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.

Table of contents (1)

• 1 Introduction

• 2 "Periodic" table of n-categories

• 3 From Hilbert spaces to 2-Hilbert spaces

• 4 Cobordism : from nCob to nCob2

• 5 From TQFT to once extended TQFT

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

• 7 Spans of Groupoids

• 8 Spans of spans

• 9 Questions

• 10 Glossary

• 11 References

• 12 Detailed table of contents

Table of contents (2)

• 1 Introduction

• 2 "Periodic" table of n-categories

  • 2.1 Sets, categories, bicategories

  • 2.2 k-tuply monoidal n-categories

  • 2.3 Braided monoidal categories

  • 2.4 Categories, knots and topology

  • 2.5 Braided monoidal bicategories

• 3 From Hilbert spaces to 2-Hilbert spaces

  • 3.1 Dagger-categories

• 4 Cobordism : from nCob to nCob2

• 5 From TQFT to once extended TQFT

  • 5.1 TQFT = topological quantum field theory

  • 5.2 Once extended TQFT

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

  • 6.1 Fock spaces

  • 6.2 2-Fock spaces

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

• 7 Spans of Groupoids

  • 7.1 Spans

  • 7.2 Creation and annihilations operators as spans of groupoids

• 8 Spans of spans

  • 8.1 Morton and Vicary's Conjecture    

  • 8.2 Hoffnung and Stay's Theorem

• 9 Questions

• 10 Glossary

  • 10.1 Categorification, by J. Baez and J. Dolan

  • 10.2 Fock spaces

  • 10.3 Heisenberg (Lie) Algebra, in nLab

  • 10.4 Span, in nLab

• 11 References

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

• 12 Detailed table of contents

Introduction

"Periodic" table of n-categories

Sets, categories, bicategories

k-tuply monoidal n-categories

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

http://en.wikipedia.org/wiki/Topological_quantum_field_theory

http://ncatlab.org/nlab/show/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

Categorification, by J. Baez and J. Dolan

Fock spaces

Heisenberg (Lie) Algebra, in nLab

Span, in nLab

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

  • 2.1 Sets, categories, bicategories

  • 2.2 k-tuply monoidal n-categories

  • 2.3 Braided monoidal categories

  • 2.4 Categories, knots and topology

  • 2.5 Braided monoidal bicategories

• 3 From Hilbert spaces to 2-Hilbert spaces

  • 3.1 Dagger-categories

• 4 Cobordism : from nCob to nCob2

• 5 From TQFT to once extended TQFT

  • 5.1 TQFT = topological quantum field theory

  • 5.2 Once extended TQFT

    • 5.2.1 2-Hilbert space of a particle

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

    • 5.2.3 Z(S2) = H

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

  • 6.1 Fock spaces

    • 6.1.1 Creation operators

    • 6.1.2 Annihilation operators

    • 6.1.3 Heisenberg algebras relations

    • 6.1.4 [ Question (returning on TQFT) ]

  • 6.2 2-Fock spaces

    • 6.2.1 Yang-Baxter equations 

    • 6.2.2 Khovanov equations (2010)

    • 6.2.3 Morton and Vicary (2012)

    • 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.

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

• 7 Spans of Groupoids

  • 7.1 Spans

  • 7.2 Creation and annihilations operators as spans of groupoids

• 8 Spans of spans

  • 8.1 Morton and Vicary's Conjecture    

  • 8.2 Hoffnung and Stay's Theorem

• 9 Questions

• 10 Glossary

  • 10.1 Categorification, by J. Baez and J. Dolan

  • 10.2 Fock spaces

  • 10.3 Heisenberg (Lie) Algebra, in nLab

  • 10.4 Span, in nLab

• 11 References

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

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

• 12 Detailed table of contents