Skeleta and categories of algebras

Abstract

We define a notion of a connectivity structure on an $\infty$-category, analogous to a $t$-structure but applicable in unstable contexts -- such as spaces, or algebras over an operad. This allows us to generalize notions of n-skeleta, minimal skeleta, and cellular approximation from the category of spaces. For modules over an Eilenberg-Mac Lane spectrum, these are closely related to the notion of projective amplitude. We apply these to ring spectra, where they can be detected via the cotangent complex and higher Hochschild homology with coefficients. We show that the spectra $Y(n)$ of chromatic homotopy theory are minimal skeleta for $H\mathbb{F}_2$ in the category of associative ring spectra. Similarly, Ravenel's spectra $T(n)$ are shown to be minimal skeleta for $BP$ in the same way, which proves that these admit canonical associative algebra structures.