# 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].

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