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Concept
Groupoid
Summary
In mathematics, especially in 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;
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. 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. introduced groupoids implicitly via Brandt semigroups.
A groupoid is an algebraic structure consisting of a non-empty set and a binary partial function '' defined on .
A groupoid is a set with a unary operation and a partial function . Here * is not a binary operation because it is not necessarily defined for all pairs of elements of . The precise conditions under which is defined are not articulated here and vary by situation.
The operations and −1 have the following axiomatic properties: For all , , and in ,
Associativity: If and are defined, then and are defined and are equal. Conversely, if one of or is defined, then they are both defined (and they are equal to each other), and and are also defined.
Inverse: and are always defined.
Identity: If is defined, then , and . (The previous two axioms already show that these expressions are defined and unambiguous.)
Two easy and convenient properties follow from these axioms:
If is defined, then .
A groupoid is a in which every morphism is an isomorphism, i.
Official source
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