nLab spectrum with G-action

Idea

For $G$ a topological group, a spectrum with $G$-action is a spectrum equipped with an action of $G$. In other words, it is a functor from $BG$ to the (infinity,1)-category of spectra. The stable homotopy theory of spectra with $G$-action is part of the subject of equivariant stable homotopy theory.

Spectra with $G$-action are sometimes called “doubly naive” $G$-spectra. They are even more naive than naive G-spectra, and can be identified with the full subcategory of G-spectra consisting of “Borel-complete” $G$-spectra (see Prop. 6.17 in MNN or Thm. II.2.7 in NS).

See also

• naive G-spectra, G-spectra

• G-spaces

• equivariant stable homotopy theory

References

• Akhil Mathew, Niko Naumann, Justin Noel, Nilpotence and descent in equivariant stable homotopy theory, Adv. Math. 305 (2017), 994–1084, arXiv:1507.06869.

• Thomas Nikolaus, Peter Scholze, On topological cyclic homology, arXiv:1707.01799.