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

This page is to record the reference:

• Michael Hopkins, Jacob Lurie:

  
  

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

  
  

  (2013)

  pdf, pdf

on bilimits in K(n)-local stable homotopy theory and generally on ambidextrous adjunctions and their un-twisted Wirthmüller isomorphisms for ∞-module bundles in semiadditive (∞,1)-categories.

(The untwisted Wirthmüller isomorphism is the map $\mu$ in Construction 4.0.7 and then the norm map in Remark 4.1.12. The induced integration map considered in Construction 4.0.7, Notation 4.1.6 there is also discussed (for the general twisted case) in the article Quantization via Linear homotopy types, see section 4.3 there for details.)

The material was the basis of a talk at Notre Dame Graduate Summer School on Topology and Field Theories and Harvard lecture 2012 (video part 1, part 2, part 3 part 4, pdf lecture notes by Chris Elliott)

which appeared in expository form as:

• Gijs Heuts, Jacob Lurie: Ambidexterity, in: Topology and Field Theories, Contemporary Mathematics 613, AMS (2014) 79-110 [ISBN:978-1-4704-1015-5, doi:10.1090/conm/613, pdf]

The discussion in the article is apparently motivated as part of what it takes to make precise the discussion of quantization in sections 3 and 8 of

• Daniel Freed, Mike Hopkins, Jacob Lurie, Constantin Teleman: Topological Quantum Field Theories from Compact Lie Groups, in P. R. Kotiuga (ed.), A celebration of the mathematical legacy of Raoul Bott, AMS (2010) 367-403 [arXiv:0905.0731, doi:10.1090/crmp/050, ISBN:978-0-8218-4777-0]

For the 2014 installment of UOregon’s Moursund Lecture Series, Jacob Lurie gave three (video recorded) lectures on spectral algebraic geometry, one of which is

• Jacob Lurie, Ambidexterity

1. Multiplicative aspects of Dieudonne Theory

2. The Morava $K$-theory of Eilenberg-MacLane Spaces

• Morava K-theory

3. Alternating powers of $p$-Divisible groups

4. Ambidexterity

4.1 Beck-Chevalley fibrations and Norm maps

• Beck-Chevalley condition

• pointwise (∞,1)-Kan extension

4.2 Properties of the norm

4.3 Local systems

• local system

• ambidextrous space

4.4 Examples

• semiadditive (∞,1)-category

5. Ambidexterity of $K(n)$-Local stable homotopy theory

5.1 Ambidexterity and duality

5.2 The main theorem

• K(n)-local stable homotopy theory

5.3 Cartier duality

• Cartier duality

5.4 The global sections functor

category: reference