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• Tags
  • #homotopy/stable-homotopy/equivariant
• Refs:
  • #todo/add-references
• Links:
  • equivariant functor
  • RAPC
  • Yoneda

group actions on categories

Categorifying a group action: regard \(G\) as a groupoid with one object, then a functor \(F\in [G, {\mathsf{Top}}]\) is precisely the data of a group action, \(\colim F\cong {\mathrm{Orb}}_G(X)\) is the orbit space, and \(\lim F = \mathrm{Fix} _G(X)\)a are the fixed points. A functor \(F\in [G, { \mathsf{Vect} }_{/ {k}} ]\) is a \(k{\hbox{-}}\)representation of \(G\), i.e. a \(kG{\hbox{-}}\)module.

Equivalently, a categorical fibration over \({\mathbf{B}}G\).