What is "groupoid"

n. 1 (context algebra English) A magma: a set with a total binary operation. 2 (context algebra and category theory English) A set with a partial binary operation that is associative and has inverses and identity.

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.

Groupoids are often used to reason about geometrical objects such as manifolds. introduced groupoids implicitly via Brandt semigroups.