Pair Groupoid

Given any set B, the associated pair groupoid is the set B×B with the maps alpha(a,b)=a and beta(a,b)=b, and multiplication (a,b)·(b,c)=(a,c). The inverse is (a,b)^(-1)=(b,a). The left and right identity elements for (a,b) in G are lambda_((a,b))=(a,a) and rho_((a,b))=(b,b), as is readily checked.

Any equivalence relation defines a subgroupoid E of the pair groupoid B×B, with (a,b) in E if and only if a∼b. The orbits of E are then the equivalence classes.

Given any groupoid G over B, the map alpha×beta:G->B×B is a morphism of groupoids.

See also

Groupoid, Pair Group

This entry contributed by James Montaldi

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Weinstein, A. "Groupoids: Unifying Internal and External Symmetry." Not. Amer. Math. Soc. 43, 744-752, 1996.

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Pair Groupoid

Cite this as:

Montaldi, James. "Pair Groupoid." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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