nLab N-∞ operad

Idea

A $G$-$\infty$-operad is an $\mathcal{N}_\infty$-operad if it is infinitely connected (or equivalently, $0$-truncated), 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 $\mathbb{E}_\infty$.

Definition

Let $\underline{\mathbb{F}}_G^\infty \subset \underline{\mathbb{F}}_G$ be the $G$-indexing system whose $H$-sets are those finite $H$-sets with trivial action, and let $I^\infty \subset \mathbb{F}_G$ be the corresponding indexing category. Let $\mathbb{E}_\infty^{\otimes} = \left( \mathrm{Span}_{I^\infty}(\mathbb{F}_G) \rightarrow \mathrm{Span}(\mathbb{F}_G) \right)$ be the corresponding fibration. This turns out to be a G-∞-operad.

Definition

A weak $\mathcal{N}_\infty$-operad for $G$ is a $G$-0-operad. An $\mathcal{N}_\infty$-operad for $G$ is a weak $\mathcal{N}_\infty$-operad for $G$ $\mathcal{O}^{\otimes}$ admitting a map $\mathbb{E}_\infty^{\otimes} \rightarrow \mathcal{O}^{\otimes}$.

Properties

Relationship to indexing systems/arity support

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 $\varphi^{-1}([G/H_i])$.

Definition

Given $\mathcal{O}^{\otimes}$ a $G$-operad, the arity support of $\mathcal{O}^{\otimes}$ is 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, and to weak indexing systems in Stewart 24:

Theorem

For all G-∞-operads $\mathcal{O}^{\otimes}$, $A \mathcal{O}$ is a weak indexing category, and the associated functor

$$A:\mathrm{Op}_G \rightarrow \mathrm{wIndex}_G$$

attains a fully faithful faithful right adjoint whose image is the weak $\mathcal{N}_\infty$-operads for $G$; the image of the subposet $\mathrm{Index}_G \subset \mathrm{wIndex}_G$ is the $\mathcal{N}_\infty$-operads for $G$.

As sub-terminal $G$-$\infty$-operads

Let $\mathcal{N}_{(-)\infty}^{\otimes}:\mathrm{wIndex}_G \rightarrow \mathrm{Op}_G$ be the right adjoint to $A$. The adjoint relationship implies that, for all $G$-$\infty$-operads $\mathcal{P}^{\otimes}$, we have

$$\Map(\mathcal{P}^{\otimes}, \mathcal{N}_{I\infty}^{\otimes}) = \begin{cases} * & A\mathcal{P} \subset I,\\ \emptyset & \mathrm{otherwise}. \end{cases}$$

In other words, $\mathcal{N}_{I \infty}^{\otimes} \in \mathrm{Op}_G$ is a subterminal object classifying the arity support condition $A(-) \leq I$. We refer to the resulting full subcategory $\mathrm{Op}_I \coloneqq \mathrm{Op}_{G, \mathcal{N}_{I \infty}^{\otimes}} \subset \mathrm{Op}_G$ as the $I$-operads.

On the other hand, it is shown in Stewart 24 that the functor $\mathcal{O}^{\otimes} \mapsto \mathcal{O}(S)$ is corepresentable, so if $\mathcal{O}^{\otimes}$ is a subterminal $G$-$\infty$-operad, its $S$-ary operation space $\mathcal{O}(S)$ is either empty or contractible for all $S$; unwinding definitions, this shows that a $G$-$\infty$-operad is subterminal if and only if it’s a weak $\mathcal{N}_\infty$-operad for $G$.

As $\otimes$-idempotent $G$-$\infty$-operads.

The Boardman-Vogt tensor product naturally extends to a tensor product on $G$-operads via the formula

$$\mathcal{O}^{\otimes} \otimes^{BV} \mathcal{P}^{\otimes} \coloneqq L_{\mathrm{Op}_G} \left( \mathcal{O}^{\otimes} \times \mathcal{P}^{\otimes} \rightarrow \mathrm{Span}(\mathbb{F}_G) \times \mathrm{Span}(\mathbb{F}_G) \xrightarrow{\wedge} \mathrm{Span}(\mathbb{F}_G)) \right),$$

where $\wedge:\mathrm{Span}(\mathbb{F}_G) \times \mathrm{Span}(\mathbb{F}_G) \rightarrow \mathrm{Span}(\mathbb{F}_G)$ is induced by the cartesian product of finite $G$-sets. The following is shown in Stewart 24.

Theorem

There exists an equivalence $\mathcal{N}_{I\infty}^{\otimes} \otimes^{BV} \mathcal{N}_{I \infty}^{\otimes} \simeq \mathcal{N}_{I \infty}^{\otimes}$ if and only if $I$ is an aE-unital weak indexing system, in which case a reduced $G$-$\infty$-operad $\mathcal{O}^{\otimes}$ satisfies

$$\mathcal{O}^{\otimes} \otimes^{BV} \mathcal{N}_{I \infty}^{\otimes} \simeq \mathcal{O}^{\otimes}$$

if and only if the G-∞-category $\underline{\mathrm{Alg}}_{\mathcal{O}}(\underline{\mathcal{S}}_G)$ of $\mathcal{O}$-algebras in G-spaces is I-semiadditive.

As a corollary, Stewart 24 concludes the following.

Corollary

If $I$ and $J$ are unital weak indexing systems, then there is a (unique) equivalence

$$\mathcal{N}_{I\infty}^{\otimes} \otimes^{BV} \mathcal{N}_{J \infty}^{\otimes} \simeq \mathcal{N}_{I \vee J \infty}^{\otimes},$$

where $I \vee J$ denotes the join of $I$ and $J$ in the poset of weak indexing systems.

The same theorem extends to aE-unital weak indexing systems, but the statement is somewhat more complicated; since indexing systems are a join-closed full sub-poset of weak indexing systems, this specializes to a theorem on the level of indexing systems, by omitting the words “unital weak.”

In the language of §6.3 of Blumberg-Hill 13, this confirms that in the homotopy-coherent setting every action of an $\mathcal{N}_\infty$-operad interchanges with itself, and every pair of interchanging $I$- and $J$-commutative algebra structures agrees on $I \cap J$ and is restricted from an $I \vee J$-commutative algebra structure.

Algebras over $\mathcal{N}_\infty$-operads

As incomplete Mackey functors in the cartesian setting

The following theorem was proved in the setting of graph $G$-operads and for $\mathcal{C} = \underline{\mathcal{S}}_G$ in Marc 24, and in the setting of G-∞-operads in Stewart 24.

Theorem

If $\mathcal{C}^{\otimes}$ is a I-symmetric monoidal ∞-category whose indexed tensor products are indexed products, then there is a canonical equivalence

$$\mathrm{Alg}_{\mathcal{N}_{I \infty}}(\mathcal{C}) \simeq \mathrm{CMon}_I(\mathcal{C})$$

over $\mathcal{C}$. In particular, if $\mathcal{C}$ is the $G$-$\infty$-category of coefficient systems in an ∞-category $\mathcal{V}$ with its Cartesian structure, then there is an equivalence

$$\mathrm{Alg}_{\mathcal{N}_{I \infty}}(\mathcal{C}) \simeq \mathrm{Fun}^{\times}(\mathrm{Span}(\mathbb{F}_G, \mathcal{V}).$$

As incomplete Tambara functors

In CHLL 24, it is shown that the equivariant Day convolution G-symmetric monoidal structure structure on the equivariant functor category $\mathrm{Fun}(\mathrm{Span}(\underline{\mathbb{F}_G}), \mathcal{C})$ restricts to a $G$-symmetric monoidal structure on the G-∞-category $\underline{\mathrm{CMon}}_G(\mathcal{C})$ of G-commutative monoids in $\mathcal{C}$. Then, CHLL 24 Theorem B concludes that $\mathcal{N}_{I\infty}$-algebras are homotopy-coherent incomplete Tambra functors.

Theorem

There is a fully faithful functor

$$\mathrm{CAlg}_I(\underline{\mathrm{Sp}}_G^{\otimes}) \hookrightarrow \mathrm{Fun}^{\times}(\mathrm{BiSpan}_I(\mathbb{F}_G), \mathcal{S})$$

whose image consists of the functors whose $H$-value “additive” commutative monoid is grouplike for all $H \subset G$.

As normed algebras

(under construction…)

Let $p$ be a prime number and $C_p$ the cyclic group of order $p$. Let $(-)^e\colon \mathrm{Sp}_G \rightarrow \mathrm{Sp}^{BG}$ be the underlying spectrum with $G$-action, and let $N_e^G$ be the Hill-Hopkins-Ravanel norm $G$-spectrum on a spectrum.

Given $A \in \mathrm{CAlg}(\mathrm{Sp}^{BC_p})$ a (highly structured) commutative ring spectrum with $C_p$-action, we let $A^{\otimes^{\Delta} p} \in \mathrm{CAlg}(\mathrm{Sp}^{BC_p})$ be the commutative ring spectrum with diagonal $C_p$-action

$$\sigma(a_1 \otimes \cdots \otimes a_p) = \sigma(a_p) \otimes \sigma(a_1) \otimes \cdots \otimes \sigma(a_{p-1}).$$

Furthermore, we let $A^{\otimes^{\tau} p} \in \mathrm{CAlg}(\mathrm{Sp}^{BC_p})$ be the commutative ring spectrum with transpotiion $C_p$-action

$$\sigma(a_1 \otimes \cdots \otimes a_p) = a_p \otimes a_1 \otimes \cdots \otimes a_{p-1}.$$

Definition

A normed $\mathbb{E}_\infty$-algebra in $C_p$-spectra is the data of

• an $\mathbb{E}_\infty$-algebra in $C_p$,

• a morphism of $\mathbb{E}_\infty$-rings $n_A\colon N^{C_p}_e A^e_{h_{C_p}} \rightarrow A$, and

• a homotopy making the following diagram $\mathcal{O}_{C_p} \rightarrow \mathbb{E}_\infty \mathrm{Alg}(\mathrm{Sp})^{B C_p}$ commute:

We denote the category of normed $\mathbb{E}_\infty$-algebras in $C_p$-spectra by $\mathrm{NAlg}_{C_p}(\mathrm{Sp}_{C_p})$.

Yang 23 constructs a forgetful functor $U\colon \mathrm{CAlg}_{C_p}(\mathrm{Sp}_{C_p}) \rightarrow \mathrm{NAlg}_{C_p}(\mathrm{Sp}_{C_p})$. The main result of Yang 23 is that this is an equivalence:

Theorem

The forgetful functor

$$U\colon \mathrm{CAlg}_{C_p}(\mathrm{Sp}_{C_p}) \rightarrow \mathrm{NAlg}_{C_p}(\mathrm{Sp}_{C_p})$$

is an equivalence.

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)

• Natalie Stewart: On tensor products of equivariant commutative operads (draft) [pdf]

Presentation of algebras in various cases:

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

• Gregoire Marc, A higher Mackey functor description of algebras over an $\mathcal{N}_\infty$-operad, (2024) (arXiv:2402.12447)

• Bastiaan Cnossen, Rune Haugseng, Tobias Lenz, Sil Linskens: Normed equivariant ring spectra and higher Tambara functors (arXiv:2407.08399)