The definition of Lie group bundle is provided below:
Definition A Lie group bundle, or LGB, is a smooth fibre bundle (K,q,M) in which each fibre K_m=q^{-1}(m), and the fibre type G, has a Lie group structure, and for which there is an atlas {\psi_i: U_i \times G\to K_m} such that each \psi_{i,m}: G \to K_m, m\in U_i , is an isomorphism of Lie groups.
and
A morphism of LGBs from (K,q,M) to (K',q',M') is a morphism (F, f) of fibre bundles such that each F_m : K_m —> K'_{f(m)} is a morphism of Lie groups.
Hey Romeo,
The categorical relation between Lie group bundles (LGB) and Lie groupoid bundles can be understood in terms of their properties and morphisms within the category of fiber bundles.
A Lie group bundle (LGB) is a smooth fiber bundle (K, q, M) in which each fiber K_m=q^{-1}(m) is equipped with a Lie group structure. Essentially, it's a bundle where the fibers are Lie groups, and there are compatible charts (atlas) that allow us to work with these groups smoothly.
On the other hand, a Lie groupoid bundle is a more generalized structure. A Lie groupoid bundle consists of a bundle of groupoids over a base space. Here, the fibers are Lie groupoids, which are groupoids (a category with group-like properties) equipped with smooth structures.
The categorical relation between these two can be understood as follows:
1. Lie Group Bundles (LGBs) are a special case of Lie Groupoid Bundles, where the Lie groupoids involved are actually groups. In other words, Lie group bundles are Lie groupoid bundles with a special type of fiber, namely Lie groups.
2. Morphisms of LGBs correspond to morphisms of Lie groupoid bundles. In both cases, the morphisms respect the smooth structure of the fibers, and in the case of Lie group bundles, they also respect the group structure.
In summary, Lie group bundles are a specific instance of Lie groupoid bundles, where the fibers are Lie groups. The categorical relationship arises because Lie group bundles can be seen as a subcategory within the broader category of Lie groupoid bundles, where the morphisms and structures align with those of Lie groups.
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