$ω$-equifibrations between strict and weak $ω$-categories

TL;DR

This work defines $\omega$-equifibrations as the weak $\omega$-categorical analogue of isofibrations and establishes a robust right lifting property (RLP) framework for them via a generating set $J$ built from a coherently generated weak $\omega$-category $\mathcal{E}^1$ and its relation to Ozornova–Rovelli's coherent walking $\omega$-equivalence $\widehat{\omega\mathcal{E}}$. It develops a marked weak $\omega$-category approach to internalize lifting witnesses and proves that, in the strict setting, $\omega$-equifibrations coincide with folk fibrations; it then provides two explicit, equivalent constructions of $\mathcal{E}_{\mathcal{F}}^n$ and shows the strict reflection of $\mathcal{E}_{\mathcal{F}}^1$ is isomorphic to $\widehat{\omega\mathcal{E}}$, aligning with HLOR’s contractibility results. The paper also connects the weak and strict theories via suspension, clarifies presentations of the witnesses, and demonstrates how the folk model structure on strict $\omega$-categories captures $\omega$-equifibrations. Overall, it bridges higher-categorical coherence, model-categorical fibrations, and the folk model structure in the realm of $\omega$-categories, extending homotopical methods to the weak setting.

Abstract

We study $ω$-equifibrations between weak $ω$-categories in the sense of Batanin--Leinster. We define $ω$-equifibrations as a natural weak $ω$-categorical analogue of isofibrations between categories, and show that they can be characterised via the right lifting property with respect to a suitable set $J$ of strict $ω$-functors. The definition of $J$ involves the construction of a certain weak $ω$-category $\mathcal{E}^1$ which, roughly speaking, is freely generated by an equivalence 1-cell in a ``coherent'' manner. We show that the strict version of $\mathcal{E}^1$ coincides with Ozornova and Rovelli's coherent walking $ω$-equivalence $\widehat{ω\mathcal{E}}$. The $ω$-equifibrations between strict $ω$-categories coincide with the fibrations in the folk model structure.

Figures (3)

• Figure 1: Lifting v
• Figure 2: Lifting v
• Figure 3: Lifting q (with ˇλ, α and ρ omitted)

Theorems & Definitions (137)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3: FHM1; see Lafont_Metayer_Worytkiewicz_folk_model_str_omega_cat for the strict case
• Proposition 4: FHM1; see Lafont_Metayer_Worytkiewicz_folk_model_str_omega_cat for the strict case
• Proposition 5: FHM1
• Proposition 6: FHM1
• Proposition 7: FHM1
• Proposition 8: FHM1; see Lafont_Metayer_Worytkiewicz_folk_model_str_omega_cat for the strict case