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Coverings and crossed modules of topological groups with operations

In this paper we generalize this result to a large class of algebraic objects called topological groups with operations, including topological groups. We also prove that the crossed modules and internal categories within topological groups with operations are equivalent. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2014) 38: 833 – 845 ¨ ITAK ˙ c TUB ⃝ doi:10.3906/mat-1307-53 Coverings and crossed modules of topological groups with operations Osman MUCUK∗, Tun¸ car S ¸ AHAN Department of Mathematics, Erciyes University, Kayseri, Turkey Received: 24.07.2013 • Accepted: 11.05.2014 • Published Online: 01.07.2014 • Printed: 31.07.2014 Abstract: It is a well-known result of the covering groups that a subgroup G of the fundamental group at the identity of a semilocally simply connected topological group determines a covering morphism of topological groups with characteristic group G . In this paper we generalize this result to a large class of algebraic objects called topological groups with operations, including topological groups. We also prove that the crossed modules and internal categories within topological groups with operations are equivalent. This equivalence enables us to introduce the cover of crossed modules within topological groups with operations. Finally, we draw relations between the coverings of an internal groupoid within topological groups with operations and those of the corresponding crossed module. Key words: Covering groups, universal cover, crossed module, group with operations, topological groups with operations 1. Introduction The theory of covering spaces is one of the most interesting theories in algebraic topology. It is well known that e → X is a simply connected covering map and ˜0 ∈ X e is such if X is a topological group, say additive, p : X e becomes a topological group with identity ˜0 such that p is a morphism of topological that p(˜0) = 0 , then X groups (see, for example, [9]). The problem of universal covers of nonconnected topological groups was first studied in [25]. Taylor proved that a topological group X determines an obstruction class kX in H 3 (π0 (X), π1 (X, 0)), and that the vanishing of kX is a necessary and sufficient .