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| − | A [[Universal algebra|universal algebra]] with one binary operation. It is the broadest class of such algebras: groups, semi-groups, quasi-groups — all these are groupoids of a special type. An important concept in the theory of groupoids is that of isotopy of operations. On a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g0453601.png" /> let there be defined two binary operations, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g0453602.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g0453603.png" />; they are isotopic if there exist three one-to-one mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g0453604.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g0453605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g0453606.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g0453607.png" /> onto itself such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g0453608.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g0453609.png" />. A groupoid that is isotopic to a [[Quasi-group|quasi-group]] is itself a quasi-group; a groupoid with a unit element that is isotopic to a group, is also isomorphic to this group. For this reason, in group theory the concept of isotopy is not used: For groups isotopy and isomorphism coincide.
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| − | A groupoid with cancellation is a groupoid in which either of the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536011.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536012.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536015.png" /> are elements of the groupoid. Any groupoid with cancellation is imbeddable into a quasi-group. A homomorphic image of a quasi-group is a groupoid with division, that is, a groupoid in which the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536017.png" /> are solvable (but do not necessarily have unique solutions).
| + | ''for the general algebraic structure, see [[Magma]]'' |
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| − | A set with one partial binary operation (i.e. one not defined for all pairs of elements) is said to be a partial groupoid. Any partial subgroupoid of a free partial groupoid is free. | + | 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 |
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| − | ====References====
| + | 1) $gg^{-1}$ and $g^{-1}g$ are always defined; |
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> P.M. Cohn, "Universal algebra" , Reidel (1981)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> O. Boruvka, "Foundations of the theory of groupoids and groups" , Wiley (1976) (Translated from German)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> R.H. Bruck, "A survey of binary systems" , Springer (1958)</TD></TR></table>
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| − | ====Comments====
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| − | There is another, conflicting, use of the term "groupoid" in mathematics, which was 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536018.png" /> equipped with a unary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536019.png" /> and a partial binary operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536020.png" /> satisfying
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| − | 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536022.png" /> are always defined; | |
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| − | 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536023.png" /> is defined if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536024.png" />; | + | 2) $gh$ is defined if and only if $g^{-1}g=hh^{-1}$; |
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| − | 3) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536026.png" /> are defined, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536028.png" /> are defined and equal; | + | 3) if $gh$ and $hk$ are defined, then $(gh)k$ and $g(hk)$ are defined and equal; |
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| − | 4) each of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536031.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536032.png" /> is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045360/g04536033.png" /> 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. |
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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]]], different geometry [[#References|[a3]]] and topology [[#References|[a4]]], [[#References|[a5]]]. | + | 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]]]. |
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| | ====References==== | | ====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> | + | <table> |
| | + | <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) {{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">[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> |
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 |