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

Doctoral Thesis


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University of Cape Town

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.