nLab N-∞ operad

Idea

A $G$-$\infty$-operad is an $\mathcal{N}_\infty$-operad if it is infinitely connected, unital, and prescribes binary multiplications on fixed points for all subgroups.

These are meant to model the equivariant commutative operads which contain a non-genuine version of $\mathbb{E}_\infty$.

Properties

Fix $S = \coprod_{i \leq n} [G/H_i] \in \mathbb{F}_G$ a $G$-set. Recall that $\mathrm{Ind}_{H_i}^{G}:\mathbb{F}_{H_i} \rightarrow \mathbb{F}_{G, /[G/H_i]}$ is an equivalence; given $\varphi:T \rightarrow S$ an equivariant function of $G$-sets, write $T_i$ for the $H_i$-set corresponding with $\varphi^{-1}([G/H_i])$.

Given $\mathcal{O}^{\otimes}$ a $G$-operad, we define the subcategory

$$A \mathcal{O} \coloneqq \left\{T \rightarrow S \;\;\; \mid \;\;\; \forall [G/H_i], \in \mathrm{Orbit}(S), \;\; \mathcal{O}(T_i) \neq \emptyset \right\} \subset \mathbb{F}_{G}.$$

Let $\mathrm{Op}_G^{\Gamma}$ be the (∞,1)-category presented by the graph model structure on $G$-operads, and let $\mathcal{N}_\infty-\mathrm{Op}_G^{\Gamma} \subset \mathrm{Op}_G^{\Gamma}$ be the full subcategory spanned by $\mathcal{N}_\infty$-operads.

Theorem

The functor $A$ restricts to an equivalence

$$A:\mathcal{N}_\infty-\mathrm{Op}_G^{\Gamma} \xrightarrow\sim \mathrm{Index}_G,$$

the latter denoting the poset of indexing systems.

Fully-faithfullness in the graph model category of $G$-operads, was proved in Blumberg-Hill 13, followed by independent proofs in 2017 by Rubin, Gutiérrez-White, and Bonventre-Pereira.

Subsequently, this was generalized to the orbital setting in Nardin-Shah 22.

Related concepts

• Parametrized Higher Category Theory and Higher Algebra

• equivariant symmetric monoidal category, G-∞-operad

• indexing system

• algebraic pattern

References

Originally,

• Andrew Blumberg, Michael Hill, Operadic multiplications in equivariant spectra, norms, and transfers, (arXiv:1309.1750)

Classification via indexing systems (each independently proves this):

• Jonathan Rubin, Combinatorial $N_\infty$ operads (2017), (arXiv:1705.03585)

• Javier Gutiérrez, David White, Encoding Equivariant Commutativity via Operads (2017), (arXiv:1707.02130)

• Peter Bonventre, Luis Pereira, Genuine equivariant operads, (2017) (1707.02226)

• Denis Nardin, Jay Shah, Parametrized and equivariant higher algebra, (2022) (arxiv:2203.00072)

Presentation of algebras in the $C_p$-equivariant case:

• Lucy Yang, On normed $\mathbb{E}_\infty$-rings in genuine equivariant $C_p$-spectra, (2023) (arXiv:2308.16107)