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SemiAbelian Categories
, 2000
"... The notion of semiabelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abeliangroup and module theory. In modern terms, semiabelian categories ar ..."
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Cited by 80 (9 self)
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The notion of semiabelian category as proposed in this paper is designed to capture typical algebraic properties valid for groups, rings and algebras, say, just as abelian categories allow for a generalized treatment of abeliangroup and module theory. In modern terms, semiabelian categories are exact in the sense of Barr and protomodular in the sense of Bourn and have finite coproducts and a zero object. We show how these conditions relate to "old" exactness axioms involving normal monomorphisms and epimorphisms, as used in the fifties and sixties, and we give extensive references to the literature in order to indicate why semiabelian categories provide an appropriate notion to establish the isomorphism and decomposition theorems of group theory, to pursue general radical theory of rings, and how to arrive at basic statements as needed in homological algebra of groups and similar nonabelian structures. Mathematics Subject Classification: 18E10, 18A30, 18A32. Key words:...
TRIPLES, ALGEBRAS AND COHOMOLOGY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2003
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A DIXMIERDOUADY THEORY FOR STRONGLY SELFABSORBING C∗ALGEBRAS II: THE BRAUER GROUP
"... Abstract. We have previously shown that the isomorphism classes of orientable locally trivial fields of C∗algebras over a compact metrizable space X with fiber D ⊗ K, where D is a strongly selfabsorbing C∗algebra, form an abelian group under the operation of tensor product. Moreover this group is ..."
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Abstract. We have previously shown that the isomorphism classes of orientable locally trivial fields of C∗algebras over a compact metrizable space X with fiber D ⊗ K, where D is a strongly selfabsorbing C∗algebra, form an abelian group under the operation of tensor product. Moreover this group is isomorphic to the first group Ē1D(X) of the (reduced) generalized cohomology theory associated to the unit spectrum of topological Ktheory with coefficients in D. Here we show that all the torsion elements of the group Ē1D(X) arise from locally trivial fields with fiber D ⊗Mn(C), n ≥ 1, for all known examples of strongly selfabsorbing C∗algebras D. Moreover the Brauer group generated by locally trivial fields with fiber D ⊗Mn(C), n ≥ 1 is isomorphic to Tor(Ē1D(X)). 1.
Five interpretations of Faa ̀ di Bruno’s formula
"... Dedicated to JeanLouis Loday Abstract. In these lectures we present five interpretations of the Faa ̀ di Bruno formula which computes the nth derivative of the composition of two functions of one variable: in terms of groups, Lie algebras and Hopf algebras, in combinatorics and within operads. ..."
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Dedicated to JeanLouis Loday Abstract. In these lectures we present five interpretations of the Faa ̀ di Bruno formula which computes the nth derivative of the composition of two functions of one variable: in terms of groups, Lie algebras and Hopf algebras, in combinatorics and within operads.