A strengthened $(\infty, n)$-categorical pasting theorem

TL;DR

This paper generalizes Campion's pasting theorem for $( abla, n)$-categories to broader classes of polygraphs, including semi-simplicial sets and regular 3-polygraphs, enhancing the understanding of higher categorical structures.

Contribution

It extends the pasting theorem to directed complexes with frame-acyclic molecules and compares them with Henry's regular polygraphs, showing their equivalence up to dimension 3.

Findings

Pasting theorem applies to polygraphs presented by semi-simplicial sets.

Compatibility with Gray tensor product for certain directed complexes.

Pasting theorem also applies to regular 3-polygraphs.

Abstract

We extend Campion's pasting theorem for $(\infty, n)$-categories to a larger class of polygraphs, called the directed complexes with frame-acyclic molecules. It follows, for instance, that this pasting theorem applies to any polygraph presented by a semi-simplicial set, and that a large subclass of directed complexes with frame-acyclic molecules is compatible with the Gray tensor product. We also set up a comparison between directed complexes and Henry's regular polygraphs, and show that they coincide up to dimension $3$. As a corollary of our main results, the pasting theorem also applies to the class of regular $3$-polygraphs.