# Operadic categories and d\'ecalage

## TL;DR

This paper reconstructs operadic categories using the de9calage comonad, providing a new categorical framework that characterizes these structures as coalgebras and algebras related to this comonad.

## Contribution

It offers a novel description of operadic categories via de9calage, connecting them to comonads and monads in category theory.

## Key findings

- Unary operadic categories are D-coalgebras.
- Operadic categories are algebras for a monad on D-coalgebras.
- The framework generalizes to a modified de9calage comonad.

## Abstract

Batanin and Markl's operadic categories are categories in which each map is endowed with a finite collection of "abstract fibres" -- also objects of the same category -- subject to suitable axioms. We give a reconstruction of the data and axioms of operadic categories in terms of the d\'ecalage comonad D on small categories. A simple case involves unary operadic categories -- ones wherein each map has exactly one abstract fibre -- which are exhibited as categories which are, first of all, coalgebras for the comonad D, and, furthermore, algebras for the monad induced on the category of D-coalgebras by the forgetful-cofree adjunction. A similar description is found for general operadic categories arising out of a corresponding analysis that starts from a "modified d\'ecalage" comonad on the arrow category of Cat.