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Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms
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...
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| Format: | Thesis |
| Language: | English |
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Department of Mathematics and Applied Mathematics
2016
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| _version_ | 1867613170836701184 |
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| access_status_str | Open Access |
| author | Ramasu, Pako |
| author2 | Janelidze, George |
| author_browse | Janelidze, George Ramasu, Pako |
| author_facet | Janelidze, George Ramasu, Pako |
| author_sort | Ramasu, Pako |
| collection | Thesis |
| description | 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. |
| format | Thesis |
| id | oai:open.uct.ac.za:11427/20248 |
| institution | University of Cape Town (South Africa) |
| language | eng |
| last_indexed | 2026-06-10T12:31:53.390Z |
| license_str | Not specified — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from UCTD — University of Cape Town Open Access Repository |
| publishDate | 2016 |
| publishDateRange | 2016 |
| publishDateSort | 2016 |
| publisher | Department of Mathematics and Applied Mathematics |
| publisherStr | Department of Mathematics and Applied Mathematics |
| record_format | dspace |
| source_str | UCTD — University of Cape Town Open Access Repository |
| spelling | oai:open.uct.ac.za:11427/20248 Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms Ramasu, Pako Janelidze, George Mathematics 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. 2016-07-07T09:52:07Z 2016-07-07T09:52:07Z 2015 Doctoral Thesis Doctoral PhD http://hdl.handle.net/11427/20248 eng application/pdf Department of Mathematics and Applied Mathematics Faculty of Science University of Cape Town |
| spellingShingle | Mathematics Ramasu, Pako Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms |
| thesis_degree_str | Doctoral |
| title | Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms |
| title_full | Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms |
| title_fullStr | Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms |
| title_full_unstemmed | Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms |
| title_short | Internal monoid actions in a cartesian closed category and higher-dimensional group automorphisms |
| title_sort | internal monoid actions in a cartesian closed category and higher dimensional group automorphisms |
| topic | Mathematics |
| url | http://hdl.handle.net/11427/20248 |
| work_keys_str_mv | AT ramasupako internalmonoidactionsinacartesianclosedcategoryandhigherdimensionalgroupautomorphisms |