Redirected from "G-∞-operad".
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Idea
--operads are to (∞,1)-operads as equivariant symmetric monoidal categories are to symmetric monoidal categories
Definition
G--operads may be defined as fibrous patterns over the Burnside category . Explicitly:
Definition
A --operad is a functor such that
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has -cocartesian lifts for backwards maps
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For every , cocartesian transport along -cocartesian lifts lying over the inclusions implement an equivalence
where .
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For every map of orbits and pair of objects , writing , postcomposition with the -cartesian lifts yields an equivalence
where .
A morphism of --operads is a functor over preserving cocartesian lifts for backwards maps.
Definition
If are --operads, an -algebra in is a morphism of --operads ; we denote by
the full subcategory spanned by -algebras in .
In particular, if is a -symmetric monoidal -category, then -algebras in are defined to be -algebras in the underlying --operad of .
The underlying -category and -symmetric sequence
Definition
Given a --operad, the underlying --category of is .
Explicitly, the -value of is the fiber , and the restriction functor is pullback. We say that has one object if is the terminal G-∞-category.
Definition
If is a one-object -operad, then its underlying G-symmetric sequence is the functor defined by
This is significant largely due to the following proposition.
Proposition
A map of -operads is an equivalence if and only if, for all subgroups and all finite -sets , the induced map is an equivalence.
Examples
References
Originally,
In terms of the Burnside category,