Actions and semi-direct products in categories of groups with action
Tamar Datuashvili, Tun\c{c}ar \c{S}ahan

TL;DR
This paper defines derived actions and semi-direct products in categories of groups with action, addressing key problems and establishing conditions for action representability in these categories.
Contribution
It introduces derived actions, semi-direct product constructions, and explores properties of reduced groups with action, advancing understanding of action representability in these categories.
Findings
Derived actions are characterized in categories of groups with action.
Semi-direct products correspond to derived actions within these categories.
Properties of reduced groups with action facilitate future studies on action representability.
Abstract
Derived actions in the category of groups with action on itself 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 of 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 and and it is proved that an action is a derived action in (resp. ) if and only if the corresponding semi-direct product is and object…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology · Finite Group Theory Research
