3 × 3 lemma for star-exact sequences

Abstract

A regular category is said to be normal when it is pointed and every regular epimorphism in it is a normal epimorphism. Any abelian category is normal, and in a normal category one can define short exact sequences in a similar way as in an abelian category. Then, the corresponding 3 × 3 lemma is equivalent to the so-called subtractivity, which in universal algebra is also known as congruence 0-permutability. In the context of non-pointed regular categories, short exact sequences can be replaced with "exact forks" and then, the corresponding 3 × 3 lemma is equivalent, in the universal algebraic terminology, to congruence 3-permutability; equivalently, regular categories satisfying such 3 × 3 lemma are precisely the Goursat categories. We show how these two seemingly independent results can be unified in the context of star-regular categories recently introduced in a joint work of A. Ursini and the first two authors.

Higher extensions in exact Mal'tsev categories: distributivity of congruences and the $3^n$-Lemma

“…For instance, if all rows and two out of three of the columns including the middle one are exact forks, then the third column is an exact fork as well [3,Theorem 3.1]. However, this only works if the context is sufficiently strong: the article [3] treats the case of a regular Mal'tsev category, but variations on this theme have been considered in more general environments [20,18,13]; in [12], the pointed and unpointed cases are even studied in a single framework.…”

On 2-star-permutability in regular multi-pointed categories

“…This implies that N -trivial objects are closed under strong quotients. One says that a multi-pointed category C has enough trivial objects [8] when N is a closed ideal [14], i.e. any morphism in N factors through an N -trivial object and, moreover, the class of N -trivial objects is closed under subobjects and squares, where the latter property means that, for any Ntrivial object X, the object X 2 = X × X is N -trivial.…”

Effective descent morphisms in star-regular categories

“…This property is referred to as having enough trivial objects in [9] (see also [12], and the references therein, for the related notion of a closed ideal of morphisms). There are several equivalent conditions defining when a category has enough trivial objects (see Proposition 3.5 in [9]). For the purpose of the present article, the following will be the most suitable one:…”

“…Recall also that a finitely complete category is quasi-pointed [3] if it has an initial object 0, a terminal object 1, and the unique arrow 0 → 1 is a monomorphism. As explained in [9] any quasi-pointed category provides an example of proto-pointed context: it suffices to choose for N the class of morphisms that factor through the initial object. Also, in the quasi-pointed context, C clearly has enough trivial objects.…”