nLab N-∞ operad




A G G - \infty -operad is an 𝒩 \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 𝔼 \mathbb{E}_\infty.


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

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

A𝒪{TS[G/H i],Orbit(S),𝒪(T i)}𝔽 G. 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 Op G Γ\mathrm{Op}_G^{\Gamma} be the (∞,1)-category presented by the graph model structure on GG-operads, and let 𝒩 Op G ΓOp G Γ\mathcal{N}_\infty-\mathrm{Op}_G^{\Gamma} \subset \mathrm{Op}_G^{\Gamma} be the full subcategory spanned by 𝒩 \mathcal{N}_\infty-operads.


The functor AA restricts to an equivalence

A:𝒩 Op G ΓIndex G, 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 GG-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.



Classification via indexing systems (each independently proves this):

Presentation of algebras in the C pC_p-equivariant case:

Last revised on May 10, 2024 at 18:21:36. See the history of this page for a list of all contributions to it.