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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.
4
votes
Algebraic stacks as (étale) groupoid algebraic spaces/schemes
The notion of "torsor under a groupoid in a topos" already appears in
Breen, Lawrence Tannakian categories. Motives (Seattle, WA, 1991), 337–376, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc …