2021/02/19 by Camell Kachour, Kachour, Camell
Mathematics · #18C10 #18C15 #18C30 #18G55 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2102.09787
openalex publication_date 2021/02/19 · openalex created_date 2023/02/24 · openalex updated_date 2026/07/28
In these notes we describe models of globular weak (∞,m)-categories (m∈ℕ) in the Grothendieck style, i.e for each m∈ℕ we define a globular coherator Θ∞_\mathbbMm whose set-models are globular weak (∞,m)-categories. Then we describe the combinatorics of the small category Θ0 whose objects are cubical pasting diagrams and whose morphisms are morphisms of cubical sets. This provides an accurate description of the monad, on the category of cubical sets (without degeneracies and connections), of cubical strict ∞-categories with connections. We prove that it is a cartesian monad, solving a conjecture in \citecamark-cub-1. This puts us in a position to describe the cubical coherator Θ∞W whose set-models are cubical weak ∞-categories with connections and the cubical coherator Θ∞W0 whose set-models are cubical weak ∞-groupoids with connections.