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Groupoids

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 $X$ with one object $x$ is uniquely determined by its automorphism group $G = \text{Aut}(x)$; 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 $G$ is often denoted by $BG$. This notation is also used for the classifying space of $G$ because taking the geometric realization of the nerve of $BG$ the category gives a model for $BG$ the space.

Example. Let $X$ be a set equipped with an equivalence relation $\sim$. From this data we can construct a groupoid with object set $X$ and a unique morphism $x \to y$ exactly when $x \sim y$; 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 $G$ be a group acting on a set $S$. From this data we can construct a groupoid, the action groupoid $S//G$, whose objects are the elements of $S$ and whose morphisms $x \to y$ are pairs $(g, x) \in G \times S$ such that $gx = y$, with composition induced by multiplication in $G$.

The action groupoid may be regarded as a version of the quotient $S/G$ which keeps more data; for that reason it is sometimes called the weak quotient.

Example. Let $X$ be a topological space. Its fundamental groupoid $\Pi_1(X)$ is the groupoid whose objects are the points of $X$ and whose morphisms $x \to y$ are homotopy classes of continuous paths $[0, 1] \cong I \to X$ beginning at $x$ and ending at $y$, 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 $p : Y \to X$ is a covering map of spaces, it induces a functor

$\Pi_1(X) \to \text{Set}$

given by sending a point $x \in X$ to the fiber $p^{-1}(x) \subset Y$ over $X$ and sending a path $x \to y$ in $X$ to the unique function $p^{-1}(x) \to p^{-1}(y)$ given by lifting paths to a point in $p^{-1}(x)$ and following them in $Y$ until they end at a point in $p^{-1}(y)$. If $X$ 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 $X$ 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 $F$, its absolute Galois groupoid $\Pi_1(\text{Spec } F)$ is the groupoid whose objects are the separable closures of $F$ and whose morphisms are $F$-isomorphisms between these. If $F \to L$ is a separable extension, it induces a functor

$\Pi_1(\text{Spec } F) \to \text{Set}$

sending an algebraic closure $F_s$ to the set $\text{Hom}_F(L, F_s)$ and sending a morphism $F_{s,1} \to F_{s,2}$ to the induced map $\text{Hom}_F(L, F_{s,1}) \to \text{Hom}_F(L, F_{s,2})$. This assignment is not an equivalence of categories for two reasons: in general there is a profinite topology on $\Pi_1(\text{Spec } F)$ that needs to be taken into account, and by considering only algebraic extensions we obtain only the “connected covers” of $\text{Spec } F$. 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 $C$to its core $\text{core}(C)$, which consists of the same objects in $C$ but has morphisms only the isomorphisms in $C$.

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 $\text{Gpd} \to \text{Set}$ has both a left and a right adjoint. The left adjoint sends a set to the discrete groupoid on $X$, whose objects are the elements of $X$ and whose morphisms are the identity morphisms. The right adjoint sends a set to the indiscrete groupoid on $X$, whose objects are the elements of $X$, 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 $\text{Top} \to \text{Set}$.

The 2-category of groupoids and homotopy theory

Since groupoids are categories, they naturally organize themselves not into a category but into a $2$-category of groupoids, functors, and natural transformations between functors. This structure can be elegantly described as follows. There is a distinguished groupoid, the interval $I$, which has two objects $0, 1$ which are isomorphic via a unique isomorphism.

Proposition: Let $X, Y$ be groupoids. The data of a functor $H : X \times I \to Y$ is precisely the data of a natural transformation between the pair of functors $F, G : X \to Y$ given by restriction to the subcategories $X \times 0, X \times 1$ of $X \times I$.

Proof. This is just a verification that the definitions match. The restriction of $H$ to the subcategories $X \times 0, X \times 1$ of objects of the form $(x, 0)$ resp. objects of the form $(x, 1)$ certainly describes two functors $F, G : X \to Y$. The extra data contained in $H$ is what it does to morphisms $(x, 0) \to (x, 1)$ induced by the identity $\text{id}_x : x \to x$ and the unique isomorphism $0 \to 1$, since what $H$ does to all of the remaining morphisms in $C \times I$ is uniquely and freely determined by this data, and the answer is that $H$ necessarily assigns a collection of morphisms

$\displaystyle \eta_x : F(x) \to G(x)$

such that $\eta_x$ describes a natural transformation $F \to G$ by functoriality. (Note that since $Y$ is a groupoid, a natural transformation is automatically a natural isomorphism.) $\Box$

The corresponding statement for categories remains true if we replace $I$ with a category that only has a morphism $0 \to 1$ 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 $\text{Top}$, and in fact the analogy is so strong that there is a functor exhibiting it, namely the fundamental groupoid functor $\Pi_1$. In particular, $\Pi_1$ 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 $X$ is a groupoid, then its skeleton may be written

$\displaystyle \bigsqcup_{x \in \pi_0(X)} B\text{Aut}(x)$

where $\sqcup$ denotes the disjoint union (the coproduct of groupoids) and $\pi_0(X)$ denotes the set of isomorphism classes of elements of $X$. 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 $X$ (in more topological terms, basepoints in each connected component of $X$), 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 $X, Y$ 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 $X, Y$ both have one object $x, y$, then a natural transformation between two functors $F, G : X \to Y$ is precisely the choice of an element $h \in \text{Aut}(y)$ such that $h F h^{-1} = G$; in other words, rather than considering the set

$\displaystyle \text{Hom}(\text{Aut}(x), \text{Aut}(y))$

of homomorphisms between $\text{Aut}(x)$ and $\text{Aut}(y)$, we should consider the groupoid of functors, which is the action groupoid or weak quotient

$\displaystyle \text{Hom}(\text{Aut}(x), \text{Aut}(y))//\text{Aut}(y)$

of the set of homomorphisms by the conjugation action of $\text{Aut}(y)$.

John Baez has written some interesting things about groupoids.

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 $X/G$ in which $G$ does not necessarily act freely on $X$.

Weinstein’s Groupoids: Unifying Internal and External Symmetry is a nice survey article on groupoids.

7 Responses

1. Does your proposition about natural transformations as functors from products with the interval groupoid actually bear out in general, as you’ve stated it? It seems to work if D is a groupoid, but otherwise doesn’t this force every component of the natural transformation to be iso, and so only cover natural equivalences?

• Right, I’ve misstated things slightly; for arbitrary categories you need to use a directed interval (so there’s just a morphism $0 \to 1$). Thanks for the catch.

2. You wrote:

>the category of one-object groupoids is isomorphic (not just equivalent!) to the category of groups

This is not correct. The forgetful functor from one-object groupoids to groups really does forget something: the object. For example (picking your favourite trival group *), the groupoid consisting of the object ∅ equipped with * as automorphism group is not equal to the groupoid consisting of the object {∅} equipped with * as automorphism group; but the group * is equal to the group *.

If this seems like pointless pedantry, then that is only because asking categories to be isomorphic rather than merely equivalent is also pointless pedantry.

3. Same thing here:

>the category of groupoids whose objects have no nontrivial automorphisms is isomorphic (again, not just equivalent!) to the category of equivalence relations

Only now it’s on the other end: the forgetful functor from discrete (up to equivalence) groupoids to (sets equipped with) equivalence relations forgets the morphisms (and remembers only whether a morphism exists).

• Thanks for the corrections. I’ll fix those.

4. [...] nice groupoids have a numerical invariant attached to them called groupoid cardinality. Groupoid cardinality is [...]

5. I’d just like to point out that for me an amazing aspect of groupoids was gradually seen as that whereas group objects in the category of goups are just abeliann groups, groupoid objects in the category of groupoids are much more complicated. So, given the importance of group theory in mathematics and physics, even engineering, the question was: how important should groupoids be seen in the future? There is that lovely question: What if?

This question was further stimulated when G.W.Mackey introducved himself to me in 1967 at a conference in Swansea and told me of his use of groupoids in ergodic theory. Since we had come to grouoids from two quite different directions, this suggested that there was more in the concept than met the eye. At that time, I also did not know of the extensive work of C. Ehresmann on the use of groupoids in geometry, for example in foliation theory.

Looking at my proof of the groupoid van Kampen theorem in 1965, it suddenly occurred to me that the proof should generalise to higher dimensions, using squares and cubes, if one had the right homotopical gadget generalising the fundamental groupoid. Following Ehresmann, one needed a homotopical double groupoid.

I tried to find this for 9 years, getting rather confused. Meanwhile, Chris Spencer and I had worked out notions of double groupoids with connection, and their relation to crossed mopdules, and Phiiip Higgins had worked out ways of computing colimits of crossed modules. In 1974 Philip and I analysed the seemingly unfortunsate situation and concluded:

1. J.H.C. Whitehead had a theorem (1941-49) on free crossed modules and relative homotopy groups which showed that a universal property did exist in dimension 2 homotopy theory.

2. If our putative van Kampen theorem ws to be any good it should have Whitehead’s thoerem as a Corollary.

3. So we needed a notion of homotopy double groupoid for the relative theory.

4. But a possibility for a homotopy double groupoid of a pair $(X,A)$ was easy to see: take homotopy classes of maps of a square to $X$ which took the edges into $A$ and the vertices to the base point.

5. This construction actually worked!

The passage to publication of the resulting paper was obstructed by stiff opposition, for reasons never made specific, and generalisations were pursued as one quote had it “in the teeth of opposition”. . But one part of the resulting story of groupoids in higher homotopy theory is in our 2011 EMS Tract vol 15 on “Nonabelian algebraic topology” (pdf available on my web site). Some parts of this such as the use of use of chain complexes with a grouopid (rather than just group) of operators in discussing cellular chains of CW-complexes, are still not widely recognised,

I hope there is lot more fun to be had in this area! What if?