Goursat Categories Research Articles

Anticommutativity and n-schemes

The purpose of this paper is two-fold. A first and more concrete aim is to give new characterizations of equivalence distributive Goursat categories (which extend 3-permutable varieties) through variations of the little Pappian Theorem involving reflexive and positive relations. A second and more abstract aim is to show that every finitely complete category E satisfying the n-scheme is locally anticommutative.

Facets of congruence distributivity in Goursat categories

We give new characterisations of regular Mal'tsev categories with distributive lattice of equivalence relations through variations of the so-called Triangular Lemma and Trapezoid Lemma in universal algebra. We then give new characterisations of equivalence distributive Goursat categories (which extend 3-permutable varieties) through variations of the Triangular and Trapezoid Lemmas involving reflexive and positive relations.

Monoidal characterisation of groupoids and connectors

We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal category of relations. Similarly, connectors can equivalently be described as Frobenius structures with a ternary multiplication. We study such ternary Frobenius structures and the relationship to binary ones, generalising that between connectors and groupoids.

Variations of the Shifting Lemma and Goursat categories

We prove that Mal'tsev and Goursat categories may be characterised through stronger variations of the Shifting Lemma, that is classically expressed in terms of three congruences $R$, $S$ and $T$, and characterises congruence modular varieties. We first show that a regular category $\mathcal C$ is a Mal'tsev category if and only if the Shifting Lemma holds for reflexive relations on the same object in $\mathcal C$. Moreover, we prove that a regular category $\mathcal C$ is a Goursat category if and only if the Shifting Lemma holds for a reflexive relation $S$ and reflexive and positive relations $R$ and $T$ in $\mathcal C$. In particular this provides a new characterisation of $2$-permutable and $3$-permutable varieties and quasi-varieties of universal algebras.

A note on admissibility of closed Galois structures

We present a new admissibility theorem for Galois structures in the sense of G. Janelidze. It applies to relative exact categories satisfying a suitable relative modularity condition, and extends the known admissibility theorem in the theory of generalized central extensions. We also show that our relative modularity condition holds in every relative exact Goursat category.

Beck–Chevalley condition and Goursat categories

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.

The Cuboid Lemma and Mal’tsev Categories

We prove that a regular category ℂ is a Mal’tsev category if and only if a strong form of the denormalised 3 × 3 Lemma holds true in ℂ. In this version of the 3 × 3 Lemma, the vertical exact forks are replaced by pullbacks of regular epimorphisms along arbitrary morphisms. The shape of the diagram it determines suggests to call it the Cuboid Lemma. This new characterisation of regular categories that are Mal’tsev categories (= 2-permutable) is similar to the one previously obtained for Goursat categories (= 3-permutable). We also analyse the “relative” version of the Cuboid Lemma and extend our results to that context.

A good theory of ideals in regular multi-pointed categories

By a multi-pointed category we mean a category C equipped with an ideal of null morphisms, i.e. a class N of morphisms satisfying f∈N∨g∈N⇒fg∈N for any composable pair f,g of morphisms. Such categories are precisely the categories enriched in the category of pairs X=(X,N) where X is a set and N is a subset of X, whereas a pointed category has the same enrichment, but restricted to those pairs X=(X,N) where N is a singleton. We extend the notion of an “ideal” from regular pointed categories to regular multi-pointed categories, and having “a good theory of ideals” will mean that there is a bijection between ideals and kernel pairs, which in the pointed case is the main property of ideal determined categories. The study of general categories with a good theory of ideals allows in fact a simultaneous treatment of ideal determined and Barr exact Goursat categories: we prove that in the case when all morphisms are chosen as null morphisms, the presence of a good theory of ideals becomes precisely the property for a regular category to be a Barr exact Goursat category. Among other things, this allows to obtain a unified proof of the fact that lattices of effective equivalence relations are modular both in the case of Barr exact Goursat categories and in the case of ideal determined categories.

Relative Goursat categories

We define relative Goursat categories and prove relative versions of the equivalent conditions defining regular Goursat categories. These include 3-permutability of equivalence relations, preservation of equivalence relations under direct images, a condition on so-called Goursat pushouts, and the denormalised 3×3 Lemma. This extends recent work by Gran and Rodelo on a new characterisation of Goursat categories to a relative context.

A New Characterisation of Goursat Categories

We present a new characterisation of Goursat categories in terms of special kind of pushouts, that we call Goursat pushouts. This allows one to prove that, for a regular category, the Goursat property is actually equivalent to the validity of the denormalised 3-by-3 Lemma. Goursat pushouts are also useful to clarify, from a categorical perspective, the existence of the quaternary operations characterising 3-permutable varieties.

Descent for Regular Epimorphisms in Barr Exact Goursat Categories

We show that the category of regular epimorphisms in a Barr exact Goursat category is almost Barr exact in the sense that (it is a regular category and) every regular epimorphism in it is an effective descent morphism.

The 3-by-3 lemma for regular Goursat categories

We prove a 3 £ 3 lemma for regular Goursat categories. The classical “3 £ 3 lemma” in abelian category theory involves short exact sequences. In more general contexts — in particular in a category which is not pointed — it makes sense to replace short exact sequences by “exact forks”: these are diagrams R r1 / r2 / A r / B

Relations in operational categories

There are two well-known methods for constructing the ‘connecting homomorphism’ in homological algebra. Either one constructs it as a binary relation which is shown to be universally defined and single-valued, or one makes use of the so-called two-square lemma, which provides an isomorphism between two invariants associated with adjacent commutative squares. Both constructions generalize to arbitrary ‘Goursat categories’, namely operational categories satisfying the condition that the relative product of any relation with its converse is transitive. By an ‘operational category’ we here mean a category ß accompanied by a category of setvalued functors from ß. In order to state the results, one has to define ‘relations’ in operational categories and one has to generalize the notion of ‘exactness’ from short sequences to forks and the notion of ‘commutativity’ to squares in which two arrows are doubled. The proof of the general two square lemma involves a construction which closely resembles that of PER in theoretical computer science and contains the latter as a special case.

Some remarks on Maltsev and Goursat categories

Our aim is to analyze and to publicize two interesting properties — well known in universal algebra for varieties — that a regular category, and in particular an exact category, may possess: theMaltsev property, asserting the permutabilitySR=RS of equivalence relations on any object, and the weakerGoursat property, asserting only thatSRS=RSR. We investigate these properties, give various equivalent forms of them, and develop some of their useful consequences.

3 × 3 lemma for star-exact sequences

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.