dorsal/arxiv
View SchemaUnstable synthetic deformations II: Infinitesimal extensions
| Authors | William Balderrama, Piotr Pstrągowski |
|---|---|
| Categories | |
| ArXiv ID | 2601.08812vv1 |
| URL | https://arxiv.org/abs/2601.08812 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
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 $\mathrm{Model}_{\mathcal{P}}\to\cdots\to\mathrm{Model}_{h\mathcal{P}}$.
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"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.\n 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 $\\tau$\u0027\u0027 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 $\\mathrm{Model}_{\\mathcal{P}}\\to\\cdots\\to\\mathrm{Model}_{h\\mathcal{P}}$.",
"arxiv_id": "2601.08812",
"authors": [
"William Balderrama",
"Piotr Pstr\u0105gowski"
],
"categories": [
"math.AT"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Unstable synthetic deformations II: Infinitesimal extensions",
"url": "https://arxiv.org/abs/2601.08812",
"version": "v1"
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