A combinatorial-topological shape category for polygraphs
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.
Cite
@article{arxiv.1806.10353,
title = {A combinatorial-topological shape category for polygraphs},
author = {Amar Hadzihasanovic},
journal= {arXiv preprint arXiv:1806.10353},
year = {2019}
}
Comments
v2: Major revision of v1, which contained some erroneous statements and proofs. Overhaul of terminology to clarify relation with works of Steiner and Henry. For details, see Introduction > Errata and notes on an earlier version