Diagrammatic sets as a model of homotopy types

TL;DR

This paper introduces diagrammatic sets, a new model for homotopy types based on presheaves over complex shape categories, establishing their homotopical properties and equivalences with classical models.

Contribution

It defines a cofibrantly generated model structure on diagrammatic sets and proves their Quillen equivalences with simplicial sets, advancing higher-dimensional homotopy theory.

Findings

Established a model structure on diagrammatic sets

Proved Quillen equivalences with simplicial sets

Demonstrated monoidal properties with Gray product

Abstract

Diagrammatic sets are presheaves on a rich category of shapes, whose definition is motivated by combinatorial topology and higher-dimensional diagram rewriting. These shapes include representatives of oriented simplices, cubes, and positive opetopes, and are stable under operations including Gray products, joins, suspensions, and duals. We exhibit a cofibrantly generated model structure on diagrammatic sets, as well as two separate Quillen equivalences with the classical model structure on simplicial sets. We construct explicit sets of generating cofibrations and acyclic cofibrations, and prove that the model structure is monoidal with the Gray product of diagrammatic sets.