Unstable synthetic deformations II: Infinitesimal extensions

TL;DR

This paper develops a deformation theory for Malcev theories and their models, generalizing classical deformation concepts, and applies it to analyze the structure of unstable synthetic homotopy theory and moduli space decompositions.

Contribution

It introduces a well-behaved deformation theory for Malcev theories, extending classical deformation concepts to $ty$-categorical algebraic theories and their models.

Findings

Postnikov tower of a Malcev theory is a tower of square-zero extensions

Deformation structures are preserved in $ty$-categories of models

New decompositions of moduli spaces of lifts are derived

Abstract

This paper is the second in a series devoted to the study of unstable synthetic deformations through the lens of Malcev theories: certain $\infty$-categorical algebraic theories $\mathcal{P}$ with well-behaved $\infty$-categories $\mathrm{Model}_{\mathcal{P}}$ of models. In this paper, we show that Malcev theories and their models admit a well-behaved deformation theory, generalizing the classical deformation theory of rings and modules. As our main example, we prove that the Postnikov tower of a Malcev theory $\mathcal{P}$ is a tower of square-zero extensions, and that all of this structure is preserved by passage to $\infty$-categories of models. This allows us to control the difference between the $\infty$-categories $\mathrm{Model}_{h_{n+r}\mathcal{P}}$ and $\mathrm{Model}_{h_n\mathcal{P}}$ for $r \leq n$, and forms the basis of a ``cofibre of $\tau$'' formalism in our approach to unstable synthetic homotopy theory. As an application, we derive from this a variety of new Blanc--Dwyer--Goerss style decompositions of moduli spaces of lifts along the tower $\mathrm{Model}_{\mathcal{P}}\to\cdots\to\mathrm{Model}_{h\mathcal{P}}$.