Categorical structures and commutator theory
by
Marino Gran
Université du Littoral Côte d'Opale, Laboratoire de Mathématiques Pures et Appliquées, Calais, France
Coauthors: Dominique Bourn

The purpose of this paper is to present some recent developments in the categorical approach to centrality and to commutator theory, and their applications in the categorical theory ofcentral extensions.

The theory of commutators can be considered as an extension of the classical commutator for groups to more general varieties of algebras. Maltsev categories provide an appropriate level of generality in which the main properties of commutators hold [18] [19] [4]. The study of these properties becomes simpler by adopting the internal notion of connector of two equivalence relations introduced in [3]. This notion can be defined in any category C with pullbacks: if R and S are two equivalence relations on the same object X in C and R ×_X S denotes the pullback of the codomain arrow d₁ \colon R --> X along the domain arrow d₀ \colon S --> X, a connector between R and S is an arrow p \colon R ×_X S --> X such that

1. p(x, x, y)=y            1.* p(x, y, y)=x 2. xSp(x, y, z)            2.* zRp(x, y, z) 3. p(x, y, p(y, u, v))=p(x, u, v)            3.* p(p(x, y, u), u, v)=p(x, y, v).

In any Maltsev category the axioms 1. and 1.^* imply all the others and, moreover, when there is a connector p between R and S it is necessarily unique. Accordingly, the existence of a connector between R and S becomes a property, and we then say that R and S are connected. In any Maltsev variety [20] two congruence relations R and S are connected if and only if their classical commutator is trivial, i.e., [R, S] = \Delta_X. The good behaviour of the internal categorical structures allows one to establish many important centrality properties in any regular Maltsev category, thus including also the important examples of Maltsev quasi-varieties and of topological Maltsev algebras [4]. A new improvement we present here is the fact that the main basic properties of the commutator can be derived directly from the stability properties of centrality together with the general properties of adjoint functors.

Outside the realm of Maltsev varieties the existence of a connector between two equivalence relations R and S is no longer equivalent to the condition [R, S] = \Delta_X. In general congruence modular varieties, it is the more general structure of pseudogroupoid [16] that has to be considered.

In the present paper we work under the assumption that in the base category C a categorical formulation of the Shifting Lemma [11] holds: we show that this axiom simply requires that a certain kind of internal functors between internal equivalence relations are discrete fibrations. If C is a variety of universal algebras it is well-known that the validity of the Shifting Lemma is equivalent to the fact that the variety is congruence modular.

In any finitely complete category C in which the Shifting Lemma holds we reformulate the notion of pseudogroupoid by presenting it as an idempotent \sigma\colon R \square S --> R\square S on the largest double equivalence relation on R and S, which satisfies some extra conditions. We prove that a pseudogroupoid structure between two equivalence relations is unique, when it exists. Moreover, the associativity requirement in its definition can be removed, since under these assumptions it follows from the other axioms. In particular, the definition of internal connectors in these categories can be simplified: the assumptions 3. and 3.^* can be dropped. The Shifting Lemma also implies that the category Psgrd(C) of internal pseudogroupoids in C is a full subcategory of the category 2-Eq(C) of pairs of equivalence relations on the same object. A simple description of internal categories and internal groupoids can be given, which extends some results in [17] for congruence modular varieties.

When the category C is also regular, the precise relationship between connectors and pseudo-groupoids can be expressed as follows: there is a (unique) connector between R and S if and only if there is a (unique) pseudogroupoid between R and S and R o S = S o R. When the category C is also Barr-exact and pointed, then the category of connected internal groupoids in C is proved to be equivalent to the category of central extensions in C, thus extending some of the results in [4] and [10]. Further results can be obtained in the presence of the Goursat assumption [5], as for instance the exactness of the category of internal groupoids in C.

Finally, we would like to mention that these results have some applications in the categorical theory of central extensions [13] [14] [15]. It is precisely the relationship between pseudogroupoids and connectors which allowed us to prove that any algebraically central extension is categorically central [10], so that this fact and the main result in [14] imply that there is a perfect coincidence between these two notions in any congruence modular variety. A new contribution to the categorical theory of central extensions we present here is that when C is exact Maltsev and X is any admissible subcategory of C (in the sense of [13]), the so-called Galois pregroupoid associated with any extension is always an internal groupoid.

References

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Date received: May 16, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cajf-13.