Skeleta and categories of algebras

Abstract

We define a notion of a connectivity structure on an ∞-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 HF 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.The authors would like to thank Jeremy Hahn for being a close part of questions and discussions that instigated this project, and Clark Barwick, Tim Campion, Ian Coley, and Denis Nardin for help with material related to this paper.This work is an outgrowth of conversations that took place while the authors were attending the workshop "Derived algebraic geometry and chromatic homotopy theory" during the Homotopy Harnessing Higher Structures program at the Isaac Newton Institute (EPSRC grant numbers EP/K032208/1 and EP/R014604/1).

Generalized symmetries of topological field theories

“…It factors, moreover, as X → ΩΣX → Ωτ ≤n+1 ΣX. where we have seen that the first morphism is 2k-connected and where the second morphism is n-connected as discussed in §2.4 of [BL21]. So either 2k < n and the unit is n-truncated and manifestly 2k-connected, or 2k ≥ n and the unit is n-truncated and n-connected, in which case it is an isomorphism and so 2k-connected as well.…”