# Spectral algebra models of unstable v_n-periodic homotopy theory

## TL;DR

This paper surveys spectral algebra models of unstable v_n-periodic homotopy types, extending rational homotopy theory and comparing three different approaches with relevant spectral algebra concepts.

## Contribution

It introduces a generalized spectral algebra modeling framework for unstable v_n-periodic homotopy types and discusses three recent approaches, including original and conceptual methods.

## Key findings

- Provides a comprehensive survey of spectral algebra models
- Connects spectral algebra models with key concepts like Andre9-Quillen cohomology
- Highlights differences and similarities among recent approaches

## Abstract

We give a survey of a generalization of Quillen-Sullivan rational homotopy theory which gives spectral algebra models of unstable v_n-periodic homotopy types. In addition to describing and contextualizing our original approach, we sketch two other recent approaches which are of a more conceptual nature, due to Arone-Ching and Heuts. In the process, we also survey many relevant concepts which arise in the study of spectral algebra over operads, including topological Andr\'e-Quillen cohomology, Koszul duality, and Goodwillie calculus.