View source for Groupoid

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''for the general algebraic structure, see [[Magma]]''

A term introduced by H. Brandt [[#References|[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 [[#References|[a2]]], differential geometry [[#References|[a3]]] and topology [[#References|[a4]]], [[#References|[a5]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Brandt,   "Ueber eine Verallgemeinerung des Gruppenbegriffes"  ''Math. Ann.'' , '''96'''  (1926)  pp. 360–366</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.J. Higgins,   "Categories and groupoids" , v. Nostrand-Reinhold  (1971)</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  Ch. Ehresmann,   "Structures locales et catégories ordonnés" , ''Oeuvres complètes et commentées'' , ''Supplément aux Cahiers de Topologie et Géométrie Différentielle Catégoriques'' , '''Partie II'''  (1980)</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Brown,   "Elements of modern topology" , McGraw-Hill  (1968)</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Brown,   "From groups to groupoids: a brief survey"  ''Bull. London Math. Soc.'' , '''19'''  (1987)  pp. 113–134</TD></TR>
</table>