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.
- Oriented graph
Special cases include:
- Setoids, that is: sets which come with an equivalence relation;
- G-sets, sets equipped with an action of a group G.
Groupoids are often used to reason about geometrical objects such as manifolds. Heinrich Brandt introduced groupoids implicitly via Brandt semigroups in 1926.
Read more about Groupoid: Relation To Groups, Lie Groupoids and Lie Algebroids
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