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21
Functorial Factorization, Wellpointedness and Separability
"... A functorial treatment of factorization structures is presented, under extensive use of wellpointed endofunctors. Actually, socalled weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and f ..."
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Cited by 13 (4 self)
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A functorial treatment of factorization structures is presented, under extensive use of wellpointed endofunctors. Actually, socalled weak factorization systems are interpreted as pointed lax indexed endofunctors, and this sheds new light on the correspondence between reflective subcategories and factorization systems. The second part of the paper presents two important factorization structures in the context of pointed endofunctors: concordantdissonant and inseparableseparable.
Higher Hopf formulae for homology via Galois Theory, preprint math.AT/0701815
, 2007
"... 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 ..."
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Cited by 10 (3 self)
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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
The Reflectiveness of Covering Morphisms in Algebra And Geometry
, 1997
"... . Each full reflective subcategory X of a finitelycomplete category C gives rise to a factorization system (E; M) on C, where E consists of the morphisms of C inverted by the reflexion I : C ! X . Under a simplifying assumption which is satisfied in many practical examples, a morphism f : A ! B lie ..."
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Cited by 7 (5 self)
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. Each full reflective subcategory X of a finitelycomplete category C gives rise to a factorization system (E; M) on C, where E consists of the morphisms of C inverted by the reflexion I : C ! X . Under a simplifying assumption which is satisfied in many practical examples, a morphism f : A ! B lies in M precisely when it is the pullback along the unit jB : B ! IB of its reflexion If : IA ! IB; whereupon f is said to be a trivial covering of B. Finally, the morphism f : A ! B is said to be a covering of B if, for some effective descent morphism p : E ! B, the pullback p f of f along p is a trivial covering of E. This is the absolute notion of covering; there is also a more general relative one, where some class \Theta of morphisms of C is given, and the class Cov(B) of coverings of B is a subclass  or rather a subcategory  of the category C #B ae C=B whose objects are those f : A ! B with f 2 \Theta. Many questions in mathematics can be reduced to asking whether Cov(B) is re...
Pseudogroupoids and Commutators
 Theory Appl. Categ
, 2001
"... We develop a new approach to Commutator theory based on the theory of internal categorical structures, especially of so called pseudogroupoids. It is motivated by our previous work on internal categories and groupoids in congruence modular varieties. ..."
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Cited by 6 (2 self)
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We develop a new approach to Commutator theory based on the theory of internal categorical structures, especially of so called pseudogroupoids. It is motivated by our previous work on internal categories and groupoids in congruence modular varieties.
Central Extensions and Nilpotence of Maltsev Theories
"... Relationship is clarified between the notions of linear extension of algebraic theories, and central extension, in the sense of commutator calculus, of their models. Varieties of algebras turn out to be nilpotent Maltsev precisely when their theories may be obtained as results of iterated linear ext ..."
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Cited by 3 (3 self)
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Relationship is clarified between the notions of linear extension of algebraic theories, and central extension, in the sense of commutator calculus, of their models. Varieties of algebras turn out to be nilpotent Maltsev precisely when their theories may be obtained as results of iterated linear extensions by bifunctors from the so called abelian theories. The latter theories are described; they are slightly more general than theories of modules over a ring.
Baer Invariants In SemiAbelian Categories I: General Theory
 GENERAL THEORY, THEORY AND APPLICATIONS OF CATEGORIES
, 2004
"... Extending the work of Fröhlich, Lue and FurtadoCoelho, we consider the theory of Baer invariants in the context of semiabelian categories. Several exact sequences, relative to a subfunctor of the identity functor, are obtained. We consider a notion of commutator which, in the case of abelianizatio ..."
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Cited by 3 (2 self)
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Extending the work of Fröhlich, Lue and FurtadoCoelho, we consider the theory of Baer invariants in the context of semiabelian categories. Several exact sequences, relative to a subfunctor of the identity functor, are obtained. We consider a notion of commutator which, in the case of abelianization, corresponds to Smith's. The resulting notion of centrality fits into Janelidze and Kelly's theory of central extensions. Finally we propose a notion of nilpotency, relative to a Birkhoff subcategory of a semiabelian category.
Central extensions in Mal’tsev varieties
 Theory Appl. Categ
"... Abstract. We show that every algebraically–central extension in a Mal’tsev variety — that is, every surjective homomorphism f: A− → B whose kernel–congruence is contained in the centre of A, as defined using the theory of commutators — is also a central extension in the sense of categorical Galois t ..."
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Cited by 2 (0 self)
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Abstract. We show that every algebraically–central extension in a Mal’tsev variety — that is, every surjective homomorphism f: A− → B whose kernel–congruence is contained in the centre of A, as defined using the theory of commutators — is also a central extension in the sense of categorical Galois theory; this was previously known only for varieties of Ωgroups, while its converse is easily seen to hold for any congruence–modular variety. 1.
Relative commutator theory in varieties of Ωgroups
 J. Pure Appl. Algebra
"... We introduce a new notion of commutator which depends on a choice of subvariety in any variety of Ωgroups. We prove that this notion encompasses Higgins’s commutator, Fröhlich’s central extensions and the Peiffer commutator of precrossed modules. Keywords: Commutator, Ωgroup, central extension, Pe ..."
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Cited by 2 (1 self)
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We introduce a new notion of commutator which depends on a choice of subvariety in any variety of Ωgroups. We prove that this notion encompasses Higgins’s commutator, Fröhlich’s central extensions and the Peiffer commutator of precrossed modules. Keywords: Commutator, Ωgroup, central extension, Peiffer commutator 0
BAER INVARIANTS IN SEMIABELIAN CATEGORIES II: HOMOLOGY
 THEORY AND APPLICATIONS OF CATEGORIES
, 2004
"... This article treats the problem of deriving the reflector of a semiabelian ..."
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Cited by 2 (1 self)
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This article treats the problem of deriving the reflector of a semiabelian
THE THIRD COHOMOLOGY GROUP CLASSIFIES DOUBLE CENTRAL EXTENSIONS
, 2010
"... We characterise the double central extensions in a semiabelian category in terms of commutator conditions. We prove that the third cohomology group H3 (Z, A) of an object Z with coefficients in an abelian object A classifies the double central extensions of Z by A. ..."
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We characterise the double central extensions in a semiabelian category in terms of commutator conditions. We prove that the third cohomology group H3 (Z, A) of an object Z with coefficients in an abelian object A classifies the double central extensions of Z by A.