Stacky Lie groups

Abstract

Presentations of smooth symmetry groups of differentiable stacks are studied within the framework of the weak 2-category of Lie groupoids, smooth principal bibundles, and smooth biequivariant maps. It is shown that principality of bibundles is a categorical property which is sufficient and necessary for the existence of products. Stacky Lie groups are defined as group objects in this weak 2-category. Introducing a graphic notation, it is shown that for every stacky Lie monoid there is a natural morphism, called the preinverse, which is a Morita equivalence if and only if the monoid is a stacky Lie group. As example we describe explicitly the stacky Lie group structure of the irrational Kronecker foliation of the torus.

40 pages; definition of group objects in higher categories added; coherence relations for groups in 2-categories given (section 4)

References in corpus (2)

• Higher Algebraic Structures and Quantization
• Lie Groupoids and Lie algebroids in physics and noncommutative geometry

Cited by in corpus (8)

• Categorified central extensions, étale Lie 2-groups and Lie's Third Theorem for locally exponential Lie algebras
• Invariant connections and PBW theorem for Lie groupoid pairs
• Higher Gauge Theory
• Strictification of etale stacky Lie groups
• Supersymmetric field theories and the elliptic index theorem with complex coefficients
• Shifted coisotropic structures for differentiable stacks
• Differentiable groupoid objects and their abstract Lie algebroids
• Functoriality of bornological groupoid convolution