Limits and colimits of crossed groups

TL;DR

This paper explores the theory of crossed groups across various categories, analyzing their structural properties, such as limits, colimits, and classification, to deepen understanding beyond the simplicial case.

Contribution

It extends the concept of crossed groups to arbitrary categories and studies their categorical properties, including local presentability, monadicity, and classification of crossed interval groups.

Findings

Category of crossed groups is locally presentable.

Monadicity of the crossed groups functor established.

Classification of crossed interval groups provided.

Abstract

Although the notion of crossed groups was originally introduced only in the simplicial case, the definition makes sense in the other categories. For instance, Batanin and Markl studied crossed interval groups to investigate symmetries on the Hochschild cohomology in view of operads. The aim of this paper is to make a comprehensive understanding of crossed groups for arbitrary base categories. In particular, we focus on the local presentability of the category of crossed groups, monadicity, and the basechange theorem along certain sorts of functors. The paper also contains the classification of crossed interval groups, which Batanin and Markl concerned about.