  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.
