Unstable synthetic deformations II: Infinitesimal extensions

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 $\mathrm{Model}_{\mathcal{P}}\to\cdots\to\mathrm{Model}_{h\mathcal{P}}$.