nLab norm map

Idea

In representation theory a norm map is a canonical morphism from coinvariants to invariants of a given action which, in suitably well behaved cases is given by group averaging.

(e.g Lurie, constructions 6.1.6.4, 6.1.6.8, 6.1.6.18)

For ∞-actions of finite groups $G$ on objects $E$ in stable (∞,1)-categories, then the homotopy cofiber $X^{t G}$ of the norm map is called the Tate construction, sitting in a homotopy fiber sequence

$$X_G \longrightarrow X^G \longrightarrow X^{t G} \,.$$

(e.g Lurie, def. 6.1.6.24)

For the stable (∞,1)-category of spectra this is accordingly called the Tate spectrum.

Examples

Example

The comparison map between left and right induced representations is a norm map (for linear representations understood as local systems on delooping groupoids), see there.

Related concepts

• ambidextrous adjunction

• Tate spectrum

• cyclotomic spectrum

• Ambidexterity in K(n)-Local Stable Homotopy Theory

References

The general abstract construction is due to

• Jacob Lurie, section 6.1.6 of Higher Algebra

• Michael Hopkins, Jacob Lurie: Remark 4.1.12 of: Ambidexterity in K(n)-Local Stable Homotopy Theory (2014)

Review with an eye towards discussion of topological cyclic homology is in section I.1 of

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