Page 135 - Textos de Matemática Vol. 46

SOME REMARKS ON PULLBACKS IN GUMM CATEGORIES
MARINO GRAN AND DIANA RODELO
Dedicated to Manuela Sobral on the occasion of her seventieth birthday
Abstract. We extend some properties of pullbacks which are known to hold in a Mal’tsev context to the more general context of Gumm categories. The varieties of universal algebras which are Gumm categories are precisely the congruence modular ones. These properties lead to a simple alternative proof of the known property that central extensions and normal extensions coincide for any Galois structure associated with a Birkho↵ subcategory of an exact Goursat category.
1. Introduction
A categorical approach to the property of congruence modularity, well known in universal algebra, was proposed in [5, 6] via a categorical formulation of the so-called Shifting Lemma (recalled in Section 3). The categories satisfying this categorical property are called Gumm categories, since it was the mathe- matician H.P. Gumm who proved that, for a variety of universal algebras, the validity of the Shifting Lemma is equivalent to congruence modularity [13]. As examples of Gumm categories, we also have regular Mal’tsev categories [9, 8] and regular Goursat categories [8], which are defined by the property that any pair of equivalence relations R and S in C (on a same object) 3-permute: RSR = SRS.
In this context [3] D. Bourn established an interesting permutability result (see Theorem 3.2), that we use in the present paper to prove the following
Received: 6 August 2014 / Accepted: 29 October 2014.
2010 Mathematics Subject Classification. 18C05, 08B10, 08C05, 18B10, 18E10.
Key words and phrases. Regular category, Mal’tsev category, Goursat category, Gumm
category, congruence modularity, pullback properties.
The second author was supported by the Centro de Matem´atica da Universidade de Coim-
bra (CMUC), funded by the European Regional Development Fund through the program COMPETE and by the Portuguese Government through the FCT - Funda¸ca˜o para a Ciˆencia e a Tecnologia under the projects PEst-C/MAT/UI0324/2013 and PTDC/MAT/120222/2010 and grant number SFRH/BPD/69661/2010.
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