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Skeleta and categories of algebras

Jonathan Beardsley, Tyler Lawson

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.

Skeleta and categories of algebras

Abstract

We define a notion of a connectivity structure on an -category, analogous to a -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 of chromatic homotopy theory are minimal skeleta for in the category of associative ring spectra. Similarly, Ravenel's spectra are shown to be minimal skeleta for in the same way, which proves that these admit canonical associative algebra structures.

Paper Structure

This paper contains 38 sections, 75 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Main applications
  3. Outline
  4. Notation and conventions
  5. Acknowledgements
  6. Connectivity structures
  7. Definitions
  8. Examples
  9. Basic properties of connectivity
  10. Stable connectivity structures
  11. Stabilization
  12. Cartesian squares
  13. Skeleta
  14. Skeletal objects
  15. Colimits of skeletal objects
  16. ...and 23 more sections

Key Result

Proposition 2.16

Suppose that we have a retract diagram $A \to X \to A$. Then the map $A \to X$ is $k$-connected if and only if the map $X \to A$ is $(k+1)$-connected.

Theorems & Definitions (209)

  • Definition
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • ...and 199 more