This leads to the notion of an internal groupoid in a category.

If a groupoid has only one object, then the set of its morphisms forms a group.

Using the algebraic definition, such a groupoid is literally just a group.

Thus we have a groupoid in the algebraic sense.

Thus any groupoid is equivalent to a multiset of unrelated groups.

The fifteen puzzle is a classic application, though it actually involves a groupoid.

Composition is defined via compatible representatives as in the pair groupoid.

Shows the advantage of generalising from group to groupoid.

In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid.

Also there is proved there a nice normal form for the elements of the fundamental groupoid.

A subgroupoid is a subcategory that is itself a groupoid.

As an example consider the Lie groupoid cohomology.

A groupoid is a category in which every morphism is an isomorphism.

A group G can be considered a category (even a groupoid) with one object which we denote by -.

More generally, G can be a Lie groupoid.

Thus one has the fundamental groupoid instead of the fundamental group, and this construction is functorial.

Groupoid in category theory (an area of mathematics)

A groupoid can be seen as a:

This requires a generalization from the concept of symmetry group to that of a groupoid.

Any Lie group gives a Lie groupoid with one object, and conversely.

A quite general example is the Morita-morphism of the Čech groupoid which goes as follows.

A set G with a closed n-ary operation is an n-ary groupoid.

As a more explicit example consider the Lie algebroid associated to the pair groupoid .

The semidirect product is then relevant to finding the fundamental groupoid of the orbit space .

(A double groupoid can also be considered as a generalization of certain higher-dimensional groups.)