Pita factorisation in operadic categories

TL;DR

This paper introduces the pita factorisation in operadic categories, develops a theory to handle its non-orthogonal nature, and proves the coherence of the operadic nerve, with applications to simplicial approaches in operad theory.

Contribution

It defines pita factorisation in operadic categories and establishes the coherence of the operadic nerve, advancing the understanding of their simplicial structures.

Findings

Pita factorisation is unique but does not form an orthogonal system.

The pita nerve is an oplax simplicial object in Cat.

Operadic nerve is coherent, and becomes a decomposition space when quasibijections are invertible.

Abstract

In strictly factorisable operadic categories, every morphism $f$ factors uniquely as $f=\eta_f \circ \pi_f$ where $\eta_f$ is order-preserving and $\pi_f$ is a quasi\-bijection that is order-preserving on the fibres of $\eta_f$. We call it the pita factorisation. In this paper we develop some general theory to compensate for the fact that generally pita factorisations do not form an orthogonal factorisation system. The main technical result states that a certain simplicial object in Cat, called the pita nerve, is oplax (rather than strict as it would be for an orthogonal factorisation system). The main application is the result that the so-called operadic nerve of any operadic category is coherent. This result is a key ingredient in the simplicial approach to operadic categories developed in the `main paper' \cite{Batanin-Kock-Weber:mainpaper}, which motivated the present paper. We also show that in the important case where quasibijections are invertible, the pita nerve is a decomposition space.