From $(\mathrm{\infty },n)$-categories to $(\mathrm{\infty },\mathrm{\infty })$-categories

There is a well-developed theory of higher categories of finite dimension, in the form of strict $n$-categories in the strict setting and of $(\mathrm{\infty },n)$-categories (for which $n$-complicial sets provide one model) in the weak setting. For higher categories of infinite dimension, however, the picture is subtler and less well understood. In the strict setting, a well-studied notion is that of an $\omega$-category, while in the weak setting Verity introduced the notion of a complicial set.

In both contexts, the category of $n$-dimensional categories embeds into the corresponding category of infinite-dimensional categories. Moreover, this inclusion admits both a left and a right adjoint, yielding two natural ways to approximate an infinite-dimensional category by one of finite dimension. In this talk, we address the natural question of whether an infinite-dimensional category can be recovered from the family of its finite-dimensional approximations.

In the strict setting, an $\omega$-category can be recovered both from its left truncations and from its right truncations. More precisely, the category of $\omega$-categories is equivalent both to the limit of the tower of left truncations and to the limit of the tower of right truncations. The weak setting exhibits a markedly different behaviour. A complicial set can be recovered from its right truncations, but not in general from its left truncations. More precisely, the category of complicial sets is equivalent to the limit of the tower of right truncations [2, 4]. By contrast, the limit of the tower of left truncations embeds as a proper reflective subcategory of the category of complicial sets [1, 3].