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Difference between revisions of "Groupoid"

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''for the general algebraic structure, see [[Magma]]''
 
''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|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
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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;
 
1) $gg^{-1}$ and $g^{-1}g$ are always defined;
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4) each of $g^{-1}gh$, $hg^{-1}g$, $gg^{-1}h$, and $hgg^{-1}$ is equal to $h$ if it is defined.
 
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]]], different geometry [[#References|[a3]]] and topology [[#References|[a4]]], [[#References|[a5]]].
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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====
 
====References====
 
<table>
 
<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>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Brandt,  "Ueber eine Verallgemeinerung des Gruppenbegriffes"  ''Math. Ann.'' , '''96'''  (1926)  pp. 360–366 {{ZBL|52.0110.09}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.J. Higgins,  "Categories and groupoids" , v. Nostrand-Reinhold  (1971)</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  P.J. Higgins,  "Categories and groupoids" , v. Nostrand-Reinhold  (1971) {{ZBL|0226.20054}}</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">[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">[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>
 
<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>
 
</table>

Latest revision as of 20:56, 16 March 2023


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 Zbl 52.0110.09
[a2] P.J. Higgins, "Categories and groupoids" , v. Nostrand-Reinhold (1971) Zbl 0226.20054
[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=36918
This article was adapted from an original article by V.D. Belousov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article