Reedy categories which encode the notion of category actions

TL;DR

This paper introduces new categorical structures called Reedy and generalized Reedy categories that encode actions of categories on other categories and operads, providing a novel perspective on group actions within category theory.

Contribution

It defines categories that encode actions of categories on categories and operads, establishing their Reedy and generalized Reedy structures, respectively.

Findings

Reedy category structure for categories acting on categories

Generalized Reedy structure for categories acting on operads

New framework for understanding group actions in category theory

Abstract

We study a certain type of action of categories on categories and on operads. Using the structure of the categories {\Delta} and {\Omega} governing category and operad structures, respectively, we define categories which instead encode the structure of a category acting on a category, or a category acting on an operad. We prove that the former has the structure of an elegant Reedy category, whereas the latter has the structure of a generalized Reedy category. In particular, this approach gives a new way to regard group actions on categories and on operads.