In mathematics, especially in category theory and homotopy theory, a groupoid is a simultaneous generalisation of a group, a setoid (a set equipped with an equivalence relation), and a -set (a set equipped with an action of a group ).
Groupoids are often used to capture information about geometrical objects such as manifolds.
Groupoids were first developed by Heinrich Brandt in 1926.
A groupoid is a set
with two operations: a partially defined binary operation
and a total (everywhere defined) function
, which satisfy the following three conditions on elements
- Associativity: For all , and in , , if either product is defined.
- Identity: Where is defined and , uniquely.
- Inverse: and are always defined.
Category theory definition
From a more abstract point of view, a groupoid is simply a small category
in which every morphism
is an isomorphism
(that is, invertible).
To be explicit, a groupoid
- a set of objects;
- for each pair of objects and in , a set of morphisms (or arrows) from to — we write to indicate that is an element of ;
- an element of ;
- for each triple of objects , , and , a binary function from to — we write for , where , ;
- a function from to ;
- if , then and ;
- if , , and , then ;
- if , then and .
Comparison of the definitions
The relation between these definitions is as follows:
Given a groupoid in the category-theoretic sense, let
be the disjoint union
of all of the sets
become partially defined operations on
will in fact be defined everywhere; so we define
. This gives a groupoid in the algebraic definition.
Explicit reference to
(and hence to
) can be dropped.
On the other hand, given a groupoid in the algebraic sense, let be the set of all elements of the form for elements of .
In other words, the objects are identified with the identity morphisms, and is just .
Let be the set of all elements such that is defined.
Then and break up into several functions on the various , which may be called and , respectively.
While we have referred to sets in the definitions above, one may instead want to use classes, in the same way as for other categories.
The category whose objects are groupoids and whose morphisms are groupoid
homomorphisms is called the groupoid category
, or the category of groupoids
Given a field
, the general linear groupoid
consists of all invertible matrices
with entries from
, with composition given by matrix multiplication
contains a copy of the set of natural numbers
, since there is one identity matrix
for each natural number
contains other matrices.
, in which case it is the set of
Start with a topological space
be the set
The morphisms from the point
to the point
are equivalence classes
of continuous paths
, with two paths being considered 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
An important extension of this idea is to consider the fundamental groupoid where is a subset of . Here, one considers only paths whose endpoints belong to . It is a sub-groupoid of . The set of base points may be chosen according to the geometry of the situation at hand.
is a set and
is an equivalence relation
, then we can form a groupoid representing this equivalence relation as follows:
The objects are the elements of
, and for any two elements
, there is a single morphism from
to if and only if
If the group acts
on the set
, then we can form a groupoid representing this group action as follows:
The objects are the elements of
, and for any two elements
, there is a morphism from
for every element
Composition of morphisms is given by the group operation in
. Another way to describe
-sets is the functor category
is the groupoid (category) with one element and isomorphic to the group
. Indeed, every functor
of this category defines a set
and for every
(i.e. morphism in
) induces a bijection
. The categorical structure of the functor
assures us that
-action on the set
. The (unique) 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
(i.e. the element
) to the permutation
of the set
. We deduce from the Yoneda Embedding that the group
is isomorphic to the group
which is a subgroup of the group of permutations of
The symmetries of the Fifteen puzzle
form a groupoid (not a group, as not all moves can be composed).
This groupoid acts
Relation to groups
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.
Many concepts of group theory
can be generalized to groupoids, with the notion of group homomorphism
being replaced by that of functor
If is an object of the groupoid , then the set of all morphisms from to forms a group .
If there is a morphism from to , then the groups and are isomorphic, with an isomorphism given by mapping to .
Every connected groupoid (that is, one in which any two objects are connected by at least one morphism) is isomorphic to a groupoid of the following form:
Pick a group and a set (or class) .
Let the objects of the groupoid be the elements of .
For elements and of , let the set of morphisms from to be .
Composition of morphisms is the group operation of .
If the groupoid is not connected, then it is isomorphic to a disjoint union of groupoids of the above type (possibly with different groups per connected component).
Thus, any groupoid may be given (up to isomorphism) by a set of ordered pairs .
Note that the isomorphism described above is not unique, and there is no natural choice.
Choosing such an isomorphism for a connected groupoid essentially amounts to picking one object , a group isomorphism from to , and for each other than a morphism in from to .
In category-theoretic terms, each connected component of a groupoid is equivalent (but not isomorphic) to a groupoid with a single object, that is, a single group.
Thus any groupoid is equivalent to a multiset of unrelated groups.
In other words, for equivalence instead of isomorphism, you don’t have to specify the sets , only the groups .
Consider the examples in the previous section.
The general linear groupoid is both equivalent and isomorphic to the disjoint union of the various general linear groups .
On the other hand, the fundamental groupoid of is equivalent to the collection of the fundamental groups of each path-connected component of , but for an isomorphism you must also specify the set of points in each component.
The set with the equivalence relation is equivalent (as a groupoid) to one copy of the trivial group for each equivalence class, but for an isomorphism you must also specify what each equivalence class is.
Finally, the set equipped with an action of the group is equivalent (as a groupoid) to one copy of for each orbit of the action, but for an isomorphism you must also specify what set each orbit is.
The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it’s not natural.
Thus when groupoids arise in terms of other structures, as in the above examples, it can be helpful to maintain the full groupoid.
If you don’t, then you must choose a way to view each in terms of a single group, and this can be rather arbitrary.
In our example from topology, you would have to make a coherent choice of paths (or equivalence classes of paths) from each point to each point in the same path-connected component.
As a more illuminating example, the classification of groupoids with one endomorphism does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification of vector spaces with one endomorphism is non trivial.
Morphisms of groupoids come in more kinds than those of groups: we have, for example, fibrations, covering morphisms, universal morphisms, quotient morphisms. Thus a subgroup of a group yields an action of on the set of cosets of in and hence a covering morphism from say to where is a groupoid with vertex groups isomorphic to . In this way, presentations of the group can be lifted to presentations of the groupoid , and this is a useful way of obtaining information on presentations of the subgroup . For further information, see the books by Higgins and by Brown listed below.
Another useful fact is that the category of groupoids, unlike that of groups, is cartesian closed.
Lie groupoids and Lie algebroids
When studying geometrical objects, the arising groupoids often carry some differentiable structure, turning them into Lie groupoids.
These can be studied in terms of Lie algebroids, in analogy to the relation between Lie groups and Lie algebras.
- Ronald Brown, From groups to groupoids: a brief survey, Bull. LMS, 19 (1987) 113-134, gives some of the history of groupoids, namely the origins in work of Brandt on quadratic forms, and an indication of later work up to 1987, with 160 references. These have been updated slightly in the downloadable version, available as
- Alan Weinstein, Groupoids: unifying internal and external symmetry, available as Groupoids.ps or weinstein.pdf
- Part VI of Geometric Models for Noncommutative Algebras, by A. Cannas da Silva and A. Weinstein PDF file.
- Higher dimensional group theory is a web article with lots of references explaining how the groupoid concept has to led to notions of higher dimensional groupoids, not available in group theory, with applications in homotopy theory and in group cohomology.
- General theory of Lie groupoids and Lie algebroids, K.C.H. Mackenzie, CUP, 2005
- Topology and groupoids, Ronald Brown, Booksurge 2006 revised and extended edition of a book previously published in 1968 and 1988. e-version available.
- Categories and groupoids, P.J. Higgins, downloadable reprint of van Nostrand Notes in Mathematics, 1971, which deal with applications of groupoids in group theory and topology.
- Galois theories, F. Borceux, G. Janelidze, CUP, 2001 shows how generalisations of Galois theory lead to Galois groupoids.
- M. Golubitsky, I. Stewart, `Nonlinear dynamics of networks: the groupoid formalism', Bull. Amer. Math. Soc. 43 (2006), 305-364