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Pontrjagin duality on multiplicative Gerbes

Jaider Blanco, Bernardo Uribe, Konrad Waldorf

Abstract

We use Segal-Mitchison's cohomology of topological groups to define a convenient model for topological gerbes. We introduce multiplicative gerbes over topological groups in this setup and we define its representations. For a specific choice of representation, we construct its category of endomorphisms and we show that it induces a new multiplicative gerbe over another topological group. This new induced group is fibrewise Pontrjagin dual to the original one and therefore we called the pair of multiplicative gerbes `Pontrjagin dual'. We show that Pontrjagin dual multipliciative gerbes have equivalent categories of representations and moreover, we show that their monoidal centers are equivalent. Examples of Pontrjagin dual multiplicative gerbes over finite and discrete, as well as compact and non-compact Lie groups are provided.

Pontrjagin duality on multiplicative Gerbes

Abstract

We use Segal-Mitchison's cohomology of topological groups to define a convenient model for topological gerbes. We introduce multiplicative gerbes over topological groups in this setup and we define its representations. For a specific choice of representation, we construct its category of endomorphisms and we show that it induces a new multiplicative gerbe over another topological group. This new induced group is fibrewise Pontrjagin dual to the original one and therefore we called the pair of multiplicative gerbes `Pontrjagin dual'. We show that Pontrjagin dual multipliciative gerbes have equivalent categories of representations and moreover, we show that their monoidal centers are equivalent. Examples of Pontrjagin dual multiplicative gerbes over finite and discrete, as well as compact and non-compact Lie groups are provided.

Paper Structure

This paper contains 46 sections, 18 theorems, 223 equations.

Table of Contents

  1. Segal-Mitchison cohomology
  2. Coefficients
  3. Cohomology of spaces
  4. Cohomology of simplicial spaces
  5. Cohomology of topological groups
  6. Segal-Mitchison cohomology as a derived functor
  7. Lyndon-Hochschild spectral sequence
  8. Segal-Mitchison gerbes
  9. Objects
  10. Morphisms
  11. 2-Morphisms
  12. Monoidal structure
  13. Multiplicative Segal-Mitchison gerbes
  14. Objects
  15. Morphisms
  16. ...and 31 more sections

Key Result

Lemma 1.3

Suppose that $X_\bullet$ is a simplicial paracompact space and $A$ is a contractible topological group, or alternatively suppose that the simplicial space $X_\bullet$ is homotopy equivalent to a simplicial set. Then, in either case, the forgetful homomorphism induces an isomorphism of cohomologies:

Theorems & Definitions (38)

  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • Definition 1.4
  • Lemma 1.5
  • proof
  • Lemma 1.6
  • proof
  • Proposition 1.7
  • proof
  • ...and 28 more