I want to talk about an interesting 2-category of topological groupoids that I’ve been thinking about recently. Let’s start with the basics; what is a groupoid? Well a group can be thought of as a category with one object and with every morphism invertible. A groupoid is the same thing, except that there can be multiple objects. Why are these interesting? well groupoids generalize three different notions at once: sets, equivalence relations, and groups. We’ve seen how groups enter. Equivalence relations on sets can also be viewed as categories where there are exactly one or zero morphisms between two objects: one if the objects are equivalent, zero otherwise.
Obvious Fact: Since groupoids are categories we can talk about functors between groupoids and natural transformations between these. Hence groupoids form a 2-category.
What happens if we add topology? What if we want a space of objects and a space of morphisms? (source, target and identiy maps are countinuous, of course). Well things start to break down.
The main problem is this: suppose that we have a space that is the union of two open subspaces,
For notation, let be the intersection of and .
Then we can form two topological groupoids out of this:
- X viewed as a space of objects with only the identity morphisms.
- U, which has objects the disjoint union of and , and where the morphisms are the disjoint union of and . The source and target maps are the obvious inclusions.
U is a special case of an equivalence relation: two points in and are equivalent if they correspond to the same point in X.
Moreover there is a natural map/functor from U to X which sends a point of U to the corresponding point in X.
If we forget the topology, then this is an equivalence of categories/groupoids, and morally we want this to hold if we have topology around too. These two topological categories should be equivalent.
Therein lies the problem. If we keep the topology around (and require our functors to be continuous) then the obvious map is not in general an equivalence. For example suppose that X is connected. Then to have the “inverse” of the natural functor from U to X, we would need a functor from X to U which is surjective on equivalence classes of objects. As long as neither is all of X, then this is impossible since we would need a continuous map from a connected space (X) whose image lies in distict components of a disconnected space (the objects of U). No such map can exist.
What are we to do?
Well fortunately there is a pretty nice solution. The idea is to replace functors with “bibundles”. First, let’s back up a little and talk about ordinary groupoids again.
When I said that groupoids generalize spaces and groups, I did what every category theorist would consider a big no-no; I talked about the objects and said nothing about the morphisms. Since it’s Monday and I’m lazy I’ll leave it as a exercise to check that if you consider two sets as groupoids then the functors between them are the same as just maps of sets and the natural transformations are identities. Similarly if you take two groups and view them as groupoids, then functors between them correspond to homomorphisms between the groups (natural trans. are more interesting here).
Now I’m going to start tweaking our 2-category of groupoids until we have something that will work for topological groupoids.
Let’s set some notation. Groupoids will be denoted as pairs , where is the set of objects and is the set of morphisms. All of the structure maps are going to be implicit. Thus if S is a set then (S,S) is S viewed as a groupoid with only identity morphisms and if G is a group, (pt, G) is G viewed as a groupoid.
There is an important definition which we’ll need: a right action of a groupoid os a set consists of a map , together with an action map , compatible with the composition of .
This is sort of difficult to visualize at first, so let’s do some examples.
If our groupoid is just an ordinary group, then is just a point. Thus the first map is no data. The second map just becomes an ordinary action map. So we see in this case it is just the usual notion of an action of G on S.
If our groupoid is a set (Y,Y), then the first map is a map of sets from S to Y, but the second map becomes trivial. It reduces to a map from S to S, but since it’s “compatible with composition” it must be the identity map.
Of course we define left actions in a totally analogous way. If we have two groupoids then we can define biactions on a set to be a left action of one groupoid and a right action of the other which commute. There are also obvious notions of “equivariant maps” of sets with groupoid actions.
Now we can use this to “change” our 2-category of groupoids in a sneaky way. We will replace functors with certain sets with biactions.
Let’s consider the case where we have two groups, G and H, which we are considering as groupoids. Note that the set H has a natural right action by the groupoid H (the map to the point is for free and the action map is also for free). Notice that the group H acts on this set H from the left as well, and that this commutes with the right action. Thus we can turn the set H into a set with a G-H biaction. The map to the objects of G is again free and G acts on the left of H via the map of groups from G to H.
Summarizing: Out of a functor between groups we can construct a set with a G-H biaction, and such that H acts freely and transitively on the fiber over = pt. We’ll call this set a bibundle for reason that will be obvious later.
Exercises: Show that there is a natural notion of composition of bibundles (between groups) and that it matches up with the composition of functors. Also show that natural transformations induce equivariant maps of bibundles.
Similarly suppose that we have two sets, X and Y, viewed as groupoids and a functor f between them. We have a similar story where we can build a space P which has a left X action and a right Y action. Recall that this is just a map to the set X and a map to the set Y. Let’s take P=X, with the identity map to X and the map f from P to Y. Note that the fibers of the map to the objects of the goupoid X are just single elements of P. But since Y is just a space, it acts “free and transitively” on these fibers. The key new ingrediant that we have from this example is that the map from P to (the objects of) X is surjective.
Definition: A right principle bibundle between two groupoids G and H is a set P with a G-H biaction such that the map from P to the objects of G is surjective and such that the fibers are acted on by H freely and transitively.
This extends the above construction from groups to spaces and in fact we can extend it to all groupoids by taking
where the map from to is the functor on objects. The value of the functor on morphisms yields the action on P.
Exercise: Show that this construction yields a (weak) 2-functor from the 2-category of (groupoids, functors, nat. trans) to the bicategory of (groupoids, bibndles, equivariant maps). Show that this is an equivalence of bicategories.
Whew! what a detour, eh? Well now we are all set to fix the category of topological groupoids. If we return to the bad example above where X is the union of two open sets, we see that the problem that we ran into was a local one. If we could shrink X enough (say to one of the open sets ) the we can get an equivalence of topological categories (the shrunken X is equivalent to its preimage category in U).
Based on our bibundle ideas we define a new bicategory where the 1-morphisms are topological bibundles (all maps are continuous). It turns out that this includes too many morphisms so we also require that our 1-morphisms are locally trivial (in the source). This means that if we shrinch the source groupoid enough, then the bibundle becomes isomorphic to one of the form
I think this is a really neat bicategory. First of all, it fixes the problem we had before, U and X become equivalent objects in this bicategory. It is a place where things which are “locally a functor” make sense. It also includes lots of interesting geometric stuff. For example, suppose that G is a topological group and X is a topological space. What is the hom category hom(X, G)?
It’s the category of right G-principle bundles over X! Wow!