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Deformation theory and cotangent complex of dg operads

Yonatan Harpaz, Truong Hoang

TL;DR

The paper develops a comprehensive cotangent- and deformation-theory framework for differential graded operads. By modeling the operadic cotangent complex as an infinitesimal bimodule, it provides a uniform description of Quillen cohomology for operad algebras and explicit models for key operads: the dg $E_infty$-operad via the Pirashvili functor and the dg $E_n$-operad via Hochschild cohomology. It establishes deep links between deformation theory and Quillen cohomology, formulating first-order deformation spaces as HQ$^1$ and showing that the entire deformation theory of operads is governed by a dg Lie algebra in the formal moduli context framework. The results yield explicit cofiber/fiber sequences connecting HQ and HH for $E_n$-operads, and they provide a robust, computable approach to understand deformations of dg operads, including the $E_n$-family, with broad consequences for the homotopy theory of $E_n$-algebras and related tangent-cotangent structures.

Abstract

In the first part, we give an explicit description of the cotangent complex of differential graded (dg) operads, modeled as an operadic infinitesimal bimodule. This leads to a uniform formula for the Quillen cohomology of their associated algebras. We further show that the cotangent complex of the dg $E_\infty$-operad is represented by the Pirashvili functor, while that of the dg $E_n$-operad is conveniently described via its Hochschild complex. In the second part, we establish an explicit relation between deformation theory and (spectral) Quillen cohomology for various types of algebraic objects. Combining these results, we obtain a formulation of the space of first-order deformations of dg operads, which is particularly convenient in the case of dg $E_n$-operads.

Deformation theory and cotangent complex of dg operads

TL;DR

The paper develops a comprehensive cotangent- and deformation-theory framework for differential graded operads. By modeling the operadic cotangent complex as an infinitesimal bimodule, it provides a uniform description of Quillen cohomology for operad algebras and explicit models for key operads: the dg -operad via the Pirashvili functor and the dg -operad via Hochschild cohomology. It establishes deep links between deformation theory and Quillen cohomology, formulating first-order deformation spaces as HQ and showing that the entire deformation theory of operads is governed by a dg Lie algebra in the formal moduli context framework. The results yield explicit cofiber/fiber sequences connecting HQ and HH for -operads, and they provide a robust, computable approach to understand deformations of dg operads, including the -family, with broad consequences for the homotopy theory of -algebras and related tangent-cotangent structures.

Abstract

In the first part, we give an explicit description of the cotangent complex of differential graded (dg) operads, modeled as an operadic infinitesimal bimodule. This leads to a uniform formula for the Quillen cohomology of their associated algebras. We further show that the cotangent complex of the dg -operad is represented by the Pirashvili functor, while that of the dg -operad is conveniently described via its Hochschild complex. In the second part, we establish an explicit relation between deformation theory and (spectral) Quillen cohomology for various types of algebraic objects. Combining these results, we obtain a formulation of the space of first-order deformations of dg operads, which is particularly convenient in the case of dg -operads.
Paper Structure (22 sections, 37 theorems, 186 equations)

This paper contains 22 sections, 37 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Some terminology and notations
  3. Cotangent complex of dg operads
  4. Deformation theory and Quillen cohomology
  5. Background and conventions
  6. Operads and various types of modules over an operad
  7. Infinitesimal bimodules over an operad
  8. Endomorphism constructions
  9. Tangent categories and Quillen cohomology
  10. Some base categories and operadic model structures
  11. More notations
  12. Quillen cohomology of dg operads
  13. Cotangent complex of dg operads
  14. Operadic modules of Kähler differentials and reduced Quillen cohomology
  15. Cotangent complex of the dg $\mathop{\mathrm{E}}\nolimits_\infty$-operad
  16. ...and 7 more sections

Key Result

Theorem 1.2.1

(t:cotan) Under the equivalence $\mathcal{T}_\mathcal{P}\mathop{\mathrm{Op}}\nolimits(\mathcal{C}(\textbf{k})) \simeq \mathop{\mathrm{IbMod}}\nolimits(\mathcal{P})$, the cotangent complex $\mathop{\mathrm{L}}\nolimits_\mathcal{P}$ is identified with $\overline{\mathop{\mathrm{L}}\nolimits}_\mathcal{ the $m$-fold direct sum of the complex of operations $\mathcal{P}(c_1,\cdots,c_m;c)$. Consequently,

Theorems & Definitions (138)

  • Theorem 1.2.1
  • Corollary 1.2.2
  • Remark 1.2.3
  • Theorem 1.2.4
  • Remark 1.2.5
  • Theorem 1.2.6
  • Remark 1.2.7
  • Corollary 1.2.8
  • Example 1.3.1
  • Theorem 1.3.2
  • ...and 128 more