Categories of operators and actions of group operads

TL;DR

This paper introduces a new model for multicategories with symmetries based on Zhang's group operads, utilizing embeddings into crossed interval groups and representing symmetric structures as internal presheaves in double categories.

Contribution

It develops a novel framework connecting multicategories, group operads, and double categories, expanding the understanding of symmetries in categorical structures.

Findings

Established a fully faithful embedding of group operads into crossed interval groups.

Demonstrated that each multicategory induces a fibration over a quotient of group operads.

Represented symmetric structures as internal presheaves in a double category.

Abstract

We propose a new model for multicategories with symmetries with respect to Zhang's group operads. The fully faithful embedding of the category of group operads into that of crossed interval groups is made use of, and it is shown that every multicategory gives rise to a fibration, in a sense, over a quotient of the total category of group operads. The symmetric structures can be presented as structures of internal presheaves over a category internal to the category of small categories, in other words a double category.