives at the notion of an abelian bicategory, which is a predecessor of the present-day notion of an abelian category [10, 20, 36]. Although there were attempts to look at the case of non-abelian groups, and in particular by Wyler [46] (see also the work of Terlikowska-Os lowska [40]), this direction of research did not per- sist (instead, it inspired a non-dual axiomatic development—see [25] and the references there), while Wyler himself shifted his interest to categorical topol- ogy, where he introduced topological theories [47], which up to the so-called Grothendieck construction are the same as what today are known as topo- logical functors (see [9] and references there)—faithful Grothendieck bifibra- tions whose fibres are complete lattices. The thesis of this series of papers is that the structure of a topological theory, and its natural variations, pro- vide a convenient language for work on the “duality for groups hypothesis”: homomorphism theorems for groups can be established in a category-theoretic Presented by J. Adamek. Received May 19, 2014; accepted in final form June 12, 2015. 2010 Mathematics Subject Classification : Primary: 18C99; Secondary: 08A30, 08B05, 08C05, 18A20, 18A32, 18D30, 18G50. Key words and phrases : abelian category, axiomatic duality, exact form, Grothendieck fi- bration, homological category, form of subobjects, ideal-determined variety, normal category, 0-permutable variety, protomodular category, 0-regular variety, semi-abelian category, sub- object fibration, subtractive category, subtractive variety, universalizer, variety with ideals. Supported by the South African National Research Foundation (both authors) and the MIH Media Lab at Stellenbosch University (second author). Algebra Univers. 77 (2017) 1–28 DOI 10.1007/s00012-017-0422-7 Published online January 17, 2017 © Springer International Publishing 2017