Spans and the Categorified Heisenberg Algebra, by John Baez (May 21, 2014)

Prise de vue, mise en ligne et en page : S. Dugowson



John Baez is a mathematical physicist at the University of California, Riverside and at the Centre for Quantum Technologies of Singapore.





Introduction


"Periodic" table of n-categories

Sets, categories, bicategories

k-tuply monoidal n-categories

 

Vidéo YouTube

 

Braided monoidal categories

Categories, knots and topology

Braided monoidal bicategories

From Hilbert spaces to 2-Hilbert spaces

Dagger-categories


 
 

Cobordism : from nCob to nCob2


 
 

From TQFT to once extended TQFT

TQFT = topological quantum field theory




Once extended TQFT

2-Hilbert space of a particle


  


Bartlett, Douglas, Schommer-Pries, Vicary,...

Z(S2) = H


 
 


Collections of particles : from Fock spaces to 2-Fock spaces

Fock spaces

Creation operators

Annihilation operators

Heisenberg algebras relations

                    

[ Question (returning on TQFT) ]

2-Fock spaces


 
 

Yang-Baxter equations 



Khovanov equations (2010)

            

Morton and Vicary (2012)

There is one more way to create a particle and then annihilate one, than to annihilate a particle and then create one.

            


The groupoid of finite sets and the creation operator (but annihilation operators ?)

  



Spans of Groupoids

Spans

  



Creation and annihilations operators as spans of groupoids


 
 


 



 




Spans of spans

  


 
        


Morton and Vicary's Conjecture    


    

Hoffnung and Stay's Theorem

        

Questions

Glossary




References

Spans and the Categorified Heisenberg Algebra, by J. Baez, on http://math.ucr.edu/home/baez/spans/

Slides for this talk : http://math.ucr.edu/home/baez/spans/spans.pdf


Detailed table of contents

  1. Introduction
  2. "Periodic" table of n-categories
    1. 2.1 Sets, categories, bicategories
    2. 2.2 k-tuply monoidal n-categories
    3. 2.3 Braided monoidal categories
    4. 2.4 Categories, knots and topology
    5. 2.5 Braided monoidal bicategories
  3. From Hilbert spaces to 2-Hilbert spaces
    1. 3.1 Dagger-categories
  4. Cobordism : from nCob to nCob2
  5. From TQFT to once extended TQFT
    1. 5.1 TQFT = topological quantum field theory
    2. 5.2 Once extended TQFT
      1. 5.2.1 2-Hilbert space of a particle
      2. 5.2.2 Bartlett, Douglas, Schommer-Pries, Vicary,...
      3. 5.2.3 Z(S2) = H
  6. Collections of particles : from Fock spaces to 2-Fock spaces
    1. 6.1 Fock spaces
      1. 6.1.1 Creation operators
      2. 6.1.2 Annihilation operators
      3. 6.1.3 Heisenberg algebras relations
      4. 6.1.4 [ Question (returning on TQFT) ]
    2. 6.2 2-Fock spaces
      1. 6.2.1 Yang-Baxter equations 
      2. 6.2.2 Khovanov equations (2010)
      3. 6.2.3 Morton and Vicary (2012)
      4. 6.2.4 There is one more way to create a particle and then annihilate one, than to annihilate a particle and then create one.
    3. 6.3 The groupoid of finite sets and the creation operator (but annihilation operators ?)
  7. Spans of Groupoids
    1. 7.1 Spans
    2. 7.2 Creation and annihilations operators as spans of groupoids
  8. Spans of spans
    1. 8.1 Morton and Vicary's Conjecture    
    2. 8.2 Hoffnung and Stay's Theorem
  9. Questions
  10. 10 Glossary
    1. 10.1 Categorification, by J. Baez and J. Dolan
    2. 10.2 Fock spaces
    3. 10.3 Heisenberg (Lie) Algebra, in nLab
    4. 10.4 Span, in nLab
  11. 11 References
    1. 11.1 Spans and the Categorified Heisenberg Algebra, by J. Baez, on http://math.ucr.edu/home/baez/spans/
      1. 11.1.1 Slides for this talk : http://math.ucr.edu/home/baez/spans/spans.pdf
  12. 12 Detailed table of contents


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