# Coverings of a space and coverings of a groupoid

In algebraic topology, there is the well-known notion of coverings of a space. It is very nice, it has many properties, but I find it frustrating that:
1) some hypotheses are needed for them to work perfectly, like semi-locally simply-connectedness for the existence of coverings,
2) those covering maps don't seem to correspond to anything interesting from a categorical point-of-view.

On the other hand, there is this notion of coverings of a groupoid (see for example Brown's book). In my opinion, those are much nicer, since to any property of a covering of a space corresponds an analogue property in groupoids by applying the fundamental groupoid functor, and since the two objections disappear. For example, a covering of a groupoid can be defined as a functor that as right lifting properties with respect to some simple functors, so this notion makes sense categorically.

So my (naive) questions are the following. Are there particular reasons to stick to this historical notion of coverings of a space? What would remain of the theory if we replace this notion by defining, for example, a covering as a continuous function which induces a covering between the fundamental groupoids? Do you know if there is some work on that direction?

• Well, covering maps are local homeomorphisms, so covers of manifolds are still manifolds and so forth. So there are reasons not to want to work with a homotopy-invariant notion of cover (which does exist) and actually want covers on the nose. – Qiaochu Yuan Jan 10 '18 at 10:11
• I agree that the algebraic theory of covering morphisms of groupoids should be developed independently of the topological theory, but with that theory in mind, and the relation between the two should be studied by the fundamental groupoid functor $\pi_1$. I expect there is also more to be done on the orbit space/orbit groupoid relation then is in my book. – Ronnie Brown Jan 13 '18 at 15:44
• You can take a look at this paper, which explains how topological covers and groupoid covers are actually instances of the same phenomenon. In particular, you also get that topological covers are the maps having RLP with respect to some simple maps (namely continuous maps with connected homotopy fibers). And you have an a posteriori justification of the semi-locally simply-connectedness : for a continuous $f : X\to Y$, it gives a left adjoint to $f^\ast : \mathsf{Sh}_{\rm loc}(Y) \to \mathsf{Sh}_{\rm loc}(X)$. (The "loc" subscript means locally constant.) – Pierre Cagne Jan 18 '18 at 10:41

As you will see this is a very rich area, so my answer can do no more than give a sketch. The basic idea is that a good category of coverings, Cov(X), say, has certain good categorical properties and then one can show that those properties give the existence of a groupoid such that functors from that groupoid, G, to sets form a category equivalent to Cov(X). That G is then defined to be the fundamental groupoid.

The problem you identify of `semi-locally bla-bla' is merely to make sure there enough paths in X so that the G you get by this other route can be identified with the fundamental groupoid defined as classically. You can check that this sort of approach extends the covering groupoid approach as Ronnie Brown's books so clearly show.

Going slightly more widely there is an extensive categorical theory which is given in the book by Borceux and Janelidze which looks at 'covering space' theory as part of a wider general Galois theory. There are also more classical approaches (e.g. in a book by Douady and Douady) which give the link between covering spaces and Galois theory. There is an AMS memoir by Joyal and Tierney giving another categorical perspective and, of course, the SGA1 seminars by Grothendieck sketch out an approach the allows one to define fundamental groups and fundamental groupoids via the category of coverings.

• Thanks for your answer. It answers at least some of my questions. I need some time to process that and to reformulate my remaining broad ideas. – Jeremy Jan 12 '18 at 6:18

1) some hypotheses are needed for them to work perfectly, like semi-locally simply-connectedness for the existence of coverings

Tim Porter already said this but I'll say it again with a slightly different emphasis: these hypotheses are needed to connect covering space theory to the fundamental group as defined using loops. If a space doesn't admit many maps from $\mathbb{R}$ there's no hope of its covering space theory being related to its fundamental group in the usual sense.

For any space you can try to develop its covering space theory, or maybe better its theory of locally constant sheaves, and then use Grothendieck's Galois theory to recover a group. The point-set hypotheses are needed to compare this group to the usual $\pi_1$; in general it is a different notion of $\pi_1$. See fundamental group of a topos for some details along these lines.

2) those covering maps don't seem to correspond to anything interesting from a categorical point-of-view.

I'm not really sure what you mean by this. They correspond to locally constant sheaves; aren't those interesting from a categorical point of view?

For example, a covering of a groupoid can be defined as a functor that as right lifting properties with respect to some simple functors, so this notion makes sense categorically.

I don't know what you mean by "makes sense categorically" here. If you mean "invariant under equivalence of categories," covering maps of groupoids are not invariant under equivalence of categories, in the same way that covering maps of spaces are not invariant under homotopy equivalence.

(Because of this, it's ambiguous what "continuous map which induces a covering map on fundamental groupoids" means, because the answer will depend on which basepoints you choose for the fundamental groupoids.)

The homotopy invariant way to say "covering map" is "map whose homotopy fibers are discrete." But there are reasons to explicitly want the point-set notion of covering as mentioned in my comment, so that covers of manifolds are manifolds, covers of complex manifolds are complex manifolds, etc.

• I do not understand the suggestion about base points in Qiaochu's answer as the fundamental groupoid does not have a base point chosen. In the book "Topology and Groupoids" Section 9.5, a local condition involving $\pi_1(p)$ is given on $X$ for $p:\widetilde{X} \to X$ to be a covering map. . – Ronnie Brown Jan 12 '18 at 12:28
• In answer to a part of Jeremy's question we could define a map $p: Y \to X$ to be a $\pi_1$-fibration (covering) if $\pi_1(p)$ is a fibration (covering) of groupoids. In the fibration case this would give the exact sequences of the fibration in low dimensions, as given in T&G 7.2.9. – Ronnie Brown Jan 12 '18 at 12:37
• @Ronnie: generally I think of the fundamental groupoid as an object only well-defined up to equivalence, and I exploit the freedom to use any set of basepoints containing at least one basepoint in each path component to define it (e.g. for a path connected space I exploit my freedom to use exactly one basepoint to get the fundamental group back). But since the definition of covering map of groupoids is not invariant under equivalence I now have to make a particular choice of basepoints. I guess you want to pick the maximal choice given by every point. – Qiaochu Yuan Jan 12 '18 at 20:24
• @RonnieBrown: thank you. That is the kind of answers I was ultimately expecting. Maybe a follow up would be: How far from a covering a $\pi_1$-covering could be? And to join Tim's answer: How far from a good category of coverings the category of $\pi_1$-coverings could be? – Jeremy Jan 17 '18 at 1:12
• @QiaochuYuan: as you mentioned, since there seems to be no canonical choice of basepoints, I was implicitly taking all the points. – Jeremy Jan 17 '18 at 1:12