Infinite-Dimensional Representations of 2-Groups

Introduction......Page 3
2-Groups......Page 4
2-Vector spaces......Page 5
Representations......Page 7
Applications......Page 12
Plan of the paper......Page 15
2-groups as 2-categories......Page 17
Crossed modules......Page 19
Representing groups......Page 21
Representing 2-groups......Page 22
The 2-category of representations......Page 26
From vector spaces to 2-vector spaces......Page 28
Categorical perspective on 2-vector spaces......Page 30
From 2-vector spaces to measurable categories......Page 34
Measurable fields and direct integrals......Page 35
The 2-category of measurable categories: Meas......Page 40
Construction of Meas as a 2-category......Page 53
Main results......Page 56
Invertible morphisms and 2-morphisms in Meas......Page 60
Structure of representations......Page 65
Structure of intertwiners......Page 70
Structure of 2-intertwiners......Page 77
Equivalence of representations and of intertwiners......Page 79
Direct sums and tensor products in Meas......Page 82
Direct sums and tensor products in 2Rep(G)......Page 87
Representations......Page 89
Intertwiners......Page 92
Conclusion......Page 100
Lebesgue decomposition and Radon-Nikodym derivatives......Page 103
Geometric mean measure......Page 104
Measurable groups......Page 107
Measurable G-spaces......Page 111