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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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Groupoids vs. action groupoids
I might be wrong but at first glance, I would say "yes" to Question 1. The reason is the following. If a groupoid $\Gamma \rightrightarrows Y$ acts by automorphisms on a groupoid $A\rightrightarrows X …