Category in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation, called inverse by analogy with group theory. A groupoid where there is only one object is a usual group.
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 .
A groupoid is an algebraic structure consisting of a non-empty set and a binary partial function '' defined on .
A groupoid is a 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.
and -1 have the following axiomatic properties: For all , , and in ,
Associativity: If and are defined, then and are defined and are equal. Conversely, if one of and is defined, then so are both and as well as = .
For each pair of objects x and y in G0, there exists a (possibly empty) 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).
For every object x, a designated element of G(x,x);
For each triple of objects x, y, and z, a function;
For each pair of objects x, y a function ;
satisfying, for any f : x -> y, g : y -> z, and h : z -> w:
If f is an element of G(x,y) then x is called the source of f, written s(f), and y is called the target of f, written t(f).
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 G be the disjoint union of all of the sets G(x,y) (i.e. the sets of morphisms from x to y). Then and become partial operations on G, and will in fact be defined everywhere. We define * to be and -1 to be , which gives a groupoid in the algebraic sense. Explicit reference to G0 (and hence to ) can be dropped.
Conversely, given a groupoid G in the algebraic sense, define an equivalence relation on its elements by
iff a * a-1 = b * b-1. Let G0 be the set of equivalence classes of , i.e. . Denote a * a-1 by if with .
Now define as the set of all elements f 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 x is then , and the category-theoretic inverse of f is f-1.
Sets in the definitions above may be replaced with classes, as is generally the case in category theory.
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.
Category of groupoids
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 useful that this category is, like the category of small categories, Cartesian closed. That is, we can construct for any groupoids a groupoid whose objects are the morphisms and whose arrows are the natural equivalences of morphisms. Thus if are just groups, then such arrows are the conjugacies of morphisms. The main result is that for any groupoids there is a natural bijection
This result is of interest even if all the groupoids are just groups.
Fibrations and coverings
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.
Given a topological space, let be the set . The morphisms from the point to the point are equivalence classes of continuouspaths 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 (or sometimes, ). The usual fundamental group is then the vertex group for the point . For a path-connected space, fundamental groupoid and the fundamental group coincide, and the composition operation is defined for all pairs of equivalence classes.
An important extension of this idea is to consider the fundamental groupoid where is a chosen set of "base points". Here, one considers only paths whose endpoints belong to . is a sub-groupoid of . The set may be chosen according to the geometry of the situation at hand.
If is a set with an equivalence relation denoted by infix, 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.
If the group acts on the set , then we can form the action groupoid (or transformation 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 ;
More explicitly, the action groupoid is a small category with and with source and target maps and . It is often denoted (or ). Multiplication (or composition) 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 (which is why vertex groups are also called isotropy groups).
Another way to describe -sets is the functor category, where is the groupoid (category) with one element and isomorphic to the group . Indeed, every functor of this category defines a set and for every in (i.e. for every morphism in ) induces a bijection : . The categorical structure of the functor assures us that defines a -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 of (i.e. the element 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 .
Consider the finite set , we can form the group action acting on by taking each number to its negative, so and . The quotient groupoid is the set of equivalence classes from this group action , and has a group action of on it.
On , any finite group which maps to give a group action on (since this is the group of automorphisms). Then, a quotient groupoid can be forms , which has one point with stabilizer at the origin.
Fiber product of groupoids
Given a diagram of groupoids with groupoid morphisms
where and , we can form the groupoid whose objects are triples , where , , and in . Morphisms can be defined as a pair of morphisms where and such that for triples , there is a commutative diagram in of , and the .
A two term complex
of objects in a concrete Abelian category can be used to form a groupoid. It has as objects the set and arrows where the source morphism is just the projection onto while the target morphism is the addition of projection onto composed with and projection onto . That is, given we have
Of course, if the abelian category is the category of coherent sheaves on a scheme, then this construction can be used to form a presheaf of groupoids.
^?Closure, which is used in many sources, is an equivalent axiom to totality, though defined differently.
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 generalize to groupoids, with the notion of functor replacing that of group homomorphism.
If is an object of the groupoid , then the set of all morphisms from to forms a group (called the vertex group, defined above). If there is a morphism from to , then the groups and are isomorphic, with an isomorphism given by the mapping.
Every connected groupoid - that is, one in which any two objects are connected by at least one morphism - is isomorphic to an action groupoid (as defined above) . By connectedness, there will only be one orbit under the action. If the groupoid is not connected, then it is isomorphic to a disjoint union of groupoids of the above type (possibly with different groups and sets for each connected component).
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, one need not specify the sets , only the groups For example,
The set with the equivalence relation is equivalent (as a groupoid) to one copy of the trivial group for each equivalence class, but an isomorphism requires specifying what each equivalence class is:
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 an isomorphism requires specifying 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 is 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. Otherwise, one must choose a way to view each in terms of a single group, and this choice can be 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 nontrivial.
Morphisms of groupoids come in more kinds than those of groups: we have, for example, fibrations, covering morphisms, universal morphisms, and 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 about presentations of the subgroup . For further information, see the books by Higgins and by Brown in the References.
Proof of first property: from 2. and 3. we obtain a-1 = a-1 * a * a-1 and (a-1)-1 = (a-1)-1 * a-1 * (a-1)-1. Substituting the first into the second and applying 3. two more times yields (a-1)-1 = (a-1)-1 * a-1 * a * a-1 * (a-1)-1 = (a-1)-1 * a-1 * a = a. ?
Proof of second property: since a * b is defined, so is (a * b)-1 * a * b. Therefore (a * b)-1 * a * b * b-1 = (a * b)-1 * a is also defined. Moreover since a * b is defined, so is a * b * b-1 = a. Therefore a * b * b-1 * a-1 is also defined. From 3. we obtain (a * b)-1 = (a * b)-1 * a * a-1 = (a * b)-1 * a * b * b-1 * a-1 = b-1 * a-1. ?
^J.P. May, A Concise Course in Algebraic Topology, 1999, The University of Chicago Press ISBN0-226-51183-9 (see chapter 2)
^Mapping a group to the corresponding groupoid with one object is sometimes called delooping, especially in the context of homotopy theory, see "delooping in nLab". ncatlab.org. Retrieved ..
Brandt, H (1927), "Über eine Verallgemeinerung des Gruppenbegriffes", Mathematische Annalen, 96 (1): 360-366, doi:10.1007/BF01209171
Brown, Ronald, 1987, "From groups to groupoids: a brief survey," Bull. London Math. Soc.19: 113-34. Reviews the history of groupoids up to 1987, starting with the work of Brandt on quadratic forms. The downloadable version updates the many references.
—, 2006. Topology and groupoids. Booksurge. Revised and extended edition of a book previously published in 1968 and 1988. Groupoids are introduced in the context of their topological application.
Higgins, P. J., "The fundamental groupoid of a graph of groups", J. London Math. Soc. (2) 13 (1976) 145--149.
Higgins, P. J. and Taylor, J., "The fundamental groupoid and the homotopy crossed complex of an orbit space", in Category theory (Gummersbach, 1981), Lecture Notes in Math., Volume 962. Springer, Berlin (1982), 115--122.
Higgins, P. J., 1971. Categories and groupoids. Van Nostrand Notes in Mathematics. Republished in Reprints in Theory and Applications of Categories, No. 7 (2005) pp. 1-195; freely downloadable. Substantial introduction to category theory with special emphasis on groupoids. Presents applications of groupoids in group theory, for example to a generalisation of Grushko's theorem, and in topology, e.g. fundamental groupoid.
R.T. Zivaljevic. "Groupoids in combinatorics—applications of a theory of local symmetries". In Algebraic and geometric combinatorics, volume 423 of Contemp. Math., 305-324. Amer. Math. Soc., Providence, RI (2006)