The n-Category Café

Covering spaces are determined by what happens at the level of fundamental groupoids. If the base space is path-connected and pointed, we can be slack and just use fundamental groups. As a disclaimer, I’m going to sweep all finicky considerations like “semi-locally simply connected”, and its generalisations, under the carpet, and assume for now that all spaces are nice enough (all locally contractible spaces are nice enough). It is a well-known theorem that a pointed, path-connected covering space is uniquely determined by a subgroup of the fundamental group of the base, and vice-versa. If we forget the point, then it is determined by a conjugacy class of subgroups.

Instrumental in proving this result is the path lifting property of covering spaces. If we knew straight away that a covering space $$p:Y \to X$$ is a fibration with fibre $F$, then the answer leaps out at us using the long exact sequence in homotopy: $$0 = \pi_1(F) \to \pi_1(Y) \to \pi_1(X)$$ voila! $\pi_1(p)$ is injective since $F$ is a discrete space, and hence has simply connected path components. The definition of a covering space is a long way from saying it is a fibration, so how is it we get the result? More importantly, what does this tell us about 2-covering spaces?

What we need to do is to bump the categorical dimension up a notch on everything we are working with. Thus we replace spaces with topological groupoids and groups with 2-groups. For me topological groupoids are groupoids internal to $\mathbf{Top}$, not groupoids enriched in $\mathbf{Top}$. The tricky part is getting a homotopy 2-group from a topological groupoid, since the category of topological groupoids one first thinks of does not have enough equivalences. There are ways to sort this out using intimidating words like ‘bicategory of fractions’ or ‘anafunctors’, but paths and homotopies are simple enough to sort out without such technology.

Recall that given a space $M$ with a cover $W \to M$ we can form the (topological) groupoid $W^{[2]}$ with object space $W$ and a unique morphism between any two points in the fibre over $M$. The spaces $M$ we are interested in are the interval, square and cube, $I^k$ for $k=1,2,3$. Also, we will only be considering finite closed covers by intervals/polygons/polyhedrons.

I call a groupoid $\mathfrak{p}$ arising from a closed cover of $I$ by intervals a partition groupoid, as it is determined by a partition of $I$. A path in a topological groupoid $X$ is just a functor $$\mathfrak{p} \to X.$$ Unwrapping the definition, this is precisely the sort of path for open covers that Urs has discussed, and that Moerdijk-Mrcun use to define the fundamental groupoid of a Lie groupoid. A path looks like a finite sequence of paths in the object space with the ‘jumps’ bridged by morphisms - these are the images of the morphisms of $\mathfrak{p}$.

Given a pair of paths that agree at 0 and 1 (this is unambiguous, as the fibres of the cover over 0,1 are single points), we can try to define a homotopy between them. This requires defining the right sort of cover of the square. In my slides I give a list of conditions, including one that makes all vertices at most trivalent, but it is easier just to give a picture of an example:

Let $\mathfrak{h}$ be a groupoid arising from such a cover of the square. The two subgroupoids which arise from restricting this cover to the top and bottom edges of the square are called the final edge and initial edge respectively. A functor $$\mathfrak{h} \to X$$ is a homotopy between the paths defined by the initial and final edges if the vertical edges of the square (or rather, the subgroupoids defined thereby) are mapped to $X$ through the trivial groupoid $\{*\}$. Call the images of these two edges $x$ and $y$. We can think of such a homotopy as a string diagram, since each edge is mapped to a path in the arrow space of $X$, and each polygon is mapped to the object space of $X$, the paths of arrows linking the edges of the polygons. At the trivalent vertices, a cocycle condition must be satisfied. “But this isn’t a string diagram,” I hear you cry, “it has lines escaping the edges!” But remember that each outside vertical edge is fixed at a point, so we can imagine that they are mapped to the identity arrows, and there are invisible borders on the square, mapped to $x$ and $y$. If our topological groupoid is an ordinary groupoid, then this reduces to the usual notion of string diagram. Or if $X$ has one object, then this is a string diagram for a topological group. I often drop the adjective ‘modified’, which I use in the slides.

We can do the same thing for homotopies between string diagrams, but this time it is too hard for me to draw. Suffice it to say that homotopies between string diagrams use covers that look like covers of the cube by polyhedrons coming from something like a spin foam. In calculations these are generally rectilinear, but we allow more general such things.

Homotopy classes of string diagrams, where the homotopy is fixed around the edges of the square, are called 2-tracks. A 2-track between two constant paths at a point is what is generally known as an element of the second homotopy group, at least when we are dealing with good old spaces. To introduce the fundamental 2-group $\Pi_2(X,x)$ of a topological groupoid $X$ at the point $x$, I will first describe the case when the topological groupoid is a space.

One description of the fundamental 2-group of a space, due to Cegarra and Garzon, is the fundamental groupoid of the loop space. This has based loops for objects, and (using the cartesian closed nature of your favourite version of $\mathbf{Top}$) 2-tracks for morphisms. The product in the 2-group is induced by the loop space product, which we know is homotopy associative and unital. The 2-tracks represented by the usual homotopies expressing this become the coherence morphisms in the 2-group. This description relies on the topology on the loop space, which is not something we have for the analogous object for topological groupoids. However, we can use the description using 2-tracks, which is due to Hardie-Kamps-Kieboom (used in defining the fundamental bigroupoid of a space). Simply replace paths in the space with paths in the topological groupoid, and 2-tracks in the space with 2-tracks in the topological groupoid. Again, this is not in general a strict 2-group.

A functor $f:(X,x) \to (Y,y)$ of pointed topological groupoids induces a functor $\Pi_2(X,x) \to \Pi_2(Y,y)$ of 2-groups. It is fairly immediate that the fundamental 2-group of a space, considered as a topological groupoid with only identity arrows, is equivalent to the usual one.

The point of all of this is that we want to relate the fundamental 2-group of a 2-covering space to that of the base. But I haven’t told you what a 2-covering space is, yet!

Definition: A 2-covering space is a functor of topological groupoids $p:Z \to X$ such that $X$ is a space, and there is an open cover $\coprod U_\alpha \to X$ such that there is a collection of topologically discrete groupoids $D_\alpha$, and a weak equivalence $$U \times D_\alpha \to Z\big|_{U_\alpha}$$ over $U_\alpha$.

Whoa! “Now what’s a weak equivalence?” I hear you grumble. Easy: it’s a fully faithful, essentially surjective functor such that the ‘surjective’ part admits local sections. This implies the fibres of $p$ are groupoids weakly equivalent to topologically discrete groupoids. This will help us in our task to categorify the old result about covering spaces above, since a topologically discrete groupoid is a 1-type, much as the fibres of a covering space are 0-types.

First we prove this theorem:

Theorem: If $f:D \to Y$ is a weak equivalence and $D$ is topologically discrete, $d \in D$, the induced functor $\Pi_2(D,d) \to \Pi_2(Y,f(d))$ is an equivalence.

Then we can calculate that $\Pi_2(D,d) \simeq \pi_1(D,d) = D(d,d)$, so the fibres of a 2-covering space have trivial $\pi_2$. Or, to put it another way, there is at most one 2-track between paths in a fibre.

It is possible to lift paths to a 2-covering space, but now we can only define the starting point up to an arrow (which are all ‘vertical’ - they project to identity arrows). This is reminiscent of Dold fibrations, which lifts of homotopies exist, where the starting point can only be specified up to a vertical homotopy. The following theorem corresponds to the uniqueness of path lifting for 1-covering spaces. There is a groupoids worth of paths over a given path in the base, due to the natural transformations available to us.

Theorem: The groupoid of lifts of a given path in $X$, starting at $x\in X$, is weakly equivalent to the fibre of $Z \to X$ over $x$.

It is also possible to lift homotopies: given a paracompact space $Y$ a homotopy $h:Y \times I \to X$ and an anafunctor $Y \times \{0\} \to Z$ covering $h$, we can lift the homotopy so that it agrees with the existing lift up to an ananatural transformation. If the 2-covering space trivialises over a numerable cover, it is possible to drop the paracompactness assumption on $Y$. That is a whole lot of jargon, but if you like just replace $Y$ with a square, and the anafunctor with a string diagram.

We then use this to show

Theorem: Let $p:Z \to X$ be a 2-covering space and $z \in Z$. The induced map of 2-groups $$\Pi_2(Z,z) \to \Pi_2(X,p(z))$$ is faithful.

My thanks go to Mathieu Dupont for disabusing me of a misconception I had when initially trying to prove an early, incorrect, version of this theorem.

This relates to the notion of sub-2-group discussed by others present at the categorical groups workshop, e.g. Carrasco–Garzon–Vitale.

Also, the (homotopy) quotient $G//H$ of a map of 2-groups $j:H \to G$ is (equivalent to) a groupoid if and only if the induced homomorphism $\pi_2(j)$ is injective. This is the case if $j$ is faithful. We could consider the homotopy quotient $$\Pi_2(X,p(z)) // \Pi_2(Z,z)$$ and compare it to the fibre, but that is for another time.

We therefore state this slogan:

The fundamental 2-group of a 2-covering space is a sub-2-group of the fundamental 2-group of the base.