Coextensivity and arithmeticity in categorical algebra

Several categorical-algebraic contexts possess well-behaved “measures of abelianness”, and therefore naturally produce notions that maximise or minimise such measures. The categories of abelian groups and Boolean rings, for instance, illustrate this extremism with respect to internal notions of abelian object: in the former every object is abelian, while in the latter only the terminal object is. Mal’tsev categories have a well-behaved theory of centrality of equivalence relations [2], through which naturally Mal’tsev categories [8] and (proto)arithmetical categories [9, 1] emerge, in a suitable way, as the centrality maximising and minimising extremes, respectively. From the universal-algebraic point of view, arithmetical algebraic categories are precisely those which are Mal’tsev and congruence distributive, and in other contexts congruence distributivity appears at the opposite-of-abelian end of the spectrum.

The aim of this talk is to show how several topics associated with arithmetical categories and congruence distributive varieties can be approached from the theory of (co)extensive categories [3]. For instance, using a notion of coextensive morphism developed in [5, 7], we have that a semi-abelian category $𝒞$ is arithmetical if and only if every product projection in $𝒞$ is coextensive. A Barr-exact Mal’tsev category is arithmetical if and only if every product projection in each fibre of its fibration of points ${\pi }_{𝒞}:\mathrm{Pt}(𝒞)\to 𝒞$ is coextensive. Outside any particular categorical context, universal-algebraic properties such as the strict refinement property [4], or the Fraser-Horn property, may be formulated categorically as assertions that certain canonical classes of morphisms in the algebraic category are coextensive.

The defining exactness properties of Mal’tsev and arithmetical categories may be presented as matrix properties in the sense of Z. Janelidze. The final part of this talk presents results from a joint work [6] together with P.-A. Jacqmin and Z. Janelidze which classifies finitely complete categories according to their matrix properties. One such result is that among all non-trivial matrix properties of regular categories, the matrix property corresponding to arithmetical categories is the strongest.