nLab simplicial model for weak omega-categories

Idea

While an algebraic definition of strict ∞-categories is comparatively straightforward, algebraic definitions of fully weak $\omega$-categories (aka $\infty$-categories) are difficult to work with (although some definitions exist, such as those of Batanin, Trimble, Leinster, and Penon).

However, just like strict $\omega$-categories have a simplicial nerve – a complicial set – induced by the orientals, and just like the category of strict $\omega$-categories is equivalent to the category of complicial sets, one expects that every weak $\omega$-category naturally has a simplicial nerve and that the theory of algebraically defined weak $\omega$-categories is equivalent to the theory of the simplicial sets that arise as their nerves. In fact, one can hope that the theory is simpler in the weak case: complicial sets are simplicial sets equipped with extra structure (a stratification representing the chosen ‘strict’ composites), while in the nerve of a weak $\omega$-category this structure might be reducible to a property.

In the context of simplicial models for weak $\omega$-categories the goal is to characterize those simplicial sets (or stratified simplicial sets) which should arise as nerves of algebraically defined weak $\omega$-categories and thus provide a geometric definition of higher category generalizing the familiar simplicial description of $\omega$-groupoids as Kan complexes and of $(\omega,1)$-categories as quasi-categories to general higher categories.

In effect, the goal is to define a weak $\omega$-category to be a certain sort of (stratified) simplicial set. One could then hope to prove that these are precisely the nerves of weak $\omega$-categories defined in some other way.

To distinguish the study of weak $\omega$-categories in terms of their presumed nerves from the study of their would-be algebraic descriptions people speak of simplicial weak ∞-categories, see in particular the articles by Dominic Verity referenced below. One should just beware that in this context simplicial weak ∞-category is not meant as simplicial object in the category of weak $\omega$-categories.

This program was originally begun by Ross Street and has been carried forward by Dominic Verity with the theory of weak complicial sets. It is expected that the (nerves of) weak $\omega$-categories will be weak complicial sets satisfying an extra “saturation” condition ensuring that “every equivalence is thin.”

References

• Dominic Verity, Weak complicial sets, a simplicial weak omega-category theory. Part I: basic homotopy theory (arXiv)

• Dominic Verity, Weak complicial sets, a simplicial weak omega-category theory. Part II: nerves of complicial Gray-categories (pdf)