$\ddot{I}‰$-equifibrations between strict and weak $\ddot{I}‰$-categories

Abstract

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

33 pages. Comments welcome!