My current top candidate for a mathematical concept that should be and is not (as far as I can tell) consistently taught at the advanced undergraduate / beginning graduate level is the notion of a groupoid. Today’s post is a very brief introduction to groupoids together with some suggestions for further reading.
Definition and examples
A groupoid is a small category in which every morphism is invertible. A morphism between groupoids is a functor between categories.
Example. A groupoid with one object is uniquely determined by its automorphism group ; consequently, any group may be regarded as a one-object groupoid. More precisely, the category of one-object groupoids is
isomorphic (not just equivalent!) equivalent to the category of groups.
The one-object category corresponding to a given group is often denoted by . This notation is also used for the classifying space of because taking the geometric realization of the nerve of the category gives a model for the space.
Example. Let be a set equipped with an equivalence relation . From this data we can construct a groupoid with object set and a unique morphism exactly when ; consequently, any equivalence relation may be regarded as a groupoid in which objects have no nontrivial automorphisms. More precisely, the category of groupoids whose objects have no nontrivial automorphisms is
isomorphic (again, not just equivalent!) equivalent to the category of equivalence relations. (This is the category whose objects are sets equipped with equivalence relations and whose morphisms are functions respecting the equivalence relations.)
Example. Let be a group acting on a set . From this data we can construct a groupoid, the action groupoid , whose objects are the elements of and whose morphisms are pairs such that , with composition induced by multiplication in .
The action groupoid may be regarded as a version of the quotient which keeps more data; for that reason it is sometimes called the weak quotient.
Example. Let be a topological space. Its fundamental groupoid is the groupoid whose objects are the points of and whose morphisms are homotopy classes of continuous paths beginning at and ending at , with composition given by concatenation.
The fundamental groupoid is a natural generalization of the fundamental group which is independent of the choice of basepoint, and many statements made about fundamental groups are more naturally made as statements about fundamental groupoids. For example, the classification of covering spaces may be restated as follows. If is a covering map of spaces, it induces a functor
given by sending a point to the fiber over and sending a path in to the unique function given by lifting paths to a point in and following them in until they end at a point in . If is suitably nice, the classification of covering spaces states that this assignment is an equivalence of categories. Note that we do not need to assume that is connected.
The Seifert-van Kampen theorem also has a more general form involving fundamental groupoids on a set of basepoints which allows, among other things, the direct computation of the fundamental group of a circle without passing through covering space theory. This is discussed, for example, in Ronnie Brown’s answer to my math.SE question on the subject.
Example. In the same way that groupoids let you talk about covering spaces without choosing a basepoint, groupoids also let you talk about Galois theory without choosing a separable closure. Namely, given a field , its absolute Galois groupoid is the groupoid whose objects are the separable closures of and whose morphisms are -isomorphisms between these. If is a separable extension, it induces a functor
sending an algebraic closure to the set and sending a morphism to the induced map . This assignment is not an equivalence of categories for two reasons: in general there is a profinite topology on that needs to be taken into account, and by considering only algebraic extensions we obtain only the “connected covers” of . The correct statement may be deduced from material in Szamuely’s Galois Groups and Fundamental Groups.
Example. The forgetful functor from groupoids to categories has a right adjoint sending a category to its core , which consists of the same objects in but has morphisms only the isomorphisms in .
This example is morally important in understanding moduli problems in algebraic geometry where the objects you want to describe a moduli space of can have automorphisms (because they are really objects of some category), and you want a notion of moduli space that reflects those automorphisms (so you remember at least the core of that category). This is given by the notion of a moduli stack.
Example. The forgetful functor has both a left and a right adjoint. The left adjoint sends a set to the discrete groupoid on , whose objects are the elements of and whose morphisms are the identity morphisms. The right adjoint sends a set to the indiscrete groupoid on , whose objects are the elements of , all of which are isomorphic via a unique isomorphism. This is closely analogous to the behavior of the left and right adjoints of the forgetful functor .
The 2-category of groupoids and homotopy theory
Since groupoids are categories, they naturally organize themselves not into a category but into a -category of groupoids, functors, and natural transformations between functors. This structure can be elegantly described as follows. There is a distinguished groupoid, the interval , which has two objects which are isomorphic via a unique isomorphism.
Proposition: Let be groupoids. The data of a functor is precisely the data of a natural transformation between the pair of functors given by restriction to the subcategories of .
Proof. This is just a verification that the definitions match. The restriction of to the subcategories of objects of the form resp. objects of the form certainly describes two functors . The extra data contained in is what it does to morphisms induced by the identity and the unique isomorphism , since what does to all of the remaining morphisms in is uniquely and freely determined by this data, and the answer is that necessarily assigns a collection of morphisms
such that describes a natural transformation by functoriality. (Note that since is a groupoid, a natural transformation is automatically a natural isomorphism.)
The corresponding statement for categories remains true if we replace with a category that only has a morphism and no other non-identity morphisms.
The above is very closely analogous to the way that homotopies between continuous maps can be written down in , and in fact the analogy is so strong that there is a functor exhibiting it, namely the fundamental groupoid functor . In particular, sends homotopy equivalent spaces to equivalent groupoids.
This suggests that the correct notion of isomorphism of groupoids is not isomorphism but equivalence, as it is for categories, but it also suggests that studying groupoids up to equivalence is in some sense an approximation to studying topological spaces up to homotopy equivalence.
Groupoids up to equivalence
Recall that two categories are equivalent if and only if their skeletons are isomorphic. If is a groupoid, then its skeleton may be written
where denotes the disjoint union (the coproduct of groupoids) and denotes the set of isomorphism classes of elements of . In other words, up to equivalence every groupoid is a disjoint union of groups.
This is in some sense why it is possible to get away with not using groupoids, since one can usually talk about groups instead (for example in the case of fundamental groups and Galois groups). But there are choices that need to be made to get the above description of a groupoid. In order to extract the appropriate disjoint union of groups, it is necessary to choose a representative in each isomorphism class of objects in (in more topological terms, basepoints in each connected component of ), and in general this requires the axiom of choice, so shouldn’t extend to groupoids internal to other categories, such as topological groupoids or Lie groupoids.
Even if you aren’t interested in avoiding the axiom of choice, the above description also ignores the fact that functors between groupoids can be richer than morphisms between groups, especially if one also takes natural transformations into account. That is, if are two groupoids, then one shouldn’t just consider the set of functors between them but the groupoid of functors and natural transformations (all of which are invertible) between them. In particular, unlike the category of groups, the category of groupoids is naturally enriched over itself.
More explicitly, if both have one object , then a natural transformation between two functors is precisely the choice of an element such that ; in other words, rather than considering the set
of homomorphisms between and , we should consider the groupoid of functors, which is the action groupoid or weak quotient
of the set of homomorphisms by the conjugation action of .
Some further reading
Groupoids are used to describe covering spaces in May’s A Concise Course in Algebraic Topology.
Groupoids are used to describe a general version of the Seifert-van Kampen theorem in Brown’s Topology and Groupoids, as well as to compute the fundamental group of a quotient in which does not necessarily act freely on .
Weinstein’s Groupoids: Unifying Internal and External Symmetry is a nice survey article on groupoids.