Beck–Chevalley condition and Goursat categories

Abstract

Abstract. We characterise regular Goursat categories through a specific stability property of regular epimorphisms with respect to pullbacks. Under the assumption of the existence of some pushouts this property can be also expressed as a restricted Beck-Chevalley condition, with respect to the fibration of points, for a special class of commutative squares. In the case of varieties of universal algebras these results give, in particular, a structural explanation of the existence of the ternary operations characterising 3-permutable varieties of universal algebras.A variety of universal algebras is called a Mal'tsev variety [25] when any pair of congruences R and S on the same algebra 2-permute, meaning that RS = SR. The celebrated Mal'tsev theorem asserts that the algebraic theory of such a variety is characterised by the existence of a ternary term p(x, y, z) such that the identities p(x, y, y) = x and p(x, x, y) = y hold [22]. The weaker 3-permutability of congruences RSR = SRS, which defines 3-permutable varieties, is also equivalent to the existence of two ternary operations r and s such that the identities r(x, y, y) = x, r(x, x, y) = s(x, y, y) and s(x, x, y) = y hold [17]. A nice feature of 3-permutable varieties is the fact that they are congruence modular, a condition that plays a crucial role in the development of commutator theory [12,16].Many interesting results have been discovered in regular Mal'tsev categories [11] and in regular Goursat categories [10], which can be seen as the categorical extensions of Mal'tsev varieties and of 3-permutable varieties, respectively. The interested reader will find many properties

Geometrical Methods in Goursat Categories

“…A regular category E is called a Goursat category [14] when it is 3-permutable, i.e. for any pair of equivalence relations R and S on the same object in E one has RSR = SRS.…”

Monoidal characterisation of groupoids and connectors

“…to the "objects of composable pairs of morphisms" is a regular epimorphism whenever f 0 and f 1 are regular epimorphisms (by Theorem 1.3 (ii) in [12], for instance). This implies that Gpd(C) is closed under regular quotients in RG(C) (Theorem 3.11 (ii) in [13]), so that the regular image of the factorisation in RG(C) of the internal functor (f 0 , f 1 ) : A → B is again an internal groupoid in C. That, in turn, implies that the regular epimorphismmonomorphism factorisations are pullback stable in Gpd(C), since pullbacks in Gpd(C) are computed "componentwise" at the levels of "objects", "morphisms" and "composable pairs", respectively.…”

“…To show that the functor is faithful, we need that two subobjects in Gpd(C) are isomorphic there if and only if they are isomorphic in C. For this it suffices to show that the forgetful functor Gpd(C) → C, given by A → A 1 , reflects isomorphisms. If f = (f 0 , f 1 ) : A → B is an internal functor in Gpd(C) as in (12) and f 1 is an isomorphism in C, then both f 0 and f 2 are isomorphisms. It follows that f an isomorphism in Gpd(C).…”

Central reflections and nilpotency in exact Mal’tsev categories

“…One exception is Section 1.6 which establishes an "algebraic" Beck-Chevalley condition for pushouts of regular epimorphisms in exact Mal'tsev categories, dual to the familiar Beck-Chevalley condition for pullbacks of monomorphisms in elementary toposes. In recent and independent work, Gran-Rodelo [32] consider a weaker form of this condition and show that it characterises regular Goursat categories.…”

“…The following exactness result will be used at several places. In the stated generality, it is due to Diana Rodelo and the second author [14], but the interested reader can as well consult [44,31,32] for closely related statements. Proposition 2.8.…”