$\omega$-equifibrations between strict and weak $\omega$-categories

We study $\omega$-equifibrations between weak $\omega$-categories in the sense of Batanin--Leinster. We define $\omega$-equifibrations as a natural weak $\omega$-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 $\omega$-functors. The definition of $J$ involves the construction of a certain weak $\omega$-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 $\omega$-equivalence $\widehat{\omega\mathcal{E}}$. The $\omega$-equifibrations between strict $\omega$-categories coincide with the fibrations in the folk model structure.