In mathematics, especially in category theory and homotopy theory, a groupoid is a concept (first developed by Heinrich Brandt) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. They are often used to capture information about geometrical objects such as manifolds.
- A set G0 of objects;
- For each pair of objects x and y in G0, a set G(x,y) of morphisms (or arrows) from x to y -- we write f: x -> y to indicate that f is an element of G(x,y);
- An element idx of G(x,x);
- For each triple of objects x, y, and z, a binary function compx,y,z from G(x,y) and G(y,z) to G(x,z) -- we write g'f for compx,y,z(f,g);
- A function invx,y from G(x,y) to G(y,x) -- we write f-1 for invx,y(f);
- If f: x -> y, then fidx = f and idyf = f;
- If f: x -> y, g: y -> z, and h: z -> w, then (h'g)f = h(g'f);
- If f: x -> y, then f'f-1 = idy and f-1f = idx.
One can also define a groupoid as a certain algebraic structure. To be specific, let G be a set and let comp be a partially defined binary operation on G. That is, given elements f and g of G, comp(f,g) may be an element of G, or it may be undefined. We write g'f for comp(f,g). There is also a total (everywhere defined) function inv on G. We write f-1 for the inverse inv(f) of f. Then G is a groupoid if:
- Whenever f'g and g'h are both defined, then (f'g)h and f(g'h) are also defined, and they are equal;
- f-1f and f'f-1 are always defined;
- Whenever f'g is defined, then f'g'g-1 = f and f-1f'g = g -- we already know that these expressions are unambiguously defined by the previous conditions.
The relation between this definitions is as follows: Given a groupoid in the category-theoretic sense, let G be the disjoint union of all of the sets G(x,y). Then inv and comp become partially defined operations on G, and inv will in fact be defined everywhere. Explicit reference to G0 (and hence to id) can be dropped.
On the other hand, given a groupoid in the algebraic sense, let G0 be the set of all elements of the form f'f-1 for some element f of G. In other words, the objects are identified with the identity morphisms, and idx is just x. Let G(x,y) be the set all elements f such that y'f'x is defined. Then inv and comp break up into several functions on the various G(x,y).
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.
From linear algebra: Given a field K, the general linear groupoid GL*(K) consists of all invertible matrices with entries from K, with composition given by matrix multiplication. If G = GL*(K), then G0 can be identified with the set of natural numbers, since there is one identity matrix for each natural number. G(m,n) is empty unless m = n, in which case it is the set of n by n matrices.
From topology: Start with a topological space X and let G0 be the set X. The morphisms from the point p to the point q are equivalence classes of continuous paths from p to q, 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 of X, denoted Π1(X).
If X is a set and ~ is an equivalence relation on X, then we can form a groupoid representing this equivalence relation as follows: The objects are the elements of X, and for any two elements x and y in X, there is a single morphism from x to y if and only if x ~ y.
If the group G acts on the set X, then we can form a groupoid representing this group action as follows: The objects are the elements of X, and for any two elements x and y in X, there is a morphism from x to y for every element g of G such that g.x = y. Composition of morphisms is given by the group operation in G.
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 x is an object of the groupoid G, then the set of all morphisms from x to x forms a group G(x). If there is a morphism f from x to y, then the groups G(x) and G(y) are isomorphic, with an isomorphism given by mapping g to f'g'f-1.
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 G and a set (or class) X. Let the objects of the groupoid be the elements of X. For elements x and y of X, let the set of morphisms from x to y be G. Composition of morphisms is the group operation of G. If the groupoid is not connected, then it is isomorphic to a disjoint union of groupoids of the above type (possibly with different groups G per connected component). Thus, any groupoid may be given (up to isomorphism) by a set of ordered pairs (X,G).
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 x0, a group isomorphism h from G(x0) to G, and for each x other than x0 a morphism in G from x0 to x.
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 X, only the groups G.
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 GLn(F). On the other hand, the fundamental groupoid of X is equivalent to the collection of the fundamental groups of each path-connected component of X, but for an isomorphism you must also specify the set of points in each component. The set X 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 X equipped with an action of the group G is equivalent (as a groupoid) to one copy of G 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 G(x) 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 p to each point q in the same path-connected component.
An example of this phenomenon that is well known in physics is covariance in special relativity. Working with a single group corresponds to picking a specific frame of reference, and you can do all of physics in this fashion. But it's more natural to describe physics in a way that makes no mention of any particular frame of reference, and this corresponds to using the entire groupoid. (I need to go into more detail about this. It really is a precise correspondence -- the particular group involved is the [[Poincar� group]] -- but I'm not sure how best to explain it yet.)
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.
- Explain this
- 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.