Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms

 

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dc.contributor.advisor Janelidze, George en_ZA
dc.contributor.author Ramasu, Pako en_ZA
dc.date.accessioned 2016-07-07T09:52:07Z
dc.date.available 2016-07-07T09:52:07Z
dc.date.issued 2015 en_ZA
dc.identifier.citation Ramasu, P. 2015. Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms. University of Cape Town. en_ZA
dc.identifier.uri http://hdl.handle.net/11427/20248
dc.description.abstract The notion of cat¹-group which was introduced by Loday is equivalent to the notions of crossed module and of internal category in the category of groups. This notion of cat¹-groups and their morphisms admits natural generalization to catⁿ-groups, which give rise to n-fold categories in the category of groups. There is also a characterization of catⁿ-groups in terms of crossed n-cubes which was given by Ellis and Steiner. The category Catⁿ (Groups) of internal n-fold categories in the category of groups is a cartesian closed category, however given an object X in Catⁿ (Groups), calculating corresponding action representing object Aut (X) directly would require an enormous calculations. The main purpose of the thesis is to describe that object avoiding such calculations as much as possible. The main tool used in the thesis, apart from the theory of cartesian closed categories, is Loday's theory of catⁿ-groups. We de ne a catⁿ-group X as an additive Mₙ-group X , and then construct the corresponding Aut (X), where Mₙ is a monoid. Since the category of catⁿ-groups is equivalent to Catⁿ (Groups) and since the cartesian closed category Sets Mₙ of Mₙ-sets is much easier to handle than the cartesian closed category of n-fold categories, we shall work just with catⁿ-groups. To assert that, Aut (X) is an action representing object in Sets Mₙ , is to as- sert that, there is a canonical bijection between B-actions of catⁿ-group B on X and the internal group homomorphism B --> Aut (X). Thus, we confirm the construction of Aut (X) by establishing that bijection. Finally, as one of the results of this work, we give the comparison between our cat¹-group Aut (X) and Norrie's actor crossed module (D (G;Z) ;Aut (Z;G;p) w) of a crossed module (Z;G;p) in dimension one. en_ZA
dc.language.iso eng en_ZA
dc.subject.other Mathematics en_ZA
dc.title Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms en_ZA
dc.type Doctoral Thesis
uct.type.publication Research en_ZA
uct.type.resource Thesis en_ZA
dc.publisher.institution University of Cape Town
dc.publisher.faculty Faculty of Science en_ZA
dc.publisher.department Department of Mathematics and Applied Mathematics en_ZA
dc.type.qualificationlevel Doctoral
dc.type.qualificationname PhD en_ZA
uct.type.filetype Text
uct.type.filetype Image
dc.identifier.apacitation Ramasu, P. (2015). <i>Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms</i>. (Thesis). University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/20248 en_ZA
dc.identifier.chicagocitation Ramasu, Pako. <i>"Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms."</i> Thesis., University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2015. http://hdl.handle.net/11427/20248 en_ZA
dc.identifier.vancouvercitation Ramasu P. Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms. [Thesis]. University of Cape Town ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2015 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/20248 en_ZA
dc.identifier.ris TY - Thesis / Dissertation AU - Ramasu, Pako AB - The notion of cat¹-group which was introduced by Loday is equivalent to the notions of crossed module and of internal category in the category of groups. This notion of cat¹-groups and their morphisms admits natural generalization to catⁿ-groups, which give rise to n-fold categories in the category of groups. There is also a characterization of catⁿ-groups in terms of crossed n-cubes which was given by Ellis and Steiner. The category Catⁿ (Groups) of internal n-fold categories in the category of groups is a cartesian closed category, however given an object X in Catⁿ (Groups), calculating corresponding action representing object Aut (X) directly would require an enormous calculations. The main purpose of the thesis is to describe that object avoiding such calculations as much as possible. The main tool used in the thesis, apart from the theory of cartesian closed categories, is Loday's theory of catⁿ-groups. We de ne a catⁿ-group X as an additive Mₙ-group X , and then construct the corresponding Aut (X), where Mₙ is a monoid. Since the category of catⁿ-groups is equivalent to Catⁿ (Groups) and since the cartesian closed category Sets Mₙ of Mₙ-sets is much easier to handle than the cartesian closed category of n-fold categories, we shall work just with catⁿ-groups. To assert that, Aut (X) is an action representing object in Sets Mₙ , is to as- sert that, there is a canonical bijection between B-actions of catⁿ-group B on X and the internal group homomorphism B --> Aut (X). Thus, we confirm the construction of Aut (X) by establishing that bijection. Finally, as one of the results of this work, we give the comparison between our cat¹-group Aut (X) and Norrie's actor crossed module (D (G;Z) ;Aut (Z;G;p) w) of a crossed module (Z;G;p) in dimension one. DA - 2015 DB - OpenUCT DP - University of Cape Town LK - https://open.uct.ac.za PB - University of Cape Town PY - 2015 T1 - Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms TI - Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms UR - http://hdl.handle.net/11427/20248 ER - en_ZA


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