nLab
universal covering space

Contents

Standard definition

Let XX be a topological space which is well-connected in that it is

Then there is a connected and simply connected covering space X (1)XX^{(1)} \to X with the universal property that for any other covering space X˜X\widetilde{X} \to X there is a map of covering spaces X (1)X˜X^{(1)} \to \widetilde{X}.

There is a functorial construction of a universal covering space of a pointed space

Top * wcCov * Top_*^{wc} \to Cov_*

where Top * wcTop_*^{wc} is the full subcategory of Top *Top_* with objects the well-connected spaces and CovCov is the subcategory of Top * 2Top_*^2 of pointed maps of spaces with objects the covering space maps.

Specifically, if XX is a space with basepoint x 0x_0, we define X (1)X^{(1)} to be the space whose points are homotopy classes of paths in XX starting at x 0x_0, with the projection X (1)XX^{(1)}\to X projecting to the endpoint of a path. We can equip this set X (1)XX^{(1)}\to X with a topology coming from XX so that it becomes a universal covering space as above. As described at covering space, under the correspondence between covering spaces and Π 1(X)\Pi_1(X)-actions, the space X (1)X^{(1)} corresponds to the “regular representation” of Π 1(X)\Pi_1(X).

As the homotopy fiber of XΠ 1(X)X \rightarrow \Pi_1(X)

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 XX is the (homotopy) fiber of the map XΠ 1(X)X\to \Pi_1(X) from XX to its fundamental groupoid. We can think of this as a way of precisely saying “make Π 1\Pi_1 trivial in a universal way.” There are at least two slightly different ways of making this precise in the language of (,1)(\infty,1)-toposes, depending on whether we view XX as a little topos or as an object of a big topos.

In terms of big toposes

In this section we work in the big (∞,1)-topos of sheaves on TopBallsTopBalls, the site of topological balls with the good open cover coverage. We call objects of this (,1)(\infty,1)-topos “topological ∞-groupoids.”

Now, to a topological space XX is associated the topological groupoid Π 1(X)\Pi_1(X) – its fundamental groupoid. With XX regarded as a categorically discrete topological groupoid, there is a canonical morphism

XΠ(X) X \to \Pi(X)

that includes XX as the collection of constant paths.

Proposition

Let XX be a suitably well behaved pointed space. The universal cover X (1)X^{(1)} of XX is (equivalent to) the homotopy fiber of XΠ(X)X \to \Pi(X) in the (∞,1)-category H=Sh (,1)(Top cg)\mathbf{H} = Sh_{(\infty,1)}(Top_{cg}) of topological ∞-groupoids.

In other words, the principal ∞-bundle classified by the cocycle XΠ 1(X)X \to \Pi_1(X) is the universal cover X (1)X^{(1)}: we have a homotopy pullback square

X (1) * X Π(X). \array{ X^{(1)} &\to& {*} \\ \downarrow && \downarrow \\ X &\to& \Pi(X) } \,.

Urs Schreiber: may need polishing.

Proof

We place ourselves in the context of topological ∞-groupoids and regard both the space XX as well as its homotopy ∞-groupoid? Π(X)\Pi(X) and its truncation to the fundamental groupoid Π 1(X)\Pi_1(X) as objects in there.

The canonical morphism XΠ(X)X \to \Pi(X) hence XΠ 1(X)X \to \Pi_1(X) given by the inclusion of constant paths may be regarded as a cocycle for a Π(X)\Pi(X)-principal ∞-bundle, respectively for a Π 1(X)\Pi_1(X)-principal bundle.

Let π 0(X)\pi_0(X) be the set of connected components of XX, regarded as a topological \infty-groupoid, and choose any section π 0(X)Π(X)\pi_0(X) \to \Pi(X) of the projection Π(X)π 0(X)\Pi(X) \to \pi_0(X).

Then the Π(X)\Pi(X)-principal \infty-bundle classified by the cocycle XΠ(X)X \to \Pi(X) is its homotopy fiber, i.e. the homotopy pullback

UCov(X) π 0(X) X Π(X). \array{ UCov(X) &\to& \pi_0(X) \\ \downarrow && \downarrow \\ X &\to& \Pi(X) } \,.

We think of this topological \infty-groupoid UCov(X)UCov(X) as the universal covering \infty-groupoid of XX. To break this down, we check that its decategorification gives the ordinary universal covering space:

for this we compute the homotopy pullback

UCov 1(X) * x X Π 1(X), \array{ UCov_1(X) &\to& {*} \\ \downarrow && \downarrow^{\mathrlap{x}} \\ X &\to& \Pi_1(X) } \,,

where we assume XX to be connected with chosen baspoint xx 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 Π 1(X)\Pi_1(X)-principal bundle

E xΠ 1(X)=T xΠ 1(X). \mathbf{E}_x \Pi_1(X) = T_x \Pi_1(X) \,.

(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 E xΠ 1(X)\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 E xΠ 1(X)\mathbf{E}_x \Pi_1(X) has as objects homotopy classes rel endpoints of paths in XX starting at xx and as morphisms κ:γγ\kappa : \gamma \to \gamma' it has commuting triangles

x γ γ y κ y \array{ && x \\ &{}^{\mathllap{\gamma}}\swarrow && \searrow^{\mathrlap{\gamma'}} \\ y &&\stackrel{\kappa}{\to}&& y' }

in Π 1(X)\Pi_1(X). The topology on this can be deduced from thinking of this as the pullback

E xΠ 1(X) * x Π 1(X) I d 0 Π 1(X) \array{ \mathbf{E}_x \Pi_1(X) &\to& {*} \\ \downarrow && \downarrow^{\mathrlap{x}} \\ \Pi_1(X)^I &\stackrel{d_0}{\to}& \Pi_1(X) }

in simplicial presheaves on Top. Unwinding what this means we find that the open sets in Mor(E xΠ 1(X))Mor(\mathbf{E}_x \Pi_1(X)) are those where the endpoint evaluation produces an open set in XX.

Now it is immediate to read off the homotopy pullback as the ordinary pullback

UCov 1(X) E xΠ 1(X) X Π 1(X). \array{ UCov_1(X) &\to& \mathbf{E}_x \Pi_1(X) \\ \downarrow && \downarrow \\ X &\to& \Pi_1(X) \,. }

Since XX is categorically discrete, this simply produces the space of objects of E xΠ 1(X)\mathbf{E}_x \Pi_1(X) over the points of XX, which is just the space of all paths in XX starting at xx with the projection UCov 1(X)XUCov_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

In terms of little toposes

On the other hand, we can view a space XX as the little (,1)(\infty,1)-topos Sh (,1)(X)Sh_{(\infty,1)}(X) of (,1)(\infty,1)-sheaves on XX. If XX is locally connected and locally simply connected in the “coverings” sense, then Sh(X)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)Sh_{(2,1)}(X) of sheaves (stacks) of groupoids on XX, 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 (,1)(\infty,1)-topos.

Let EE be any (2,1)-topos which is locally 1-connected. This means that in the unique global sections geometric morphism (E *,E *):EGpd(E^*,E_*)\colon E\to Gpd, the functor E *E^* has a left adjoint E !:EGpdE_!\colon E \to Gpd, which is automatically GpdGpd-indexed. The fundamental groupoid of EE is defined to be Π 1(E)E !(*)\Pi_1(E)\coloneqq E_!(*), where ** is the terminal object of EE.

As discussed here in the (,1)(\infty,1)-case, the construction of Π 1(E)\Pi_1(E) is a left adjoint to the inclusion of groupoids into locally 1-connected (2,1)-toposes (which sends GGpd/GGpd GG\mapsto Gpd/G \simeq Gpd^G). Thus we have a geometric morphism EΠ 1(E)E\to \Pi_1(E) (where we regard Π 1(E)\Pi_1(E) as the (2,1)-topos Gpd Π 1(E)Gpd^{\Pi_1(E)}).

Suppose, for simplicity, that EE is connected. Then Π 1(E)\Pi_1(E) is also connected, and so we have an essentially unique functor *Π 1(E)*\to \Pi_1(E). We define the universal cover of EE to be the pullback (2,1)-topos:

E (1) E * Π 1(E)\array{ E^{(1)} & \to & E \\ \downarrow & & \downarrow \\ * & \to & \Pi_1(E)}

(where of course ** denotes the terminal (2,1)-topos GpdGpd).

Now observe that *Π 1(E)*\to \Pi_1(E) is a local homeomorphism of toposes, since we have GpdGpd/Π 1(E)/(*Π 1(E))Gpd \simeq Gpd/\Pi_1(E)/(*\to \Pi_1(E)). Since local homeomorphisms of toposes are stable under pullback, E (1)EE^{(1)}\to E is also a local homeomorphism, i.e. there exists an object E˜E\widetilde{E}\in E and an equivalence E (1)E/E˜E^{(1)} \simeq E/\widetilde{E} over EE. Moreover, it is not hard to see that E˜\widetilde{E} can be identified with the pullback

E˜ * η * E *(Π 1(E))=E *(E !(*))\array{ \widetilde{E} & \to & * \\ \downarrow & & \downarrow \mathrlap{\eta} \\ * & \to & E^*(\Pi_1(E)) \mathrlap{= E^*(E_!(*))}}

in EE. Note that the bottom map *E *(E !(*))*\to E^*(E_!(*)) is E *E^* applied to the unique map *E !(*)*\to E_!(*), while the right-hand map is the unit of the adjunction E !E *E_!\dashv E^*.

In order to see that this is a sensible definition, we first observe that E (1)E^{(1)} is itself locally 1-connected (since it is etale over EE). Moreover, it is actually 1-connected, which is equivalent to saying that E !(E˜)=*E_!(\widetilde{E}) = *. This is because the “Frobenius reciprocity” condition for the adjunction E !E *E_!\dashv E^* (which is equivalent to saying that E !E_! is GpdGpd-indexed) applied to the defining pullback of E˜\widetilde{E} implies that we also have a pullback

E !(E˜) E !(*) id * E !(*)\array{ E_!(\widetilde{E}) & \to & E_!(*) \\ \downarrow & & \downarrow \mathrlap{id} \\ * & \to & E_!(*)}

which clearly implies that E !(E˜)=*E_!(\widetilde{E}) = *.

Thus, E (1)E^{(1)} is a connected and simply connected space with a local homeomorphism to EE, but is it a covering space? In other words, is it locally trivial? Since we have supposed that EE is locally 1-connected, as a (2,1)-category it can be generated by 1-connected objects, i.e. objects UU such that E !(U)*E_!(U)\simeq *. In particular, we have a 1-connected object UU and a regular 1-epic? U*U\to *.

We claim that if UU is any 1-connected object of EE, then E˜\widetilde{E} is trivialized (or split) over UU, in that U×E˜U\times \widetilde{E} is equivalent, over UU, to U×E *SU\times E^*S for some SGpdS\in Gpd. For pulling back the defining pullback to UU, we obtain

U×E˜ U U×η U U×E *(Π 1(E)).\array{ U\times \widetilde{E} & \to & U \\ \downarrow & & \downarrow \mathrlap{U\times \eta} \\ U & \to & U\times E^*(\Pi_1(E)).}

But U×E *(Π 1(E))(E/U) *(Π 1(E))U\times E^*(\Pi_1(E)) \cong (E/U)^* (\Pi_1(E)), so to give a map U(E/U) *(Π 1(E))U \to (E/U)^* (\Pi_1(E)) over UU is the same as to give a map (E/U) !(*)Π 1(E)(E/U)_!(*) \to \Pi_1(E) in GpdGpd. But (E/U) !(*)*(E/U)_!(*)\simeq *, since UU is 1-connected, and Π 1(E)\Pi_1(E) is connected, so there is only one such morphism. Therefore, the two maps UU×E *(Π 1(E))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/UE/U of the map *Π 1(E)*\to \Pi_1(E). Thus, U×E˜U\times \widetilde{E} is equivalent to (E/U) *(S)U×E *(S)(E/U)^*(S) \cong U\times E^*(S), where S=Ω(Π 1(E))S= \Omega(\Pi_1(E)) is the loop object of Π 1(E)\Pi_1(E), i.e. what we might call the fundamental group of the connected (2,1)-topos EE.

Therefore, since E˜\widetilde{E} is trivialized over any 1-connected object, and EE is generated by 1-connected objects, E˜\widetilde{E} is locally trivial. Moreover, since ** is a discrete object of EE, so is E˜\widetilde{E}. Thus, if we specialize all this to the case E=Sh (2,1)(X)E=Sh_{(2,1)}(X) of (2,1)-sheaves on a topological space, then we conclude that E˜\widetilde{E} is an honest 1-sheaf on XX which, when regarded as a local homeomorphism over XX, is locally trivial (hence a covering space), connected, and 1-connected—i.e. a universal cover of XX.

Higher universal covering objects

The nPOV descriptions above lend themselves easily to generalization.

Universal covering objects in a big (,1)(\infty,1)-topos

Urs Schreiber: here is something that I am thinking about.

Let H\mathbf{H} be a locally ∞-connected (∞,1)-topos HΓLConstΠGrpd\mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd. Write

Π:=LConstΠ:HH \mathbf{\Pi} := LConst \circ \Pi : \mathbf{H} \to \mathbf{H}

for the internal homotopy ∞-groupoid? functor.

For nn \in \mathbb{N} write

H nτ nH \mathbf{H}_{\leq n} \stackrel{\overset{\tau_{\geq n}}{\leftarrow}}{\overset{}{\hookrightarrow}} \mathbf{H}

for the reflective (∞,1)-subcategory of n-truncated objects and τ n\mathbf{\tau}_{\leq n} for the localization

τ n:Hτ nH nH. \mathbf{\tau}_{\leq n} : \mathbf{H} \stackrel{\tau_{\leq n}}{\to} \mathbf{H}_{\leq n} \hookrightarrow \mathbf{H} \,.

Write

Π n:Hτ nH \mathbf{\Pi}_n : \mathbf{H} \stackrel{\mathbf{\tau}_{\leq n}}{\to} \mathbf{H}

for the internal fundamental n-groupoid. For XHX \in \mathbf{H} we have the (∞,1)-Postnikov tower

Π 2(X)Π 1(X)Π 0(X). \cdots \to \mathbf{\Pi}_2(X) \to \mathbf{\Pi}_1(X) \to \mathbf{\Pi}_0(X) \,.
Definition

For XHX \in \mathbf{H}, the universal geometric nn-connected cover of XX is the homotopy fiber of XΠ n(X)X \to \mathbf{\Pi}_n(X).

We have that Π n(X)LConstτ nΠ(X)\mathbf{\Pi}_n(X) \simeq LConst \tau_{\leq n} \Pi(X).

A homotopy-commuting diagram

X (n) * X Π n(X) \array{ X^{(n)} &\to& {*} \\ \downarrow && \downarrow \\ X &\to& \mathbf{\Pi}_n(X) }

in H\mathbf{H} corresponds by the adjunction relation to diagram

Π(X (n)) * Π(X) Π n(X) \array{ \Pi(X^{(n)}) &\to& {*} \\ \downarrow && \downarrow \\ \Pi(X) &\to& {\Pi}_n(X) }

in ∞Grpd. This being universal means that Π(X (n))\Pi(X^{(n)}) is nn-connected, and universal with that property as an object over Π(X)\Pi(X).

By running this construction through the Postnikov tower for Π(X)\mathbf{\Pi}(X), we obtain the Whitehead tower in an (∞,1)-topos

X (2)X (1)X \cdots \to X^{(2)} \to X^{(1)} \to X

of XHX \in \mathbf{H}.

Universal covers of a little (,1)(\infty,1)-topos

If we instead generalize the “little topos” picture, then if EE is an (n,1)(n,1)-topos (or, more generally, an (,1)(\infty,1)-topos) which is locally nn-connected, we have an nn-groupoid Π n(E)\Pi_n(E) and we can define the universal nn-connected cover as the pullback topos

E (n) E * Π n(E)\array{ E^{(n)} & \to & E \\ \downarrow & & \downarrow \\ * & \to & \Pi_n(E)}

The same arguments as above, generalized from 1 to nn, show that E (n)EE^{(n)}\to E is a locally trivial local homeomorphism and that E (n)E^{(n)} is nn-connected.

Reference

An account of the traditional way to think of the construction of the universal covering space is

Revised on November 4, 2010 23:02:20 by Urs Schreiber (87.212.203.135)