nLab G-∞-operads

Content

Content

Idea

GG-\infty-operads are to (∞,1)-operads as equivariant symmetric monoidal categories are to symmetric monoidal categories

Definition

G-\infty-operads may be defined as fibrous patterns over the Burnside category Span(𝔽 G)\mathrm{Span}(\mathbb{F}_G). Explicitly:

Definition

A GG-\infty-operad is a functor π 𝒪:𝒪 Span(𝔽 G)\pi_{\mathcal{O}}:\mathcal{O}^{\otimes} \rightarrow \mathrm{Span}(\mathbb{F}_G) such that

  1. 𝒪 \mathcal{O}^{\otimes} has π 𝒪\pi_{\mathcal{O}}-cocartesian lifts for backwards maps

  2. For every S𝔽 𝒯S \in \mathbb{F}_{\mathcal{T}}, cocartesian transport along π\pi-cocartesian lifts lying over the inclusions (SU=UUOrb(S))(S \leftarrow U = U \mid U \in \mathrm{Orb}(S)) implement an equivalence

    𝒪 S UOrb(S)𝒪 U, \mathcal{O}_S \simeq \prod_{U \in \mathrm{Orb}(S)} \mathcal{O}_U,

    where 𝒪 S=π 𝒪 1(S)\mathcal{O}_S = \pi^{-1}_{\mathcal{O}}(S).

  3. For every map of orbits TST \rightarrow S and pair of objects (C,D)𝒪 T×𝒪 S(\mathbf{C},\mathbf{D}) \in \mathcal{O}_T \times \mathcal{O}_S, writing D=(D UUOrb(S))\mathbf{D} = (D_U \mid U \in \mathrm{Orb}(S)), postcomposition with the π\pi-cartesian lifts yields an equivalence

    Map 𝒪 TS(C,D) UOrb(S)Map 𝒪 TT UU(C,D U), \mathrm{Map}_{\mathcal{O}^{\otimes}}^{T \rightarrow S}(\mathbf{C}, \mathbf{D}) \simeq \prod_{U \in \mathrm{Orb}(S)} \mathrm{Map}_{\mathcal{O}^{\otimes}}^{T \leftarrow T_U \rightarrow U}(\mathbf{C}, D_U),

    where T UT× SUT_U \coloneqq T \times_S U.

A morphism of GG-\infty-operads is a functor over Span(𝔽 G)\mathrm{Span}(\mathbb{F}_G) preserving cocartesian lifts for backwards maps.

Proposition

A functor 𝒞 Span(𝔽 G)\mathcal{C}^{\otimes} \rightarrow \mathrm{Span}(\mathbb{F}_G) is simultaneously a GG-\infty-operad and a cocartesian fibration if and only if it is the unstraightening of a G-symmetric monoidal \infty -category

Definition

If 𝒪 ,𝒫 \mathcal{O}^{\otimes},\mathcal{P}^{\otimes} are GG-\infty-operads, an 𝒪\mathcal{O}-algebra in 𝒫 \mathcal{P}^{\otimes} is a morphism of GG-\infty-operads 𝒪 𝒫 \mathcal{O}^{\otimes} \rightarrow \mathcal{P}^{\otimes}; we denote by

Alg 𝒪(𝒫)Fun /Span(𝔽 G)(𝒪 ,𝒫 ) \mathrm{Alg}_{\mathcal{O}}(\mathcal{P}) \subset \mathrm{Fun}_{/\mathrm{Span}(\mathbb{F}_G)}(\mathcal{O}^{\otimes}, \mathcal{P}^{\otimes})

the full subcategory spanned by 𝒪\mathcal{O}-algebras in 𝒫\mathcal{P}.

In particular, if 𝒞 \mathcal{C}^{\otimes} is a GG-symmetric monoidal \infty-category, then 𝒪\mathcal{O}-algebras in 𝒞 \mathcal{C}^{\otimes} are defined to be 𝒪\mathcal{O}-algebras in the underlying GG-\infty-operad of 𝒞 \mathcal{C}^{\otimes}.

The underlying GG-category and GG-symmetric sequence

Definition

Given 𝒫 \mathcal{P}^{\otimes} a GG-\infty-operad, the underlying GG-\infty-category of 𝒫 \mathcal{P}^{\otimes} is U𝒫𝒫 × Span(𝔽 G)𝒪 G opU \mathcal{P} \coloneqq \mathcal{P}^{\otimes} \times_{\mathrm{Span}(\mathbb{F}_G)} \mathcal{O}_G^{\mathrm{op}}.

Explicitly, the HH-value of U𝒫U\mathcal{P} is the fiber 𝒫 Hπ 𝒫 1([G/H])\mathcal{P}_H \coloneqq \pi_{\mathcal{P}}^{-1}([G/H]), and the restriction functor Res K H:𝒫 H𝒫 K\Res_K^H:\mathcal{P}_H \rightarrow \mathcal{P}_K is pullback. We say that 𝒫 \mathcal{P}^{\otimes} has one object if U𝒫U\mathcal{P} is the terminal G-∞-category.

Definition

If 𝒫 \mathcal{P}^{\otimes} is a one-object GG-operad, then its underlying G-symmetric sequence is the functor 𝒫():totΣ̲ G𝒮\mathcal{P}(-):\tot \underline{\Sigma}_G \rightarrow \mathcal{S} defined by

𝒫(S)Map cP Ind H GS[G/H](Ind H GS,[G/H]). \mathcal{P}(S) \coloneqq \Map_{\cP^{\otimes}}^{\Ind_H^G S \rightarrow [G/H]}(\Ind_H^G S, [G/H]).

This is significant largely due to the following proposition.

Proposition

A map of GG-operads 𝒫 𝒬 \mathcal{P}^{\otimes} \rightarrow \mathcal{Q}^{\otimes} is an equivalence if and only if, for all subgroups HGH \subset G and all finite HH-sets S𝔽 HS \in \mathbb{F}_H, the induced map φ(S):𝒫(S)𝒬(S)\varphi(S):\mathcal{P}(S) \rightarrow \mathcal{Q}(S) is an equivalence.

Examples

References

Originally,

In terms of the Burnside category,

Last revised on July 14, 2024 at 13:49:22. See the history of this page for a list of all contributions to it.