# Groupoid

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for the general algebraic structure, see Magma

A term introduced by H. Brandt [a1]. A groupoid may conveniently be defined as a (small) category in which every morphism is an isomorphism; equivalently, it is a set \$G\$ equipped with a unary operation \$g\mapsto g^{-1}\$ and a partial binary operation \$(g,h)\mapsto gh\$ satisfying

1) \$gg^{-1}\$ and \$g^{-1}g\$ are always defined;

2) \$gh\$ is defined if and only if \$g^{-1}g=hh^{-1}\$;

3) if \$gh\$ and \$hk\$ are defined, then \$(gh)k\$ and \$g(hk)\$ are defined and equal;

4) each of \$g^{-1}gh\$, \$hg^{-1}g\$, \$gg^{-1}h\$, and \$hgg^{-1}\$ is equal to \$h\$ if it is defined.

Groupoids, as a special case of categories, play an important role in many areas of application of category theory, including algebra [a2], differential geometry [a3] and topology [a4], [a5].

#### References

 [a1] H. Brandt, "Ueber eine Verallgemeinerung des Gruppenbegriffes" Math. Ann. , 96 (1926) pp. 360–366 [a2] P.J. Higgins, "Categories and groupoids" , v. Nostrand-Reinhold (1971) [a3] Ch. Ehresmann, "Structures locales et catégories ordonnés" , Oeuvres complètes et commentées , Supplément aux Cahiers de Topologie et Géométrie Différentielle Catégoriques , Partie II (1980) [a4] R. Brown, "Elements of modern topology" , McGraw-Hill (1968) [a5] R. Brown, "From groups to groupoids: a brief survey" Bull. London Math. Soc. , 19 (1987) pp. 113–134
How to Cite This Entry:
Groupoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Groupoid&oldid=39751
This article was adapted from an original article by V.D. Belousov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article