# An Introduction to Higher Categorical Algebra

## TL;DR

This survey introduces higher categorical algebra, focusing on symmetric monoidal stable $
abla$-categories, ring spectra, modules, and deformation theory within the $
abla$-categorical framework, highlighting developments by Lurie.

## Contribution

It provides a comprehensive overview of algebraic structures in $
abla$-categories, emphasizing key concepts like spectra, modules, localization, and deformation theory, based on Lurie's work.

## Key findings

- Introduction of symmetric monoidal stable $
abla$-categories
- Explanation of ring spectra and module categories
- Overview of localization, completion, and deformation theory

## Abstract

This article is a survey of algebra in the $\infty$-categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing symmetric monoidal stable $\infty$-categories, such as the derived $\infty$-category of a commutative ring, before turning to our main example, the $\infty$-category of spectra. We then go on to consider ring spectra and their $\infty$-categories of modules, as well as basic constructions such as localization, completion, and dualizability. We conclude with a brief account of the cotangent complex and deformation theory.