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groupoids
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Definitions of groupoids in various dictionaries:
noun - a type of mathematical set
GROUPOIDS - In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of...
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| Plural form of groupoid. |
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In mathematics, especially in category theory and homotopy theory, a Groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a:* Group with a partial function replacing the binary operation; * Category in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation, called inverse by analogy with group theory. Notice that a groupoid where there is only one object is a usual group.Special cases include: * * Setoids, that is: sets that come with an equivalence relation; * G-sets, sets equipped with an action of a group * * * * G * * * {\displaystyle G} * .Groupoids are often used to reason about geometrical objects such as manifolds. Heinrich Brandt (1927) introduced groupoids implicitly via Brandt semigroups. |