nLab
universal covering space

Standard definition
As the homotopy fiber of $X ↪ Π_{1} (X)$
Universal covering objects in an $(∞,1)$ -topos
Reference

Standard definition

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 $\tilde{X} \to X$ there is a map of covering spaces $X^{(1)} \to \tilde{X}$ .

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

{Top}_{*}^{wc} \to {Cov}_{*}

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.

David Roberts: We should move the construction of the universal covering space from covering space to here, but I have limited time.

Urs Schreiber: some of it is now below. Some not.

As the homotopy fiber of $X ↪ Π_{1} (X)$

We describe now how the universal cover construction may be understood from the nPOV. In the next section this point of view is then used to conceive notions of covering spaces and higher covering spaces in more general contexts of (∞,1)-toposes.

To a topological space $X$ is associated the topological groupoid $Π_{1} (X)$ – its fundamental groupoid. With $X$ regarded as a categorically discrete topological groupoid, there is a canonical morphism

X \to Π (X)

that includes $X$ as the collection of constants paths.

Proposition

Let $X$ be a suitably well behaved pointed space. The universal cover $X^{(1)}$ of $X$ is (equivalent to) the homotopy fiber of $X \to Π (X)$ in the (∞,1)-category $H = {Sh}_{(∞,1)} ({Top}_{cg})$ of topological ∞-groupoids.

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

\begin{matrix} X^{(1)} & \to & * \\ ↓ & ↓ \\ X & \to & Π (X) \end{matrix} .

Urs Schreiber: may need polishing.

Proof

We place ourselves in the context of topological ∞-groupoids and regard both the space $X$ as well as its homotopy ∞-groupoid $Π (X)$ and its truncation to the fundamental groupoid $Π_{1} (X)$ as objects in there.

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

Let $π_{0} (X)$ be the set of connected components of $X$ , regarded as a topological $∞$ -groupoid, and choose any section $π_{0} (X) \to Π (X)$ of the projection $Π (X) \to π_{0} (X)$ .

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

\begin{matrix} UCov (X) & \to & π_{0} (X) \\ ↓ & ↓ \\ X & \to & Π (X) \end{matrix} .

We think of this topological $∞$ -groupoid $UCov (X)$ as the universal covering $∞$ -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

\begin{matrix} {UCov}_{1} (X) & \to & * \\ ↓ & ↓^{x} \\ X & \to & Π_{1} (X) \end{matrix},

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 $Π_{1} (X)$ -principal bundle

E_{x} Π_{1} (X) = T_{x} Π_{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 $E_{x} Π_{1} (X)$ achieves. Then the ordinary pullback in the category of simplicial presheaves is the homotopy pullback in $∞$ -prestacks. Then by left exactness of $∞$ -stackification, the image of that in $∞$ -stacks is still a homotopy pullback. )

The topological groupoid $E_{x} Π_{1} (X)$ has as objects homotopy classes rel endpoints of paths in $X$ starting at $x$ and as morphisms $κ : γ \to γ'$ it has commuting triangles

\begin{matrix} x \\ ^{γ} ↙ & ↘^{γ'} \\ y & \overset{κ}{\to} & y' \end{matrix}

in $Π_{1} (X)$ . The topology on this can be deduced from thinking of this as the pullback

\begin{matrix} E_{x} Π_{1} (X) & \to & * \\ ↓ & ↓^{x} \\ Π_{1} (X)^{I} & \overset{d_{0}}{\to} & Π_{1} (X) \end{matrix}

in simplicial presheaves on Top. Unwinding what this means we find that the open sets in $Mor (E_{x} Π_{1} (X))$ 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

\begin{matrix} {UCov}_{1} (X) & \to & E_{x} Π_{1} (X) \\ ↓ & ↓ \\ X & \to & Π_{1} (X) . \end{matrix}

Since $X$ is categorically discrete, this simply produces the space of objects of $E_{x} Π_{1} (X)$ 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

Covering spaces, Building a universal cover , Math Reference Project

Universal covering objects in an $(∞,1)$ -topos

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

Let $H$ be a locally contractible (∞,1)-topos $H \overset{\overset{Π}{\to}}{\overset{\overset{LConst}{\leftarrow}}{\underset{Γ}{\to}}} ∞ Grpd$ . Write

Π : = LConst \circ Π : H \to H

for the internal homotopy ∞-groupoid functor.

For $n \in ℕ$ write

H_{\leq n} \overset{\overset{τ_{\geq n}}{\leftarrow}}{\overset{}{↪}} H

for the reflective (∞,1)-subcategory of n-truncated objects and $τ_{\leq n}$ for the localization

τ_{\leq n} : H \overset{τ_{\leq n}}{\to} H_{\leq n} ↪ H .

Urs Schreiber: various $n$ s here will be off by $\pm 1$ . Too tired to straighten this out right now.

Write

Π_{n} : H \overset{τ_{\leq n}}{\to} H

for the internal homotopy $n$ -groupoid. For $X \in H$ we have the (∞,1)-Postnikov tower

\dots \to Π_{2} (X) \to Π_{1} (X) \to Π_{0} (X) .

Definition

For $X \in H$ , the universal geometric $n$ -connected cover of $X$ is the homotopy fiber of $X \to Π_{n} (X)$ .

We have that $Π_{n} (X) ≃ LConst τ_{\leq n} Π (X)$ .

A homotopy-commuting diagram

\begin{matrix} X^{(n)} & \to & * \\ ↓ & ↓ \\ X & \to & Π_{n} (X) \end{matrix}

in $H$ corresponds by the adjunction relation to diagram

\begin{matrix} Π (X^{(n)}) & \to & * \\ ↓ & ↓ \\ Π (X) & \to & Π_{n} (X) \end{matrix}

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

By running this construction through the Postnikov tower for $Π (X)$ , we obtain the Whitehead tower in an (∞,1)-topos

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

of $X \in H$ .

Reference

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

Covering spaces, Building a universal cover , Math Reference Project

Revised on March 23, 2010 20:35:01 by Urs Schreiber (87.212.203.135)

nLab universal covering space

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