groupoid

[groo-poid]

noun

Mathematics.

1. an algebraic system closed under a binary operation.

Word History and Origins

Origin of groupoid¹

First recorded in 1940–45; group + -oid

Example Sentences

Note that contractability does not imply that there is a unique composite: indeed, as we have seen in the fundamental -groupoid, there can be a large number of composite paths.

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We can use the points and paths of a space to translate problems of topology into problems of algebra: each topological space X has an associated category π1X called the “fundamental groupoid” of X. The objects of this category are the points of the space, and the transformations are paths.

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A key advantage of the fundamental groupoid construction is that it is “functorial,” meaning that a continuous function f: X → Y between topological spaces gives rise to a corresponding transformation π1f: π1X → π1Y between the fundamental groupoids.

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The fundamental groupoid is not a complete invariant, however.

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In the fundamental groupoid of the circle, the different wiggling versions of a path between two points can be labeled by integers that record the number of times the trajectory winds around the circle and a + or sign indicating, respectively, a clockwise or counterclockwise direction of transit.

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