CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
Let $X$ be a topological space which is well-connected in that it is
Then there is a connected and simply connected covering space $X^{(1)} \to X$ with the universal property that for any other covering space $\widetilde{X} \to X$ there is a map of covering spaces $X^{(1)} \to \widetilde{X}$.
There is a functorial construction of a universal covering space of a pointed space
where $Top_*^{wc}$ is the full subcategory of $Top_*$ with objects the well-connected spaces and $Cov$ is the subcategory of $Top_*^2$ of pointed maps of spaces with objects the covering space maps.
Specifically, if $X$ is a space with basepoint $x_0$, we define $X^{(1)}$ to be the space whose points are homotopy classes of paths in $X$ starting at $x_0$, with the projection $X^{(1)}\to X$ projecting to the endpoint of a path. We can equip this set $X^{(1)}\to X$ with a topology coming from $X$ so that it becomes a universal covering space as above. As described at covering space, under the correspondence between covering spaces and $\Pi_1(X)$-actions, the space $X^{(1)}$ corresponds to the “regular representation” of $\Pi_1(X)$.
This regular representation can be seen to arise by taking a category of elements in the same way that the regular representation of a group is gotten by taking its action on itself: we can see that the universal covering groupoid $\Pi(X)^{(1)}$ in the slice category $\mathbf{Grpd}/\Pi(X)$ (see the universal covering $\infty$-groupoid below) is just the category of elements of the action of $\Pi(X)$ on itself, and can be topologized in a natural way by lifting the topology on $X$ along the canonical projection $\Pi(X)^{(1)} \to \Pi(X)$; decategorifying this yields $X^{(1)}$.
We describe now how the universal cover construction may be understood from the nPOV. The basic idea is that the universal cover of a space $X$ is the (homotopy) fiber of the map $X\to \Pi_1(X)$ from $X$ to its fundamental groupoid. We can think of this as a way of precisely saying “make $\Pi_1$ trivial in a universal way.” There are at least two slightly different ways of making this precise in the language of $(\infty,1)$-toposes, depending on whether we view $X$ as a little topos or as an object of a big topos.
In this section we work in the big (∞,1)-topos of sheaves on $TopBalls$, the site of topological balls with the good open cover coverage. We call objects of this $(\infty,1)$-topos “topological ∞-groupoids.”
Now, to a topological space $X$ is associated the topological groupoid $\Pi_1(X)$ – its fundamental groupoid. With $X$ regarded as a categorically discrete topological groupoid, there is a canonical morphism
that includes $X$ as the collection of constant paths.
Let $X$ be a suitably well behaved pointed space. The universal cover $X^{(1)}$ of $X$ is (equivalent to) the homotopy fiber of $X \to \Pi(X)$ in the (∞,1)-category $\mathbf{H} = Sh_{(\infty,1)}(Top_{cg})$ of topological ∞-groupoids.
In other words, the principal ∞-bundle classified by the cocycle $X \to \Pi_1(X)$ is the universal cover $X^{(1)}$: we have a homotopy pullback square
Urs Schreiber: may need polishing.
We place ourselves in the context of topological ∞-groupoids and regard both the space $X$ as well as its homotopy ∞-groupoid? $\Pi(X)$ and its truncation to the fundamental groupoid $\Pi_1(X)$ as objects in there.
The canonical morphism $X \to \Pi(X)$ hence $X \to \Pi_1(X)$ given by the inclusion of constant paths may be regarded as a cocycle for a $\Pi(X)$-principal ∞-bundle, respectively for a $\Pi_1(X)$-principal bundle.
Let $\pi_0(X)$ be the set of connected components of $X$, regarded as a topological $\infty$-groupoid, and choose any section $\pi_0(X) \to \Pi(X)$ of the projection $\Pi(X) \to \pi_0(X)$.
Then the $\Pi(X)$-principal $\infty$-bundle classified by the cocycle $X \to \Pi(X)$ is its homotopy fiber, i.e. the homotopy pullback
We think of this topological $\infty$-groupoid $UCov(X)$ as the universal covering $\infty$-groupoid of $X$. To break this down, we check that its decategorification gives the ordinary universal covering space:
for this we compute the homotopy pullback
where we assume $X$ to be connected with chosen baspoint $x$ just to shorten the exposition a little. By the laws of homotopy pullbacks in general and homotopy fibers in particular, we may compute this as the ordinary pullback of a weakly equivalent diagram, where the point $*$ is resolved to the universal $\Pi_1(X)$-principal bundle
(More in detail, what we do behind the scenes is this: we regard the diagram as a diagram in the global model structure on simplicial presheaves on Top. In there all our topological groupoids are fibrant, hence all we have to ensure is that one of the morphisms of the diagram becomes a fibration, which is what the passage to $\mathbf{E}_x \Pi_1(X)$ achieves. Then the ordinary pullback in the category of simplicial presheaves is the homotopy pullback in $\infty$-prestacks. Then by left exactness of $\infty$-stackification, the image of that in $\infty$-stacks is still a homotopy pullback. )
The topological groupoid $\mathbf{E}_x \Pi_1(X)$ has as objects homotopy classes rel endpoints of paths in $X$ starting at $x$ and as morphisms $\kappa : \gamma \to \gamma'$ it has commuting triangles
in $\Pi_1(X)$. The topology on this can be deduced from thinking of this as the pullback
in simplicial presheaves on Top. Unwinding what this means we find that the open sets in $Mor(\mathbf{E}_x \Pi_1(X))$ are those where the endpoint evaluation produces an open set in $X$.
Now it is immediate to read off the homotopy pullback as the ordinary pullback
Since $X$ is categorically discrete, this simply produces the space of objects of $\mathbf{E}_x \Pi_1(X)$ over the points of $X$, which is just the space of all paths in $X$ starting at $x$ with the projection $UCov_1(X) \to X$ being endpoint evaluation.
This indeed is then the usual construction of the universal covering space in terms of paths, as described for instance in
On the other hand, we can view a space $X$ as the little $(\infty,1)$-topos $Sh_{(\infty,1)}(X)$ of $(\infty,1)$-sheaves on $X$. If $X$ is locally connected and locally simply connected in the “coverings” sense, then $Sh(X)$ is locally 1-connected.
In fact, for the construction of the universal cover we require only the (2,1)-topos $Sh_{(2,1)}(X)$ of sheaves (stacks) of groupoids on $X$, so we will work in that context because it is simpler. The construction can be adapted, however, to produce a “universal cover” of any locally 1-connected $(\infty,1)$-topos.
Let $E$ be any (2,1)-topos which is locally 1-connected. This means that in the unique global sections geometric morphism $(E^*,E_*)\colon E\to Gpd$, the functor $E^*$ has a left adjoint $E_!\colon E \to Gpd$, which is automatically $Gpd$-indexed. The fundamental groupoid of $E$ is defined to be $\Pi_1(E)\coloneqq E_!(*)$, where $*$ is the terminal object of $E$.
As discussed here in the $(\infty,1)$-case, the construction of $\Pi_1(E)$ is a left adjoint to the inclusion of groupoids into locally 1-connected (2,1)-toposes (which sends $G\mapsto Gpd/G \simeq Gpd^G$). Thus we have a geometric morphism $E\to \Pi_1(E)$ (where we regard $\Pi_1(E)$ as the (2,1)-topos $Gpd^{\Pi_1(E)}$).
Suppose, for simplicity, that $E$ is connected. Then $\Pi_1(E)$ is also connected, and so we have an essentially unique functor $*\to \Pi_1(E)$. We define the universal cover of $E$ to be the pullback (2,1)-topos:
(where of course $*$ denotes the terminal (2,1)-topos $Gpd$).
Now observe that $*\to \Pi_1(E)$ is a local homeomorphism of toposes, since we have $Gpd \simeq Gpd/\Pi_1(E)/(*\to \Pi_1(E))$. Since local homeomorphisms of toposes are stable under pullback, $E^{(1)}\to E$ is also a local homeomorphism, i.e. there exists an object $\widetilde{E}\in E$ and an equivalence $E^{(1)} \simeq E/\widetilde{E}$ over $E$. Moreover, it is not hard to see that $\widetilde{E}$ can be identified with the pullback
in $E$. Note that the bottom map $*\to E^*(E_!(*))$ is $E^*$ applied to the unique map $*\to E_!(*)$, while the right-hand map is the unit of the adjunction $E_!\dashv E^*$.
In order to see that this is a sensible definition, we first observe that $E^{(1)}$ is itself locally 1-connected (since it is etale over $E$). Moreover, it is actually 1-connected, which is equivalent to saying that $E_!(\widetilde{E}) = *$. This is because the “Frobenius reciprocity” condition for the adjunction $E_!\dashv E^*$ (which is equivalent to saying that $E_!$ is $Gpd$-indexed) applied to the defining pullback of $\widetilde{E}$ implies that we also have a pullback
which clearly implies that $E_!(\widetilde{E}) = *$.
Thus, $E^{(1)}$ is a connected and simply connected space with a local homeomorphism to $E$, but is it a covering space? In other words, is it locally trivial? Since we have supposed that $E$ is locally 1-connected, as a (2,1)-category it can be generated by 1-connected objects, i.e. objects $U$ such that $E_!(U)\simeq *$. In particular, we have a 1-connected object $U$ and a regular 1-epic? $U\to *$.
We claim that if $U$ is any 1-connected object of $E$, then $\widetilde{E}$ is trivialized (or split) over $U$, in that $U\times \widetilde{E}$ is equivalent, over $U$, to $U\times E^*S$ for some $S\in Gpd$. For pulling back the defining pullback to $U$, we obtain
But $U\times E^*(\Pi_1(E)) \cong (E/U)^* (\Pi_1(E))$, so to give a map $U \to (E/U)^* (\Pi_1(E))$ over $U$ is the same as to give a map $(E/U)_!(*) \to \Pi_1(E)$ in $Gpd$. But $(E/U)_!(*)\simeq *$, since $U$ is 1-connected, and $\Pi_1(E)$ is connected, so there is only one such morphism. Therefore, the two maps $U\to U\times E^*(\Pi_1(E))$ in the pullback above are in fact the same, and in particular both are the pullback to $E/U$ of the map $*\to \Pi_1(E)$. Thus, $U\times \widetilde{E}$ is equivalent to $(E/U)^*(S) \cong U\times E^*(S)$, where $S= \Omega(\Pi_1(E))$ is the loop object of $\Pi_1(E)$, i.e. what we might call the fundamental group of the connected (2,1)-topos $E$.
Therefore, since $\widetilde{E}$ is trivialized over any 1-connected object, and $E$ is generated by 1-connected objects, $\widetilde{E}$ is locally trivial. Moreover, since $*$ is a discrete object of $E$, so is $\widetilde{E}$. Thus, if we specialize all this to the case $E=Sh_{(2,1)}(X)$ of (2,1)-sheaves on a topological space, then we conclude that $\widetilde{E}$ is an honest 1-sheaf on $X$ which, when regarded as a local homeomorphism over $X$, is locally trivial (hence a covering space), connected, and 1-connected—i.e. a universal cover of $X$.
The nPOV descriptions above lend themselves easily to generalization.
Urs Schreiber: here is something that I am thinking about.
Let $\mathbf{H}$ be a locally ∞-connected (∞,1)-topos $\mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd$. Write
for the internal homotopy ∞-groupoid? functor.
For $n \in \mathbb{N}$ write
for the reflective (∞,1)-subcategory of n-truncated objects and $\mathbf{\tau}_{\leq n}$ for the localization
Write
for the internal fundamental n-groupoid. For $X \in \mathbf{H}$ we have the (∞,1)-Postnikov tower
For $X \in \mathbf{H}$, the universal geometric $n$-connected cover of $X$ is the homotopy fiber of $X \to \mathbf{\Pi}_n(X)$.
We have that $\mathbf{\Pi}_n(X) \simeq LConst \tau_{\leq n} \Pi(X)$.
A homotopy-commuting diagram
in $\mathbf{H}$ corresponds by the adjunction relation to diagram
in ∞Grpd. This being universal means that $\Pi(X^{(n)})$ is $n$-connected, and universal with that property as an object over $\Pi(X)$.
By running this construction through the Postnikov tower for $\mathbf{\Pi}(X)$, we obtain the Whitehead tower in an (∞,1)-topos
of $X \in \mathbf{H}$.
If we instead generalize the “little topos” picture, then if $E$ is an $(n,1)$-topos (or, more generally, an $(\infty,1)$-topos) which is locally $n$-connected, we have an $n$-groupoid $\Pi_n(E)$ and we can define the universal $n$-connected cover as the pullback topos
The same arguments as above, generalized from 1 to $n$, show that $E^{(n)}\to E$ is a locally trivial local homeomorphism and that $E^{(n)}$ is $n$-connected.
See the references at covering space.