Quillen cohomology of enriched operads

TL;DR

This work develops a comprehensive framework for Quillen cohomology of operads enriched over a symmetric monoidal base category. It identifies the cotangent complex through a network of tangent-category equivalences, provides explicit formulas for Quillen cohomology of enriched operads, and introduces twisted arrow ∞-categories to represent cotangent data for simplicial operads. A key achievement is the Quillen principle for E_n-operads, yielding fiber sequences that connect deformation theories to Hochschild-type complexes and Kontsevich-type conjectures. The paper also outlines the forthcoming T-cohomology theory, a twist-based functor cohomology governed by Tw(𝒫), which specializes to Quillen cohomology in stable bases and connects to obstruction theory for operads and their algebras. Overall, these results deliver explicit computational tools and structural insights for deformation and obstruction theory in enriched and simplicial operads, with broad implications for higher algebra and homotopy theory.

Abstract

A modern insight due to Quillen, which is further developed by Lurie, asserts that many cohomology theories of interest are particular cases of a single construction, which allows one to define cohomology groups in an abstract setting using only intrinsic properties of the category (or $\infty$-category) at hand. This universal cohomology theory is known as Quillen cohomology. In any setting, Quillen cohomology of a given object is classified by its cotangent complex. The main purpose of this paper is to study Quillen cohomology of operads enriched over a general base category. Our main result provides an explicit formula for computing Quillen cohomology of enriched operads, based on a procedure of taking certain infinitesimal models of their cotangent complexes. Furthermore, we propose a natural construction of the twisted arrow $\infty$-categories of simplicial operads. We then assert that the cotangent complex of a simplicial operad can be represented as a spectrum valued functor on its twisted arrow $\infty$-category. When working in stable base categories such as chain complexes and spectra, Francis and Lurie proved the existence of a fiber sequence relating the cotangent complex and Hochschild complex of an $E_n$-algebra, from which a conjecture of Kontsevich is verified. We establish an analogous fiber sequence for the operad $E_n$ itself, in the topological setting.

Theorems & Definitions (185)

• Example 1.1.1
• Theorem 1.2.1
• Theorem 1.2.2
• Proposition 1.2.3
• Theorem 1.2.4
• Theorem 1.2.5
• Theorem 1.2.6
• Theorem 1.2.7
• Example 1.2.8
• Example 1.2.9