Equivariant operads, symmetric sequences, and Boardman-Vogt tensor products

TL;DR

This work develops a comprehensive ∞-categorical framework for G-operads and their algebras, introducing a monadic underlying G-symmetric sequence and a Boardman-Vogt tensor product to encode homotopy-coherent interchange of operadic structures. It extends the Nardin–Shah approach to a broader $I$-indexing setting, defines weak ${\mathcal{N}}_{I\infty}^{\otimes}$-operads (weak ${\mathcal{N}}_\infty$-operads) realized by weak indexing systems, and shows how algebras over a BV-tensor product ${\mathcal{O}}^{\otimes}\otimes{\mathcal{P}}^{\otimes}$ correspond to interchanging ${\mathcal{P}}$- and ${\mathcal{O}}$-algebras up to coherent homotopy. The paper also develops the inflation–fixed points adjunction compatible with BV tensor products and demonstrates that Bonventre’s genuine operadic nerve admits a conservative total right derived functor, with equivalences connecting discrete genuine G-operads and weak indexing systems. By organizing operads via algebraic patterns and fibrous patterns, the authors provide a robust foundation for $I$-symmetric monoidal ∞-categories, $I$-commutative monoids, and their monoidal envelopes, enabling a precise treatment of equivariant algebraic structures in a highly structured homotopical setting. The results pave the way for a systematic understanding of equivariant algebraic theories, their Day-convolution-like symmetric monoidal structures, and their algebraic manifestations in $G$-symmetric monoidal contexts. $

Abstract

We advance the foundational study of be Nardin-Shah's $\infty$-category of $G$-operads and their associated $\infty$-categories of algebras. In particular, we construct the underlying $G$-symmetric sequence of a (one color) $G$-operad, yielding a monadic functor; we use this to lift Bonventre's genuine operadic nerve to a conservative functor of $\infty$-categories, restricting to an equivalence between categories of discrete $G$-operads. Using this, we extend Blumberg-Hill's program concerning $\mathcal{N}_\infty$-operads to arbitrary sub-operads of the terminal $G$-operad, which we show are equivalent to weak indexing systems. We then go on to define and characterize a homotopy-commutative and closed Boardman-Vogt tensor product on $\mathrm{Op}_G$; in particular, this specializes to a $G$-symmetric monoidal $\infty$-category of $\mathcal{O}$-algebras in a $G$-symmetric monoidal $\infty$-category whose $\mathcal{P}$-algebras are objects with interchanging $\mathcal{O}$-algebra and $\mathcal{P}$-algebra structures.

Theorems & Definitions (268)

• Definition
• Example
• Example
• Example
• Example
• Theorem A
• Corollary B
• Theorem C
• proof : References
• Theorem D