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Results tagged with groupoids
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user 3473
A groupoid is a category where all morphisms are invertible. This notion can also be seen as an extension of the notion of group. A motivating example is the fundamental groupoid of a topological space with respect to several base points, compared to the usual fundamental group.
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reference request for essential equivalence of top. groupoids
For example, this is proved as:
Lemma 3.34 in Lerman, E.: Orbifolds as stacks?
Proposition 60 in Metzler, D.S.: Topological and Smooth Stacks
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What is the 2-category whose 0-objects are Lie algebroids?
We'll have to talk about morphisms between Lie algebroids over different base manifolds (these are, for instance, the objects of the Lie groupoids they come from). … Applying this functor to spans of Lie groupoids produces immediately a span of Lie algebroids, both in the directed and undirected version. …
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answers
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Are Lie groupoids just groupoids internal to smooth manifolds?
To give some background, let's recall that the difference between Lie groupoids and groupoids internal to smooth manifolds is that in the first case we require source and target maps to be submersions, … The Wikipedia page says that Ehresmann originally considered groupoids internal to smooth manifolds, and that this definition was replaced by Pradines by the current definition of Lie groupoids, because …
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7
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Example of a groupoid internal to the category of smooth manifolds that is not a Lie groupoid
This questions is about the distinction between:
Lie groupoids: we require source and target maps to be submersions. This implies that the domain of the composition map, $G_1 \;{}_s\! … Groupoids internal to the category of smooth manifolds. Here we only require that the fibre product $G_1 \;{}_s\!\times_t G_1$ exists. Recall that transversality is sufficient but not necessary. …
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