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84
Higher Hopf formulae for homology via Galois Theory
, 2007
"... We use Janelidze’s Categorical Galois Theory to extend Brown and Ellis’s higher Hopf formulae for homology of groups to arbitrary semiabelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the BarrBeck derived functors of the reflector of A on ..."
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We use Janelidze’s Categorical Galois Theory to extend Brown and Ellis’s higher Hopf formulae for homology of groups to arbitrary semiabelian monadic categories. Given such a category A and a chosen Birkhoff subcategory B of A, thus we describe the BarrBeck derived functors of the reflector of A onto B in terms of centralization of higher extensions. In case A
Internal crossed modules
 Georgian Math
"... on the occasion of his seventieth birthday Abstract. We introduce the notion of (pre)crossed module in a semiabelian category, and establish equivalences between internal reflexive graphs and precrossed modules, and between internal categories and crossed modules. ..."
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on the occasion of his seventieth birthday Abstract. We introduce the notion of (pre)crossed module in a semiabelian category, and establish equivalences between internal reflexive graphs and precrossed modules, and between internal categories and crossed modules.
Commutator theory in strongly protomodular categories, Theory and
 Applications of Categories 13
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Higher central extensions and Hopf formulae
, 2009
"... Higher extensions and higher central extensions, which are of importance to nonabelian homological algebra, are studied, and some fundamental properties are proven. As an application, a direct proof of the invariance of the higher Hopf formulae is obtained. ..."
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Cited by 12 (9 self)
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Higher extensions and higher central extensions, which are of importance to nonabelian homological algebra, are studied, and some fundamental properties are proven. As an application, a direct proof of the invariance of the higher Hopf formulae is obtained.
Protoadditive functors, derived torsion theories and homology
 J. Pure Appl. Algebra
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RELATIVE HOMOLOGICAL CATEGORIES
 JOURNAL OF HOMOTOPY AND RELATED STRUCTURES, VOL. 1(1), 2006, PP.185–194
, 2006
"... We introduce relative homological and weakly homological categories (C, E), where “relative” refers to a distinguished class E of normal epimorphisms in C. It is a generalization of homological categories, but also protomodular categories can be regarded as examples. We indicate that the relative ve ..."
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Cited by 7 (2 self)
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We introduce relative homological and weakly homological categories (C, E), where “relative” refers to a distinguished class E of normal epimorphisms in C. It is a generalization of homological categories, but also protomodular categories can be regarded as examples. We indicate that the relative versions of various homological lemmas can be proved in a relative homological category.
On satellites in semiabelian categories: Homology . . .
, 2009
"... Working in a semiabelian context, we use Janelidze’s theory of generalised satellites to study universal properties of the Everaert long exact homology sequence. This results in a new definition of homology which does not depend on the existence of projective objects. We explore the relations with ..."
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Cited by 7 (3 self)
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Working in a semiabelian context, we use Janelidze’s theory of generalised satellites to study universal properties of the Everaert long exact homology sequence. This results in a new definition of homology which does not depend on the existence of projective objects. We explore the relations with other notions of homology, and thus prove a version of the higher Hopf formulae. We also work out some examples.
Ideal determined categories
"... Dedicated to Francis Borceux at the occasion of his sixtieth birthday We clarify the role of Hofmann’s Axiom in the oldstyle definition of a semiabelian category. By removing this axiom we obtain the categorical counterpart of the notion of an idealdetermined variety of universal algebras – which ..."
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Dedicated to Francis Borceux at the occasion of his sixtieth birthday We clarify the role of Hofmann’s Axiom in the oldstyle definition of a semiabelian category. By removing this axiom we obtain the categorical counterpart of the notion of an idealdetermined variety of universal algebras – which we therefore call an idealdetermined category. Using known counterexamples from universal algebra we conclude that there are idealdetermined categories which fail to be Mal’tsev. We also show that there are idealdetermined Mal’tsev categories which fail to be semiabelian.