# Groupoid

*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