# Permutads via operadic categories, and the hidden associahedron

## TL;DR

This paper explores the structure of permutads within operadic categories, demonstrating their relation to cellular chains of permutohedra, and establishes their Koszul properties and minimal models, advancing the understanding of algebraic and topological operad theory.

## Contribution

It proves that cellular chains of permutohedra form the minimal model of the terminal permutad and shows the Koszul self-duality of the terminal Per-operad, introducing new perspectives on permutads and operadic categories.

## Key findings

- Cellular chains of permutohedra form the minimal model of the terminal permutad.
- The terminal Per-operad is Koszul self-dual.
- Strongly homotopy permutads are described as algebras of the minimal model.

## Abstract

The present article exploits the fact that permutads (aka shuffle algebras) are algebras over a terminal operad in a certain operadic category Per. In the first, classical part we formulate and prove a claim envisaged by Loday and Ronco that the cellular chains of the permutohedra form the minimal model of the terminal permutad which is moreover, in the sense we define, self-dual and Koszul. In the second part we study Koszulity of Per-operads. Among other things we prove that the terminal Per-operad is Koszul self-dual. We then describe strongly homotopy permutads as algebras of its minimal model. Our paper shall advertise analogous future results valid in general operadic categories, and the prominent role of operadic (op)fibrations in the related theory.