Group extensions and cohomology theory in cartesian closed categories.
Item
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Title
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Group extensions and cohomology theory in cartesian closed categories.
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Identifier
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AAI9130307
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identifier
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9130307
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Creator
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Danas, George.
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Contributor
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Adviser: Alex Heller
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Date
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1991
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics
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Abstract
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The classical theory of group extensions, due to S. Eilenberg and S. MacLane shows that extensions of a group G by an abelian subgroup A are given by the second cohomology group H{dollar}\sp2{dollar}(G,A). Extensions of a group G by a nonabelian subgroup H give to H the structure of an abstract G-kernel (i.e., G operates externally on H) and the elements of the third cohomology group H{dollar}\sp3{dollar}(G,A) of the group G with coefficient in the G-module A are interpreted as obstructions to extensions of the abstract kernel H by G, where H contains A as its center.;In a parallel development Huebschmann using as tools the notions of crossed modules and that of crossed n-fold extensions showed that in the category of sets the crossed n-fold extensions of A by G constitute an abelian group Opext{dollar}\sp{lcub}\rm n{rcub}{dollar}(G,A) isomorphic to H{dollar}\sp{lcub}\rm n + 1{rcub}{dollar}(G,A). Wu, studying the obstructions of group extensions and H{dollar}\sp3{dollar}(G,A) gives the treatment and the study of the classical results categorically, free of tricky cocycle calculations.;Early attempts to treat in analogous fashion extensions of topological groups led to the understanding only of special cases. Another treatment of extensions of locally compact topological groups was given by Moore. In 1972 Alex Heller studied and gave a more complete analysis of this situation in which he considered the problem of group extensions with abelian kernels and cohomology in topological and simplicial categories.;The problem that we study here is that of group extensions and their relations with cohomology theory of groups in suitable cartesian closed categories. In fact, we prove that for such categories with finite limits and countable colimits and enough projective or injective objects, for example sheaf categories (e.g. simplicial sets), all the above results still hold.;For this purpose it proved necessary to reconstruct, in the context of such categories, some of the basic results of combinatorial group theory, and to redevelop the theory of crossed n-fold extensions in categories possessing injective rather than projective objects. Results along these lines should be useful in the further investigation contemplated.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
