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Galois theory for braided tensor categories and the modular closure
 Adv. Math
, 2000
"... Given a braided tensor ∗category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗category with conjugates and an irreducible unit. (A ∗category is a category enriched over VectC ..."
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Cited by 31 (7 self)
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Given a braided tensor ∗category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗category with conjugates and an irreducible unit. (A ∗category is a category enriched over VectC with positive ∗operation.) A Galois correspondence is established between intermediate categories sitting between C and C ⋊S and closed subgroups of the Galois group Gal(C⋊S/C) = AutC(C⋊S) of C, the latter being isomorphic to the compact group associated to S by the duality theorem of Doplicher and Roberts. Denoting by D ⊂ C the full subcategory of degenerate objects, i.e. objects which have trivial monodromy with all objects of C, the braiding of C extends to a braiding of C⋊S iff S ⊂ D. Under this condition C⋊S has no nontrivial degenerate objects iff S = D. If the original category C is rational (i.e. has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category C ≡ C ⋊ D is called the modular closure of C since in the rational case it is modular, i.e. gives rise to a unitary representation of the modular group SL(2, Z). (In passing we prove that every braided tensor ∗category with conjugates automatically is a ribbon category, i.e. has a twist.) If all simple objects of S have dimension one the structure of the category C ⋊ S can be clarified quite explicitly in terms of group cohomology. 1
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 14 (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.
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 lies ..."
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Cited by 8 (6 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...
BOOLEAN GALOIS THEORIES
"... Abstract. We develop a general approach to adjunctions satisfying the admissibility condition useful for Boolean Galois Theories, i. e. for Galois Theories whose Galois (pre)groupoids are profinite. Various examples and applications are briefly described. ..."
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Abstract. We develop a general approach to adjunctions satisfying the admissibility condition useful for Boolean Galois Theories, i. e. for Galois Theories whose Galois (pre)groupoids are profinite. Various examples and applications are briefly described.
SEVERAL CONSTRUCTIONS FOR FACTORIZATION SYSTEMS DALI ZANGURASHVILI
"... Abstract. The paper develops the previously proposed approach to constructing factorization systems in general categories. This approach is applied to the problem of finding conditions under which a functor (not necessarily admitting a right adjoint) "reflects"factorization system ..."
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Abstract. The paper develops the previously proposed approach to constructing factorization systems in general categories. This approach is applied to the problem of finding conditions under which a functor (not necessarily admitting a right adjoint) &quot;reflects&quot;factorization systems. In particular, a generalization of the wellknown CassidyH'ebertKelly factorization theorem is given. The problem of relating a factorization system toa pointed endofunctor is considered. Some relevant examples in concrete categories are given. 1. Introduction The problem of relating a factorization system on a category C to an adjunction C I / / XHoo, (1.1) was thoroughly considered by C. Cassidy, M. H'ebert and G. M. Kelly in [CHK]. The wellknown theorem of these authors states that in the case of a finitely wellcomplete category C the pair of morphism classes\Gamma
SEVERAL CONSTRUCTIONS FOR FACTORIZATION SYSTEMS DALI ZANGURASHVILI
"... Abstract. The paper develops the previously proposed approach to constructing factorization systems in general categories. This approach is applied to the problem of finding conditions under which a functor (not necessarily admitting a right adjoint) “reflects” factorization systems. In particular, ..."
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Abstract. The paper develops the previously proposed approach to constructing factorization systems in general categories. This approach is applied to the problem of finding conditions under which a functor (not necessarily admitting a right adjoint) “reflects” factorization systems. In particular, a generalization of the wellknown CassidyHébertKelly factorization theorem is given. The problem of relating a factorization system to a pointed endofunctor is considered. Some relevant examples in concrete categories are
Extended Galois Theory And Dissonant Morphisms
"... For a given Galois structure on a category C and an effective descent morphism p : E!B in C we describe the category of socalled weakly split objects over (E; p) in terms of internal actions of the Galois (pre)groupoid of (E; p) with an additional structure. We explain that this generates various ..."
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For a given Galois structure on a category C and an effective descent morphism p : E!B in C we describe the category of socalled weakly split objects over (E; p) in terms of internal actions of the Galois (pre)groupoid of (E; p) with an additional structure. We explain that this generates various known results in categorical Galois theory and in particular two results of M. Barr and R. Diaconescu [BD]. We also give an elaborate list of examples and applications.
Preprint Number 10–02 COMPREHENSIVE FACTORIZATION AND UNIVERSAL ICENTRAL EXTENSIONS IN THE MAL’CEV CONTEXT
"... Abstract: We show that, under suitable left exact conditions on a reflection functor I, the construction of the associated universal Icentral extension is reduced to the comprehensive factorization of a specific internal functor. This observation produces some existence conditions which hold in pa ..."
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Abstract: We show that, under suitable left exact conditions on a reflection functor I, the construction of the associated universal Icentral extension is reduced to the comprehensive factorization of a specific internal functor. This observation produces some existence conditions which hold in particular for any reflection from a Mal’cev variety to any Birkhoff subvariety.
Galois Theory for Braided Tensor Categories and the Modular Closure
, 1999
"... Given a braided tensor Vcategory C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C < S. This construction yields a tensor Vcategory with conjugates and an irreducible unit. (A Vcategory is a category enriched over Ve ..."
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Given a braided tensor Vcategory C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C < S. This construction yields a tensor Vcategory with conjugates and an irreducible unit. (A Vcategory is a category enriched over VectC with positive Voperation.) A Galois correspondence is established between intermediate categories sitting between C and C < S and closed subgroups of the Galois group Gal(C < SC)=AutC(C < S) of C, the latter being isomorphic to the compact group associated with S by the duality theorem of Doplicher and Roberts. Denoting by D/C the full subcategory of degenerate objects, i.e., objects which have trivial monodromy with all objects of C, the braiding of C extends to a braiding of C < S iff S/D. Under this condition, C < S has no nontrivial degenerate objects iff S=D. If the original category C is rational (i.e., has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category C #C < D is called the modular closure of C since in the rational case it is modular, i.e., gives rise to a unitary representation of the modular group SL(2, Z). If all simple objects of S have dimension one the structure of the category C < S can be clarified quite explicitly in terms of group cohomology. 2000 Academic Press 1.