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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:
In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed , , say. Composition is then a total function: , so that .
Special cases include:
Groupoids are often used to reason about geometrical objects such as manifolds. Heinrich Brandt (1927) introduced groupoids implicitly via Brandt semigroups.