Contents

Definition

A covering space (or wrapping space) is a bundle $p:E\to B$ in Top which is locally trivial and with discrete fiber. That is, a map $p:E\to B$ is a covering space over $B$ if for each point $x\in B$, there exists an open neighborhood $U$ of $x$ evenly covered by $p$: the pullback of $p$ over $U$ is isomorphic to a product bundle with discrete fiber $E_x=p^{-1}(x)$:

(the square is a pullback and the isomorphism maps $(x,e\in E_x)\mapsto e$).

Covering spaces over $B$ form an evident full subcategory $\mathrm{Cov}/B\hookrightarrow \mathrm{Top}/B$. These can be put together to form a replete, wide subcategory $\mathrm{Cov}$ of $\mathrm{Top}$, so that $\mathrm{Cov}/B$ is an over category, just as the notation suggests.

Remarks

• Different points in $B$ may have non-isomorphic fibers. However, if open sets $U$ and $V$ are evenly covered by $p:E\to B$ and have nonempty intersection, then there are canonical identifications

  $$E_x\cong E_z\cong E_y$$E_x \cong E_z \cong E_y

  between typical fibers over $x\in U$, $y\in V$, and $z\in U\cap V$. If $B$ is path-connected, then all the fibers match up to isomorphism (by the unique path-lifting lemma; see below).

• Fibers may be empty. Some authors choose to forbid that, adding the condition that $p$ be a surjection, but the resulting category of covering projections over a space $B$ is not as nice as it would be without that condition.

• The terms “covering space” and “covering projection”, while traditional, are certainly not optimal: they mislead by being too close to (open) “coverings”. James Dolan has suggested “wrapping space” as an alternative (as in the image of thread wrapping around a spool, to evoke the archetypal example of a covering projection, $p:ℝ\to S^1:x\mapsto \exp (ix)$).

Relation to etale spaces

Every covering space (even in the more general sense not requiring any connectedness axiom) is an etale space, but not vice versa:

• for a covering space the inverse image of some open set in the base $B$ needs to be, by the definition, a disjoint union of homeomorphic open sets in $E$; however the ‘size’ of the neighborhoods over various $e$ in the same stalk required in the definition of étalé space may differ, hence the intersection of their projections does not need to be an open set, if there are infinitely many points in the stalk.

• even if the the stalks of the etale space are finite, it need not be locally trivial. For instance the disjoint union ${\coprod }_i\mathrm{Ui}$ of a collecton of open subsets of a space $X$ with the obvious projection $({\coprod }_i{U}_i)\to X$ is etale, but does not have a typical fiber: the fiber over a given point has cardinality the number of open sets $U_i$ that contain this particular point.

Fundamental theorem of covering spaces

The connection between covering spaces over $B$ and the fundamental group ${\pi }_1(B)$ (for $B$ a connected space) is very old and runs very deep. An updated account involves shifting attention to representations of the fundamental groupoid ${\Pi }_1(B)$ (regardless of connectedness); we give a brief outline of the theory here.

Under some technical topological assumptions on the space $B$, the fundamental theorem can be stated thus:

In fact, in the proof of this theorem one establishes an adjoint equivalence: one constructs a left adjoint to $\mathrm{Fiber}$,

via a tensor product or weighted colimit construction, namely the one that extends (by left Kan extension along the Yoneda embedding on ${\Pi }_1(B{)}^{\mathrm{op}}$) the functor

that sends each object $b$ of ${\Pi }_1(B)$ to a universal covering space ${\tilde{B}}_b$ over the path-component of $b$.

We now spell out the details.

In terms of a bar-construction

Given a space $B$, let $\mid B\mid$ be $B$ retopologized with the discrete topology, and consider the pullback in $\mathrm{Top}$

Let $\overline{\mathrm{Path}}(B)$ be the quotient of $\mathrm{Path}(B)$ by the equivalence relation “homotopy rel boundary”. We can think of $\overline{\mathrm{Path}}(B)$ as a sum of spaces

fibered in the obvious way over $\mid B\mid$ (the set of all basepoints $b$), where ${\tilde{B}}_b$ is the space of paths in $B$ which begin at $b$, modulo homotopy-rel-boundary. The space ${\tilde{B}}_b$ can be thought of the universal covering space over the connected component of a point $b\in B$, considered as a space based at $b$.

We have a span

with an obvious (contravariant) composition action $\mathrm{comp}$ of the fundamental groupoid ${\Pi }_1(B)$, itself regarded as a span

with a monad structure in the bicategory of spans. The action gives a map

of spans from $\mid B\mid$ to $B$.

Now suppose given an object $F$ of ${\mathrm{Set}}^{{\Pi }_1(B)}$, i.e., a covariant action of the fundamental groupoid, that is to say a span $F:1\to \mid B\mid$ equipped with an action $\alpha$ of the monad ${\Pi }_1(B):\mid B\mid \to \mid B\mid$ in $\mathrm{Span}(\mathrm{Top})$. The data of a right-handed action $\mathrm{comp}$ on $\overline{\mathrm{Path}}(B)$ and the left-handed action $\alpha$ on $F$ gives rise to a two-sided bar construction

which here is a simplicial object in the category of spans from $1$ to $B$, whose two face maps from degree 1 to degree 0 take the form:

The coequalizer of this pair provides a canonical augmentation of the two-sided bar construction, and may be called the tensor product

(the seemingly opposite placement of the two tensor factors, as compared against the span constructions above, is simply an artifact of the discrepancy between diagrammatic order of composition, and the traditional order in which right actions are covariant and left actions contravariant).

As a span from $1$ to $B$, that is as a bundle over $B$, this tensor product is indeed a covering space over $B$, assuming that $B$ is locally connected and semi-locally simply connected. Finally, the functor

is under these conditions quasi-inverse to the fiber functor

Special case: the universal covering space

As a special case, consider the permutation representation ${\Pi }_1(B)\to \mathrm{Set}$ given by the discrete fibration

(as a span from $1$ to $\mid B\mid$) equipped with the obvious (covariant) action of the monad ${\Pi }_1(B)$ (as a span from $\mid B\mid$ to $\mid B\mid$). This is essentially the “regular representation” of the fundamental groupoid. The tensor product of the previous section,

is a way of realizing the universal covering space over $B$.

Here is a way of thinking of this construction which links it to the description of universal bundles by Roberts and Schreiber, which is based on considering tangent spaces of the fundamental groupoid. If the fundamental groupoid $G={\Pi }_1(B)$ is connected, its universal bundle (as a fibration of groupoids) may be realized as the “tangent groupoid at $b$” or slice

for a chosen basepoint $b\in B$. Note that this slice groupoid is the pullback

with $I$ the groupoid $(0\stackrel{\sim }{\to }1)$. This is then a groupoid over $G$ by the restriction of ${\mathrm{ev}}_1$.

Since the set of arrows of $G$ is obtained as a quotient of the set of paths in $B$, it inherits naturally a topology (a quotient of the compact-open topology on $B^I$) which, together with the given topology on $G_0=B$, makes $G$ a topological groupoid. Then we recover the universal covering space $B_b^{(1)}$ (I prefer this notation for the 1-connected cover, rather than the usual $\tilde{B}$, because it generalises to $B^{(n)}$ for $n$-connected covers - DR) over $B$ by pulling back along the functor $B\to G$, where we consider $B$ as a topological groupoid with only identity arrows. The assumptions on the topology of $B$ mean that $G$ is a locally trivial groupoid? with discrete hom-spaces, which implies that $B_b^{(1)}$ is a locally trivial bundle with discrete fibres. Local path-connectedness implies that it is locally trivial, and the local condition on ${\pi }_1$ holds if and only if the fibres are discrete - this last result is due to Daniel Bliss.

In this analysis, the universal covering space $E_b$ of (path-connected) $B$ is retrieved as the quotient of the space of paths which start at the basepoint $b$, modulo homotopy-rel-boundary; the projection to $B$ takes a class of a path $\varphi$ to its terminal point $\varphi (1)$. This last description is what one would find in any textbook on algebraic topology dealing with covering spaces. This covering space is, strictly speaking, universal among connected covering spaces

More generally, if $S\subset \mid B\mid$ is a set of basepoints (Thanks, Ronnie Brown! - DR), we can form the pullback

which is again a groupoid over $G$ by restriction of ${\mathrm{ev}}_1$. Then pullback of $(S/G)\to G$ along the inclusion $B\to G$ is a covering space which is the sum

of connected, 1-connected covering spaces based at the points in $S$. Thus for not-necessarily-connected $B$, taking $S$ such that it intersects each component of $B$ once we can get a universal covering space of $B$ (universal among covering spaces $E\to B$ that induce isomophisms ${\Pi }_0(E)\to {\Pi }_0(B)$).

This construction is functorlal (for general $S\subset \mid B\mid$), since a map $(B,S)\to (B\prime ,S\prime )$ of pairs (remember we are giving $S,S\prime$ the discrete topology, not the subspace topology) induces a functor of (topological) groupoids ${\Pi }_1(B)\to {\Pi }_1(B\prime )$, which by universality of the pulbacks in the above construction gives a map

covering the given map $B\to B\prime$.

The dependence on basepoints is of course spurious; we can make this explicit by considering the colimit obtained by pasting together the universal covering spaces $B_b^{(1)}$ along isomorphisms induced by paths $b\to c$. But this is in effect how our tensor product construction of the universal covering space works: $\overline{\mathrm{Path}}(B)$ is precisely the sum

which can be viewed as a topological span from $\mid B\mid$ to $B$. The fundamental groupoid acts contravariantly on this sum, and the tensor product

is the same thing as the coequalizer of the pair of arrows

in $\mathrm{Top}/B$, where one arrow is projection and the other is given by the action of pulling back along classes of paths; this coequalizer is a precise description of the pasting colimit alluded to above. It should be noted that this coequaliser is isomorphic to the covering space $B^{(1)}⟨S⟩$ when $S$ has one point in each component of $B$, but the description as the tensor product is a priori functorial without reference to a set of basepoints.

(David or Urs: please feel free to sprinkle your own sugar over this, by adapting or even copying what David wrote below based on your paper.)

(David Roberts: unless someone feels the discussion below is essential, it can be deleted.)

In terms of homotopy fibers

We want to describe here how the universal covering space of $X$ is the homotopy fiber of the canonical morphism $X\to {\Pi }_1(X)$, hence the ${\Pi }_1(X)$-principal bundle classified by this cocycle.

We place ourselves in the context of topological ∞-groupoids and regard both the space $X$ as well as its path ∞-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 contant 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 $\mathrm{\infty }$-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 $\mathrm{\infty }$-bundle classified by the cocycle $X\to \Pi (X)$ is its homotopy fiber, i.e. the homotopy pullback

We think of this topological $\mathrm{\infty }$-groupoid $\mathrm{UCov}(X)$ as the universal covering $\mathrm{\infty }$-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 $E_x{\Pi }_1(X)$ achieves. Then the ordinary pullback in the category of simplicial presheaves is the homotopy pullback in $\mathrm{\infty }$-prestacks. Then by left exactness of $\mathrm{\infty }$-stackification, the image of that in $\mathrm{\infty }$-stacks is still a homotopy pullback. )

The topological groupoid $E_x{\Pi }_1(X)$ has as objects homotopy classes rel endpoints of paths in $X$ starting at $x$ and as morphisms $\kappa :\gamma \to \gamma \prime$ 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 $\mathrm{Mor}(E_x{\Pi }_1(X))$ 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 $E_x{\Pi }_1(X)$ over the points of $X$, which is just the space of all paths in $X$ starting at $x$ with the projection ${\mathrm{UCov}}_1(X)\to X$ being endpoint evaluation.

This indeed is then the usual construction of the universal covering space in terms of paths, as described for instance in

• Karl Dahlke?, Covering spaces, Building a universal cover , Math Reference Project

Reference

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

• Karl Dahlke?, Covering spaces, Building a universal cover , Math Reference Project