Weak $\mathrm{\infty }$-categories via fat Delta

Understanding higher categories boils down to understanding their coherence complexity. In a nutshell, coherences are equations between operations such as composition and units, and may be strict (genuine equality) or weak (up to higher homotopies). Simplicial methods provide some of the most effective tools for modelling these objects. However, when these techniques are used to encode higher categorical structures, the degeneracy relations enforce strict coherence conditions on units. While often harmless, this rigidity becomes problematic in contexts where weak unit structures are essential, notably in the study of the cobordism $\mathrm{\infty }$-category [5] and in Simpson’s conjectures [1]. To address this issue, J. Kock [2] introduced the category $\underset{¯}{\mathrm{\Delta }}$, named fat Delta, as a variant of the simplex category that allows degeneracies to be treated up to homotopy, at the cost of a localisation at the class of vertical morphisms. In this work, we develop a model of weak $\mathrm{\infty }$-categories based on $\underset{¯}{\mathrm{\Delta }}$-spaces, analogous to the role played by complete Segal spaces [3] in the simplicial setting. Pursuing J. Kock’s original idea of using $\underset{¯}{\mathrm{\Delta }}$ to approach Simpson’s conjectures, we aim to bring new insights to the study of these conjectures and more generally to the theory of weak $\mathrm{\infty }$-categories.

We introduce the notion of fat $\mathrm{\infty }$-category, defined as simplicial presheaf over $\underset{¯}{\mathrm{\Delta }}$, a.k.a. $\underset{¯}{\mathrm{\Delta }}$-space, satisfying appropriate vertical, Segal, and completeness conditions. The vertical condition plays a central role: it allows degeneracy data, and in particular units, to be reconstructed up to homotopy, therefore yielding weak unit structures. These conditions are extracted as left Bousfield localisations of the Reedy model structure on $\underset{¯}{\mathrm{\Delta }}$-spaces with respect to the Kan–Quillen model structure on simplicial sets. In particular, we can show the following.

We further explore alternative characterisations of the conditions. By adapting the notions of horn and saturation [6] to the $\underset{¯}{\mathrm{\Delta }}$-space context, we provide a reformulation of the vertical and Segal conditions in terms of horns, and of the completeness in terms of saturation.

Finally, we discuss ongoing work comparing fat $\mathrm{\infty }$-categories with other established models of weak $\mathrm{\infty }$-categories. In particular, we outline a conjectural Quillen equivalence with Harpaz’s model of quasi-unital $\mathrm{\infty }$-categories [4]. This model is based on marked semisimplicial spaces, which allows us to form a natural functor by assigning to each $\underset{¯}{\mathrm{\Delta }}$-space its underlying semisimplicial space and space of markings. Conversely, by appropriately using the $f$-degenerate marked semisimplex introduced in [4], one can construct the functor in the opposite direction.