2016/12/21 by Maxime Lucas, Lucas, Maxime
Mathematics · Medicine · #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications
paper · pdf · doi:10.48550/arxiv.1612.07050
openalex publication_date 2018/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we introduce the notion of cubical (\ω,p)-categories,\nfor p \∈ mathbb N \∪ \ω . We show that the equivalence between\nglobular and groupoid \ω-categories proven by Al-Agl, Brown and Steiner\ninduces an equivalence between globular and cubical (\ω,p)-categories for\nall p \≥ 0. In particular we recover in a more explicit fashion the\nequivalence between globular and cubical groupoids proven by Brown and Higgins.\n We also define the notion of (\ω,p)-augmented directed complexes, and\nshow that Steiner's adjunction between augmented directed complexes and\nglobular \ω-categories induces adjunctions between (\ω,p)-augmented\ndirected complexes and both globular and cubical (\ω,p)-categories.\n Combinatorially, the difficulty lies in defining the appropriate notion of\ninvertibility for a cell in a cubical \ω-category. We investigate three\nsuch possible definitions and the relationship between them. We show that\ncubical (\ω,1)-categories have a natural structure of symmetric cubical\ncategories. We give an explicit description of the notions of lax, oplax and\npseudo transfors between cubical categories, the latter making use of the\nnotion of invertible cell defined previously.\n