One of the simplest examples of a category is that of groupoid, defined as a category whose arrows or morphisms are all invertible.

The groupoid concept is important in topology.

Basic constructions, such as the fundamental group or fundamental groupoid of a topological space, can be expressed as fundamental functors to the category of groupoids in this way, and the concept is pervasive in algebra and its applications.

Thus it is meaningful to speak of a " presentation of an equivalence relation ," i. e., a presentation of the corresponding groupoid ;

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

A groupoid is a small category in which every morphism is an isomorphism, and hence invertible.

More precisely, a groupoid G is:

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

A subgroupoid is a subcategory that is itself a groupoid.

A groupoid morphism is simply a functor between two ( category-theoretic ) groupoids.

The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, denoted Grpd.

It is also true that the category of covering morphisms of a given groupoid is equivalent to the category of actions of the groupoid on sets.

This groupoid is called the fundamental groupoid of X, denoted ( X ).

An important extension of this idea is to consider the fundamental groupoid ( X, A ) where A is a set of " base points " and a subset of X.

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.

Conversely, given a groupoid G in the algebraic sense, let G < sub > 0 </ sub > be the set of all elements of the form x * x < sup >− 1 </ sup > with x varying through G and define G ( x * x < sup >-1 </ sup >, y * y < sup >-1 </ sup >) as the set of all elements f such that y * y < sup >-1 </ sup > * f * x * x < sup >-1 </ sup > exists.

If X is a set with an equivalence relation denoted by infix, then a groupoid " representing " this equivalence relation can be formed as follows:

If the group G acts on the set X, then we can form the action groupoid representing this group action as follows:

More explicitly, the action groupoid is the set with source and target maps s ( g, x )

Another way to describe G-sets is the functor category, where is the groupoid ( category ) with one element and isomorphic to the group G. Indeed, every functor F of this category defines a set X = F and for every g in G ( i. e. for every morphism in ) induces a bijection F < sub > g </ sub >: X → X.

In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, the source and target operations

This problem can be resolved by working with the fundamental groupoid on a < I > set A </ I > of base points, chosen according to the geometry of the situation.

As explained above, this theorem was extended by R. Brown to the non-connected case by using the fundamental groupoid on a set A of base points.

Applications of the fundamental groupoid on a set of base points to the Jordan curve theorem, Covering spaces, and orbit spaces are given in Ronald Brown's book.

In abstract algebra, a medial magma ( or medial groupoid ) is a set with a binary operation which satisfies the identity

It is not an axiomatic algebraic idea ; rather it defines a set of closure conditions on sets of homeomorphisms defined on open sets U of a given Euclidean space E or more generally of a fixed topological space S. The groupoid condition on those is fulfilled, in that homeomorphisms

of all partial one-one transformations of a set forms not an inverse semigroup but an inductive groupoid, in the

Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G ( x, y ) ( i. e. the sets of morphisms from x to y ).

However the group G does act on the fundamental groupoid of X, and so the study is best handled by considering groups acting on groupoids, and the corresponding orbit groupoids.

The main result is that for discontinuous actions of a group G on a Hausdorff space X which admits a universal cover, then the fundamental groupoid of the orbit space is isomorphic to the orbit groupoid of the fundamental groupoid of X, i. e. the quotient of that groupoid by the action of the group G. This leads to explicit computations, for example of the fundamental group of the symmetric square of a space.

In abstract algebra, a magma ( or groupoid ; not to be confused with groupoids in category theory ) is a basic kind of algebraic structure.

Just as a Lie groupoid can be thought of as a " Lie group with many objects ", a Lie algebroid is like a " Lie algebra with many objects ".

For example, the Lie algebroid comes from the pair groupoid whose objects are, with one isomorphism between each pair of objects.

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

* Given any manifold, there is a Lie groupoid called the pair groupoid, with as the manifold of objects, and precisely one morphism from any object to any other.

* Given a Lie group acting on a manifold, there is a Lie groupoid called the translation groupoid with one morphism for each triple with.

* Any principal bundle with structure group G gives a groupoid, namely over M, where G acts on the pairs componentwise.

In this sense we say that 2 groupoids and are Morita equivalent iff there exists a third groupoid together with 2 Morita morphisms from G to K and H to K. Transitivity is an interesting construction in the category of groupoid principal bundles and left to the reader.

In particular, if " X " is a contractible space, and " A " consists of two distinct points of X, then is easily seen to be isomorphic to the groupoid often written with two vertices and exactly one morphism between any two vertices.

constructed from an inverse semigroup, and conversely .. More precisely, an inverse semigroup is precisely a groupoid in the category of posets which is an etale groupoid with respect to its ( dual ) Alexandrov topology and whose poset of objects is a meet-semilattice.

* Magma or groupoid: S and a single binary operation over S.