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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:  11:20 - 12:00

Associativity Problems in Semi-Abelian Categories
Janelidze, George (Cape Town University and Tbilisi Mathematical Institute)

A pointed category with finite coproducts is called semi-abelian \bibref{11} if it is Barr exact and Bourn protomodular. The main result in \bibref{11} asserts that a category is semi-abelian if and only it satisfies all of the so-called old axioms. Here ``all old axioms" in fact means ``all first-order exactness axioms, invented before Barr exactness, that hold in all varieties of $\omega$-groups (= groups with multiple operators in the sense of P.\ Higgins \bibref{7})". However, on the one hand there are semi-abelian varieties of universal algebras far away from $\omega$-groups (see \bibref{4}), and on the other hand there are useful stronger axioms that do not necessarily hold in all semi-abelian categories but hold in all important varieties of $\omega$-groups, including groups, rings, (modules or) algebras over rings, and Lie algebras. The aim of this talk is to discuss these axioms and their relationship with associativity in concrete algebraic cases; a number of open problems will be stated --- hoping that their solution will lead to the discovery of a convincing associativity axiom. The discussion will refer to the following papers:

\begin{references}
\bibitem F.\ Borceux, G.\ Janelidze, and G.\ M.\ Kelly, {\em Internal object actions}, Commentationes Mathematicae Universitatis Carolinae, accepted
\bibitem D.\ Bourn, {\em Commutator theory in strongly protomodular categories}, Univ.\ Littoral Preprint 208, 2004
\bibitem D.\ Bourn and G.\ Janelidze, {\em Extensions with abelian kernels in protomodular categories}, Georgian Math.\ Journal 11, 4, 2004, 645--654
\bibitem D.\ Bourn and G.\ Janelidze, {\em Characterization of protomodular varieties of universal algebras}, Theory Applications of Categories 11, No 6, 2003, 143--147
\bibitem M.\ Gerstenhaber, {\em A categorical setting for the Baer extension theory}, Proceedings of Symposia in Pure Mathematics 17, American Math.\ Soc., Providence, RI, 1970, 50--64
\bibitem M.\ Gran, {\em Applications of categorical Galois theory in universal algebra}, Fields Institute Communications 43, 2004, 243--280
\bibitem P.\ J.\ Higgins, {\em Groups with multiple operators}, Proc.\ London Math.\ Soc.\ (3) 6 1956, 366--416
\bibitem S.\ A.\ Huq, {\em Commutator, nilpotency and solvability in categories}, Quart.\ J.\ Math.\ Oxford (2) 19, 1968, 363--389
\bibitem G.\ Janelidze, {\em Internal crossed modules}, Georgian Math.\ Journal 10, 1, 2003, 99--114
\bibitem G.\ Janelidze and L.\ Márki, {\em Radicals of rings and pullbacks}, Journal of Pure and Applied Algebra 97, 1994, 29--36
\bibitem G.\ Janelidze, L.\ Márki, and W.\ Tholen, {\em Semi-abelian categories}, Journal of Pure and Applied Algebra 168, 202, 367--386
\bibitem G.\ Janelidze and M.\ C.\ Pedicchio, {\em Pseudogroupoids and commutators}, Theory and Applications of Categories 8, No 15, 2001, 405--456
\bibitem S.\ Mac Lane, {\em Duality for groups}, Bulletin of American Math.\ Soc.\ 56, 1950, 485--516.
\bibitem L.\ Márki and R.\ Wiegandt, {\em Remarks on radicals in categories}, Lecture Notes in Mathematics 962, Springer, 1982, 190--196
\bibitem M.\ C.\ Pedicchio, {\em A categorical approach to commutator theory}, Journal of Algebra 177, 1995, 647--657
\bibitem G.\ Orzech, {\em Obstruction theory in algebraic categories I}, Journal of Pure and Applied Algebra 2 1972, 287--314
\end{references}

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