Actions and semi-direct products in categories of groups with action

Abstract

Derived actions in the category of groups with action on itself ${\mathbf{Gr}}^{\bullet }$ are defined and described. This category plays a crucial role in the solution of Loday's two problems stated in the literature. A full subcategory of reduced groups with action ${\mathbf{rGr}}^{\bullet }$ of ${\mathbf{Gr}}^{\bullet }$ is introduced, which is not a category of interest but has some properties, which can be applied in the investigation of action representability in this category; these properties are similar to those, which were used in the construction of universal strict general actors in the category of interest. Semi-direct product constructions are given in ${\mathbf{Gr}}^{\bullet }$ and ${\mathbf{rGr}}^{\bullet }$ and it is proved that an action is a derived action in ${\mathbf{Gr}}^{\bullet }$ (resp. ${\mathbf{rGr}}^{\bullet }$) if and only if the corresponding semi-direct product is and object of ${\mathbf{Gr}}^{\bullet }$ (resp. ${\mathbf{rGr}}^{\bullet }$). The results obtained in this paper will be applied in the forthcoming paper on the representability of actions in the category ${\mathbf{rGr}}^{\bullet }$.

12 pages, research paper, LaTeX2e, xypic

References in corpus (2)

• Actors in categories of interest
• Pentactions and action representability in the category of reduced groups with action

Cited by in corpus (1)

• Pentactions and action representability in the category of reduced groups with action