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Macquarie University  Department of Mathematics

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Categories in Algebra, Geometry and Mathematical Physics

Conference in honour of Ross Street's sixtieth birthday

July 11-16 2005, Macquarie University, Sydney

Fri, 15 July:  16:30 - 17:20

Torsion theories in semi-normal categories
Tholen, Walter (York University)

\thanks{Joint work with M. M. Clementino and D. Dikranjan.}
The notion of semi-abelian category combines two powerful algebraic properties, Barr-exactness and Bourn-protomodularity, and provides, among other things, an appropriate axiomatic setting for homological algebra of non-commutative structures. The category of topological groups fails to be semi-abelian since equivalence relations fail to be effective, but it is still Barr-regular and Bourn-protomodular. Clementino and Borceux called such pointed categories `homological', and more recently Bourn and Gran showed that they provide a good environment for the study of torsion theories.

In this talk we shall present a generalized and relativized notion of semi-abelian category, which captures certain properties of categories such as the category of topological groups that are lost when we view them just as homological categories. Hence we give a general setting that allows for a smooth treatment of torsion theories, which we characterize in terms of radicals and closure operators. Examples well beyond the realm of semi-abelian or even homological categories are also presented.

Typeset PDF of this abstract.