Category where every morphism is invertible; generalization of a group
This article is about groupoids in category theory. For the algebraic structure with a single binary operation, see magma (algebra).
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 on the morphisms, called inverse by analogy with group theory.[1] A groupoid where there is only one object is a usual group.
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:
Setoids: sets that come with an equivalence relation,
G-sets: sets equipped with an action of a group .
Groupoids are often used to reason about geometrical objects such as manifolds. Heinrich Brandt (1927) introduced groupoids implicitly via Brandt semigroups.[2]
^Dicks & Ventura (1996). The Group Fixed by a Family of Injective Endomorphisms of a Free Group. p. 6.
^"Brandt semi-group", Encyclopedia of Mathematics, EMS Press, 2001 [1994], ISBN 1-4020-0609-8
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