Formula omitted]-coextensive objects and the strict refinement property

Abstract

The notion of an M-coextensive object is introduced in an arbitrary category C, where M is a distinguished class of morphisms from C. This notion allows for a categorical treatment of the strict refinement property in universal algebra, and highlights its connection with extensivity in the sense of Carboni, Lack and Walters. If M is the class of product projections in a category C with finite products, then M-coextensivity is closely related to the notion of a Boolean category in the sense of E. Manes: C is co-Boolean if and only if its product projections are pushout stable, and every object is M-coextensive. We show that if C is a variety of algebras then the M-coextensive objects are precisely those algebras which have the strict refinement property, when M is the class of product projections. If M is the class of surjective homomorphisms in the variety, then the M-coextensive objects are those algebras which have directly-decomposable (or factorable) congruences. Moreover, these results are proved for any object with global support in a regular category. We also show that in exact Mal'tsev categories, every centerless object with global support is projection-coextensive, i.e., has the strict refinement property. We will also show that in every exact majority category, every object with global support has the strict refinement property.