2022/07/08 by Brandon Doherty, Doherty, Brandon · 1 citation
Computer Science · Mathematics · #18N40 #18N60 #18N65 #55U35 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2207.03636
openalex publication_date 2022/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that each of the model structures for (n-trivial, saturated) comical sets on the category of marked cubical sets having only faces and degeneracies (without connections) is Quillen equivalent to the corresponding model structure for (n-trivial, saturated) complicial sets on the category of marked simplicial sets, as well as to the corresponding comical model structures on cubical sets with connections. As a consequence, we show that the cubical Joyal model structure on cubical sets without connections is equivalent to its analogues on cubical sets with connections and to the Joyal model structure on simplicial sets. We also show that any comical set without connections may be equipped with connections via lifting, and that this can be done compatibly on the domain and codomain of any fibration or cofibration of comical sets.