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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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The groupoid of algebraic expressions and proofs

In particular, we can define many categorical notions with $\infty$-groupoids instead of sets, like $\infty$-categories (categories enriched over $\infty$-groupoids) and $\infty$-Lawvere theories. … In this simple case you don't need the full-blown theory of $\infty$-groupoids and can specify all relations explicitly. …
Anton Fetisov's user avatar