Groupoid


In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:
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:
Groupoids are often used to reason about geometrical objects such as manifolds. introduced groupoids implicitly via Brandt semigroups.

Definitions

Algebraic

A groupoid can be viewed as an algebraic structure consisting of a set with a binary partial function. Precisely, it is a non-empty 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 have the following axiomatic properties: For all,, and in,
  1. 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 and are also defined.
  2. Inverse: and are always defined.
  3. Identity: If is defined, then, and.
Two convenient properties follow from these axioms:
  • ,
  • If is defined, then.

    Category-theoretic

A groupoid is a small category in which every morphism is an isomorphism, i.e., invertible. More explicitly, a groupoid is a set of objects with
  • for each pair of objects and, a set of morphisms from to ; we write to indicate that is an element of ;
  • for each triple of objects,, and, a function that is associative. That is, for every four objects,,, and functions
  • *
  • for every object, a designated element of satisfying, for any morphism
  • * and ;
  • for each pair of objects,, a function satisfying, for any :
  • * and.
If the requirement that inverses exist is removed while keeping everything else, this is then the definition of a category. Thus, a groupoid is a category in which every morphism has an inverse.
If is an element of, then is called the source of, written, and is called the target of, written.
A groupoid is sometimes denoted as, where is the set of all morphisms, and the two arrows represent the source and the target.
More generally, one can consider a groupoid object in an arbitrary category admitting finite fiber products.

Comparing the definitions

The algebraic and category-theoretic definitions are equivalent, as we now show. Given a groupoid in the category-theoretic sense, let be the disjoint union of all of the sets . Then and become partial operations on, and will in fact be defined everywhere. We define to be and to be, which gives a groupoid in the algebraic sense. Explicit reference to can be dropped.
Conversely, given a groupoid in the algebraic sense, define an equivalence relation on its elements by
iff. Let be the set of equivalence classes of, i.e.. Denote by if with.
Now define as the set of all elements such that exists. Given and, their composite is defined as. To see that this is well defined, observe that since and exist, so does. The identity morphism on is then, and the category-theoretic inverse of is.
Sets in the definitions above may be replaced with classes, as is generally the case in category theory.

Vertex groups and orbits

Given a groupoid, the vertex groups or isotropy groups or object groups in are the subsets of the form, where is any object of. 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.
The orbit of a groupoid at a point is given by the set containing every point that can be joined to by a morphism in. If two points and are in the same orbits, their vertex groups and are isomorphic: if is any morphism from to, then the isomorphism is given by the mapping.
Orbits form a partition of the set, and a groupoid is called transitive if it has only one orbit. In that case, all the vertex groups are isomorphic.

Subgroupoids and morphisms

A subgroupoid of is a subcategory that is itself a groupoid. It is called wide or full if it is wide or full as a subcategory, i.e., respectively, if or for every.
A groupoid morphism is simply a functor between two groupoids.
Particular kinds of morphisms of groupoids are of interest. A morphism of groupoids is called a fibration if for each object of and each morphism of starting at there is a morphism of starting at such that. A fibration is called a covering morphism or covering of groupoids if further such an is unique. The covering morphisms of groupoids are especially useful because they can be used to model covering maps of spaces.
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.

Examples

Fundamental groupoid

Given a topological space, let be the set. The morphisms from the point to the point are equivalence classes of continuous paths from to, with two paths being equivalent if they are homotopic.
Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is associative. This groupoid is called the fundamental groupoid of, denoted . The usual fundamental group is then the vertex group for the point.
The orbits of the fundamental groupoid are the path-connected components of. Accordingly, the fundamental groupoid of a path-connected space is transitive, and we recover the known fact that the fundamental groups at any base point are isomorphic. Moreover, in this case, the fundamental groupoid and the fundamental groups are equivalent as categories.
An important extension of this idea is to consider the fundamental groupoid where is a chosen set of "base points". Here is a subgroupoid of, where one considers only paths whose endpoints belong to. The set may be chosen according to the geometry of the situation at hand.

Equivalence relation

If is a setoid, i.e. a set with an equivalence relation, then a groupoid "representing" this equivalence relation can be formed as follows:
  • The objects of the groupoid are the elements of ;
  • For any two elements and in, there is a single morphism from to if and only if ;
  • The composition of and is.
The vertex groups of this groupoid are always trivial; moreover, this groupoid is in general not transitive and its orbits are precisely the equivalence classes. There are two extreme examples:
  • If every element of is in relation with every other element of, we obtain the pair groupoid of, which has the entire as set of arrows, and which is transitive.
  • If every element of is only in relation with itself, one obtains the unit groupoid, which has as set of arrows,, and which is completely intransitive.

    Examples

  • If is a smooth surjective submersion of smooth manifolds, then is an equivalence relation since has a topology isomorphic to the quotient topology of under the surjective map of topological spaces. If we write, then we get a groupoid which is sometimes called the banal groupoid of a surjective submersion of smooth manifolds.
  • If we relax the reflexivity requirement and consider partial equivalence relations, then it becomes possible to consider semidecidable notions of equivalence on computable realisers for sets. This allows groupoids to be used as a computable approximation to set theory, called PER models. Considered as a category, PER models are a cartesian closed category with natural numbers object and subobject classifier, giving rise to the effective topos introduced by Martin Hyland.

    Čech groupoid

A Čech groupoidp. 5 is a special kind of groupoid associated to an equivalence relation given by an open cover of some manifold. Its objects are given by the disjoint union
and its arrows are the intersections
The source and target maps are then given by the induced maps
and the inclusion map
giving the structure of a groupoid. In fact, this can be further extended by setting
as the -iterated fiber product where the represents -tuples of composable arrows. The structure map of the fiber product is implicitly the target map, since
is a cartesian diagram where the maps to are the target maps. This construction can be seen as a model for some -groupoids. Also, another artifact of this construction is -cocycles
for some constant sheaf of abelian groups can be represented as a function
giving an explicit representation of cohomology classes.

Group action

If the group acts on the set, then we can form the action groupoid representing this group action as follows:
  • The objects are the elements of ;
  • For any two elements and in, the morphisms from to correspond to the elements of such that ;
  • Composition of morphisms interprets the binary operation of.
More explicitly, the action groupoid is a small category with and and with source and target maps and. It is often denoted . Multiplication in the groupoid is then, which is defined provided.
For in, the vertex group consists of those with, which is just the isotropy subgroup at for the given action. Similarly, the orbits of the action groupoid are the orbit of the group action, and the groupoid is transitive if and only if the group action is transitive.
Another way to describe -sets is the functor category, where is the groupoid with one element and isomorphic to the group. Indeed, every functor of this category defines a set and for every in induces a bijection :. The categorical structure of the functor assures us that defines a -action on the set. The representable functor is the Cayley representation of. In fact, this functor is isomorphic to and so sends to the set which is by definition the "set" and the morphism of to the permutation of the set. We deduce from the Yoneda embedding that the group is isomorphic to the group, a subgroup of the group of permutations of.