A combinatorial-topological shape category for polygraphs

TL;DR

This paper introduces constructible directed complexes and polygraphs, providing a new combinatorial-topological framework for higher categories, with a realisation functor linking them to omega-categories and CW complexes.

Contribution

It defines constructible directed complexes and polygraphs, extending existing structures, and establishes their relation to omega-categories and CW complexes.

Findings

Constructible directed complexes include common higher-categorical shapes.

A realisation functor from constructible polygraphs to omega-categories is developed.

The geometric realisation yields CW complexes with cells corresponding to elements.

Abstract

We introduce constructible directed complexes, a combinatorial presentation of higher categories inspired by constructible complexes in poset topology. Constructible directed complexes with a greatest element, called atoms, encompass common classes of higher-categorical cell shapes, including globes, cubes, oriented simplices, and a large sub-class of opetopes, and are closed under lax Gray products and joins. We define constructible polygraphs to be presheaves on a category of atoms and inclusions, and extend the monoidal structures. We show that constructible directed complexes are a well-behaved subclass of Steiner's directed complexes, which we use to define a realisation functor from constructible polygraphs to omega-categories. We prove that the realisation of a constructible polygraph is a polygraph in restricted cases, and in all cases conditionally to a conjecture. Finally, we define the geometric realisation of a constructible polygraph, and prove that it is a CW complex with one cell for each of its elements.