Deformations of homotopy theories via algebraic theories

TL;DR

This paper introduces a homotopical approach to algebraic theories, enabling the construction of deformation and obstruction theories to analyze homotopy data from algebraic structures.

Contribution

It develops a homotopical version of algebraic theories, providing a new framework for deformations and computations in homotopy theory.

Findings

Framework for homotopical algebraic theories

Methods for constructing obstruction theories

Spectral sequences for homotopical data computation

Abstract

We develop a homotopical variant of the classic notion of an algebraic theory as a tool for producing deformations of homotopy theories. From this, we extract a framework for constructing and reasoning with obstruction theories and spectral sequences that compute homotopical data starting with purely algebraic data.