Algebraic exponentiation and internal homology in general categories

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dc.contributor.advisorJanelidze, Gen_ZA
dc.contributor.authorGray, James Richard Andrewen_ZA
dc.date.accessioned2014-12-30T06:41:59Z
dc.date.available2014-12-30T06:41:59Z
dc.date.issued2010en_ZA
dc.descriptionIncludes bibliographical references (p. 101-102).en_ZA
dc.description.abstractWe study two categorical-algebraic concepts of exponentiation:(i) Representing objects for the so-called split extension functors in semi-abelian and more general categories, whose familiar examples are automorphism groups of groups and derivation algebras of Lie algebras. We prove that such objects exist in categories of generalized Lie algebras defined with respect to an internal commutative monoid in symmetric monoidal closed abelian category. (ii) Right adjoints for the pullback functors between D. Bourns categories of points. We introduce and study them in the situations where the ordinary pullback functors between bundles do not admit right adjoints in particular for semi-abelian, protomodular, (weakly) Maltsev, (weakly) unital, and more general categories. We present a number of examples and counterexamples for the existence of such right adjoints. We use the left and right adjoints of the pullback functors between categories of points to introduce internal homology and cohomology of objects in abstract categories.en_ZA
dc.identifier.apacitationGray, J. R. A. (2010). <i>Algebraic exponentiation and internal homology in general categories</i>. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/10519en_ZA
dc.identifier.chicagocitationGray, James Richard Andrew. <i>"Algebraic exponentiation and internal homology in general categories."</i> Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2010. http://hdl.handle.net/11427/10519en_ZA
dc.identifier.citationGray, J. 2010. Algebraic exponentiation and internal homology in general categories. University of Cape Town.en_ZA
dc.identifier.ris TY - Thesis / Dissertation AU - Gray, James Richard Andrew AB - We study two categorical-algebraic concepts of exponentiation:(i) Representing objects for the so-called split extension functors in semi-abelian and more general categories, whose familiar examples are automorphism groups of groups and derivation algebras of Lie algebras. We prove that such objects exist in categories of generalized Lie algebras defined with respect to an internal commutative monoid in symmetric monoidal closed abelian category. (ii) Right adjoints for the pullback functors between D. Bourns categories of points. We introduce and study them in the situations where the ordinary pullback functors between bundles do not admit right adjoints in particular for semi-abelian, protomodular, (weakly) Maltsev, (weakly) unital, and more general categories. We present a number of examples and counterexamples for the existence of such right adjoints. We use the left and right adjoints of the pullback functors between categories of points to introduce internal homology and cohomology of objects in abstract categories. DA - 2010 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 2010 T1 - Algebraic exponentiation and internal homology in general categories TI - Algebraic exponentiation and internal homology in general categories UR - http://hdl.handle.net/11427/10519 ER - en_ZA
dc.identifier.urihttp://hdl.handle.net/11427/10519
dc.identifier.vancouvercitationGray JRA. Algebraic exponentiation and internal homology in general categories. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2010 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/10519en_ZA
dc.language.isoengen_ZA
dc.publisher.departmentDepartment of Mathematics and Applied Mathematicsen_ZA
dc.publisher.facultyFaculty of Scienceen_ZA
dc.publisher.institutionUniversity of Cape Town
dc.subject.otherMathematics and Applied Mathematicsen_ZA
dc.titleAlgebraic exponentiation and internal homology in general categoriesen_ZA
dc.typeDoctoral Thesis
dc.type.qualificationlevelDoctoral
dc.type.qualificationnamePhDen_ZA
uct.type.filetypeText
uct.type.filetypeImage
uct.type.publicationResearchen_ZA
uct.type.resourceThesisen_ZA
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