Content Context Equivariant higher algebra Equivariant categories and symmetric monoidal categories Equivariant operads edit this sidebar Higher algebra
Content Idea A --operad is an -operad if it is infinitely connected (or equivalently, -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 . Definition Let be the -indexing system whose -sets are those finite -sets with trivial action, and let be the corresponding indexing category. Let be the corresponding fibration. This turns out to be a G-∞-operad. Definition A weak -operad for is a -0-operad. An -operad for is a weak -operad for admitting a map .
Properties Relationship to indexing systems/arity support Fix a -set. Recall that is an equivalence; given an equivariant function of -sets, write for the -set corresponding with . Definition Given a -operad, the arity support of is the subcategory
Let be the (∞,1)-category presented by the graph model structure on -operads, and let be the full subcategory spanned by -operads. Theorem The functor restricts to an equivalence the latter denoting the poset of indexing systems.
Fully-faithfullness in the graph model category of -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 , is a weak indexing category, and the associated functor attains a fully faithful faithful right adjoint whose image is the weak -operads for ; the image of the subposet is the -operads for .
As sub-terminal --operads Let be the right adjoint to . The adjoint relationship implies that, for all --operads , we have In other words, is a subterminal object classifying the arity support condition . We refer...