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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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Inverse semigroups and partial symmetries

It seems to me there should be some mathematical structure here related to the ideas of groupoids or inverse semi-groups, but I lack the mathematical sophistication to see it. … Clearly, there is a non-trivial interplay between spatial and internal symmetries going on here; looking around for what's known about such symmetries, I ran across Weinstein's paper: "Groupoids: Unifying …
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