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Hochschild and cotangent complexes of operadic algebras

Truong Hoang

TL;DR

The paper develops a unified framework for cohomology of operadic structures by combining Lurie’s spectral cotangent complex with operadic tangent categories. It defines Quillen cohomology for enriched operads and spectral Hochschild cohomology for operads and their algebras, showing these cohomologies are governed by operad-level data via End-constructions and tangent-category equivalences. In the simplicial setting, cohomologies are computed as spectrum-valued functors on the twisted arrow ∞-category Tw(𝒫), enabling explicit comparisons HQ^*(A;N) ≃ HQ^*(𝒫;End^{t}_{A,N}) and HH^*(A;N) ≃ HH^*(𝒫;End^{t}_{A,N}); a fiber sequence between Hochschild and cotangent complexes yields an unstable analogue of Francis–Lurie for E_n-spaces. The Quillen principle for E_n-algebras is formulated via cofiber/fiber sequences tying the cotangent and Hochschild data, with stable and ∞-categorical refinements, providing computationally tractable routes to deformation and obstruction theories in operadic settings.

Abstract

We make use of the cotangent complex formalism developed by Lurie to formulate Quillen cohomology of algebras over an enriched operad. Additionally, we introduce a spectral Hochschild cohomology theory for enriched operads and algebras over them. We prove that both the Quillen and Hochschild cohomologies of algebras over an operad can be controlled by the corresponding cohomologies of the operad itself. When passing to the category of simplicial sets, we assert that both these cohomology theories for operads, as well as their associated algebras, can be calculated in the same framework of spectrum valued functors on the twisted arrow $\infty$-category of the operad of interest. Moreover, we provide a convenient cofiber sequence relating the Hochschild and cotangent complexes of an $E_n$-space, establishing an unstable analogue of a significant result obtained by Francis and Lurie. Our strategy introduces a novel perspective, focusing solely on the intrinsic properties of the operadic twisted arrow $\infty$-categories.

Hochschild and cotangent complexes of operadic algebras

TL;DR

The paper develops a unified framework for cohomology of operadic structures by combining Lurie’s spectral cotangent complex with operadic tangent categories. It defines Quillen cohomology for enriched operads and spectral Hochschild cohomology for operads and their algebras, showing these cohomologies are governed by operad-level data via End-constructions and tangent-category equivalences. In the simplicial setting, cohomologies are computed as spectrum-valued functors on the twisted arrow ∞-category Tw(𝒫), enabling explicit comparisons HQ^*(A;N) ≃ HQ^*(𝒫;End^{t}_{A,N}) and HH^*(A;N) ≃ HH^*(𝒫;End^{t}_{A,N}); a fiber sequence between Hochschild and cotangent complexes yields an unstable analogue of Francis–Lurie for E_n-spaces. The Quillen principle for E_n-algebras is formulated via cofiber/fiber sequences tying the cotangent and Hochschild data, with stable and ∞-categorical refinements, providing computationally tractable routes to deformation and obstruction theories in operadic settings.

Abstract

We make use of the cotangent complex formalism developed by Lurie to formulate Quillen cohomology of algebras over an enriched operad. Additionally, we introduce a spectral Hochschild cohomology theory for enriched operads and algebras over them. We prove that both the Quillen and Hochschild cohomologies of algebras over an operad can be controlled by the corresponding cohomologies of the operad itself. When passing to the category of simplicial sets, we assert that both these cohomology theories for operads, as well as their associated algebras, can be calculated in the same framework of spectrum valued functors on the twisted arrow -category of the operad of interest. Moreover, we provide a convenient cofiber sequence relating the Hochschild and cotangent complexes of an -space, establishing an unstable analogue of a significant result obtained by Francis and Lurie. Our strategy introduces a novel perspective, focusing solely on the intrinsic properties of the operadic twisted arrow -categories.
Paper Structure (24 sections, 40 theorems, 206 equations)

This paper contains 24 sections, 40 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. From tangent categories to spectral cotangent complexes
  3. Cohomologies of enriched operads and algebras over them
  4. Main statements
  5. Formulations and comparison theorem
  6. Simplicial operads and algebras over them
  7. Quillen principle for $\mathop{\mathrm{E}}\nolimits_n$-spaces
  8. Background and conventions
  9. Enriched operads
  10. Modules over an operad
  11. Semi-model categories
  12. Operadic model structures
  13. Spectral cotangent complex and Quillen cohomology
  14. More conventions and notations
  15. Operadic tangent categories and cotangent complex of enriched operads
  16. ...and 9 more sections

Key Result

Theorem 1.1

(co:bialg, t:opmain) Suppose given a fibrant object $N\in \mathcal{T}_{A^{\mathop{\mathrm{e}}\nolimits}} \mathop{\mathrm{Mod}}\nolimits_\mathcal{P}^A$. (i) There is a natural weak equivalence between the Quillen cohomology of $A$ with coefficients in $N$ and the loop space of Quillen cohomology of $\mathcal{P}$ with coefficients in $\textrm{End}^{\mathop{\mathrm{t}}\nolimits}_{A,N} \in \mathcal{T

Theorems & Definitions (135)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • ...and 125 more