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Unstable synthetic deformations II: Infinitesimal extensions

William Balderrama, Piotr Pstrągowski

TL;DR

The paper develops an unstable deformation theory for Malcev theories and their $ty$-categories of models, generalizing classical ring–module deformation theory to a homotopical, unstable setting. It introduces the spiral tower and a robust Postnikov framework that yields cartesian square-zero/linear extensions at the level of theories and transfers them to the associated $ty$-categories of models, enabling refined moduli-space decompositions and obstruction theories. A general clutching/derived-extension formalism is developed, with elementary modifiers and spiral systems encoding the entire Postnikov/spiral data and ensuring convergence via a homology Whitehead-type theorem. These tools unify and extend Blanc–Dwyer–Goerss type decompositions to synthetic, non-discrete objects and provide a flexible framework for constructing and analyzing moduli spaces of realizations and mappings in unstable homotopy theory. The results have broad applicability to how unstable deformations interact with model categories, providing new pathways to study deformations of both connective and nonconnective homotopy theories.

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

Unstable synthetic deformations II: Infinitesimal extensions

TL;DR

The paper develops an unstable deformation theory for Malcev theories and their -categories of models, generalizing classical ring–module deformation theory to a homotopical, unstable setting. It introduces the spiral tower and a robust Postnikov framework that yields cartesian square-zero/linear extensions at the level of theories and transfers them to the associated -categories of models, enabling refined moduli-space decompositions and obstruction theories. A general clutching/derived-extension formalism is developed, with elementary modifiers and spiral systems encoding the entire Postnikov/spiral data and ensuring convergence via a homology Whitehead-type theorem. These tools unify and extend Blanc–Dwyer–Goerss type decompositions to synthetic, non-discrete objects and provide a flexible framework for constructing and analyzing moduli spaces of realizations and mappings in unstable homotopy theory. The results have broad applicability to how unstable deformations interact with model categories, providing new pathways to study deformations of both connective and nonconnective homotopy theories.

Abstract

This paper is the second in a series devoted to the study of unstable synthetic deformations through the lens of Malcev theories: certain -categorical algebraic theories with well-behaved -categories 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 is a tower of square-zero extensions, and that all of this structure is preserved by passage to -categories of models. This allows us to control the difference between the -categories and for , and forms the basis of a ``cofibre of '' 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 .
Paper Structure (31 sections, 71 theorems, 283 equations)

This paper contains 31 sections, 71 theorems, 283 equations.

Table of Contents

  1. Introduction
  2. The spiral tower of a Malcev theory
  3. Applications to moduli problems
  4. Infinitesimal extensions of Malcev theories
  5. Elementary modifiers and spiral systems
  6. Modifications of infty-categories
  7. Enriched infty-category theory
  8. The modification of an infty-category
  9. Limits of modifications
  10. Modifications of Malcev theories
  11. Elementary modifiers
  12. Postnikov decompositions revisited
  13. Postnikov squares of spaces
  14. Postnikov squares of infty-categories
  15. Postnikov modifiers are elementary
  16. ...and 16 more sections

Key Result

Theorem 1.1.3

Let $\EuScript{P}$ be a Malcev theory and fix integers $1\leq r \leq n < \infty$. Then the truncation is a linear extension of $\infty$-categories in the following sense: there is a spectrum object in $\infty$-categories over $\mathrm{h}_r\EuScript{P}$, linear over the truncation $\mathbf{S}_{< r}$ of the sphere spectrum, together with cartesian Postnikov squares

Theorems & Definitions (216)

  • Definition 1.1.1: Informal
  • Example 1.1.2
  • Theorem 1.1.3: §\ref{['ssec:categoricalpostnikovsquares']}
  • Example 1.1.4
  • Example 1.1.5
  • Theorem 1.1.6
  • Example 1.1.8
  • Example 1.1.9
  • Theorem 1.2.1: \ref{['thm:modulidecomposition']}
  • Example 1.2.2
  • ...and 206 more