Representable diagrammatic sets as a model of weak higher categories

Abstract

Developing an idea of Kapranov and Voevodsky, we introduce a model of weak omega-categories based on directed complexes, combinatorial presentations of pasting diagrams. We propose this as a convenient framework for higher-dimensional rewriting. We define diagrammatic sets to be presheaves on a category of directed complexes presenting pasting diagrams with spherical boundaries. Diagrammatic sets have structural face and degeneracy operations, but no structural composition. We define a notion of equivalence cell in a diagrammatic set, and say that a diagrammatic set is representable if all pasting diagrams with spherical boundaries are connected to individual cells -- their weak composites -- by a higher-dimensional equivalence cell. We develop the basic theory of representable diagrammatic sets (RDSs), and prove that equivalence cells satisfy the expected properties in an RDS. We study nerves of strict omega-categories as RDSs and prove that 2-truncated RDSs are equivalent to bicategories. Finally, we connect the combinatorics of diagrammatic sets and simplicial sets, and prove a version of the homotopy hypothesis for "groupoidal" RDSs: there exists a geometric realisation functor for diagrammatic sets, inducing isomorphisms between combinatorial and classical homotopy groups, which has a homotopical right inverse.

arXiv commentv1: 90 pages

References in corpus (9)

• The Classification of Two-Dimensional Extended Topological Field Theories
• On the Unicity of the Homotopy Theory of Higher Categories
• The algebra of entanglement and the geometry of composition
• Polygraphs of finite derivation type
• Weak model categories in classical and constructive mathematics
• Globular: an online proof assistant for higher-dimensional rewriting
• Regular polygraphs and the Simpson conjecture
• Associative $n$-categories
• Positive Opetopes with Contractions form a Test Category