On some adjunctions in equivariant stable homotopy theory

Abstract

We investigate certain adjunctions in derived categories of equivariant spectra, including a right adjoint to fixed points, a right adjoint to pullback by an isometry of universes, and a chain of two right adjoints to geometric fixed points. This leads to a variety of interesting other adjunctions, including a chain of 6 (sometimes 7) adjoints involving the restriction functor to a subgroup of a finite group on equivariant spectra indexed over the trivial universe.

Khovanov Spectra for Tangles

“…Remark 2.58. The functor Cob e → B from [25,40] actually did not lift the Khovanov functor V but rather its opposite. That ensured that the cohomology of the space constructed in [41,25,40] was isomorphic to the Khovanov homology.…”

“…In a previous paper, we built a framed flow category combinatorially and then used the second step of the Cohen-Jones-Segal program to define a Khovanov stable homotopy type [41]. Hu-Kriz-Kriz gave another construction of a Khovanov stable homotopy type using the Elmendorf-Mandell infinite loop space machine [25]. In another previous paper we were able to show that these two constructions give equivalent invariants [40].…”

“…If the monoidal category happened to be a symmetric monoidal category, as in the case of abelian groups Ab, graded abelian groups Ab * or chain complexes Kom, then the corresponding multicategory is a symmetric multicategory. (These are examples of Hu-Kriz-Kriz's -categories [25].) Many of our multicategories will be enriched in groupoids.…”

“…Hu-Kriz-Kriz [25] observed that the Khovanov functor V : Cob → Ab lifts to a lax 2-functor V HKK : Cob e → B:…”

On weight complexes, pure functors, and detecting weights

“…1. The compactness of the spectrum S 0 H for any closed subgroup H of G is given by Corollary A.3 of [HKS18]; cf. also Lemma I.5.3 of [LMSC86].…”