Is there a classifying space for covering maps?

Question

It is often said that a sheaf on a topological space $X$ is a "continuously-varying set" over $X$, but the usual definition does not reflect this because a sheaf is not a continuous map from $X$ to some "space of sets". (Such a space must have a proper class of points!) However, I recently had the epiphany that this can be made to work, if one is willing to give up some generality and focus on locally constant sheaves, a.k.a. covering maps.

Let $X$ be a connected CW complex. If I understand correctly, an $n$-fold covering map of $X$ is the same thing as a $S_n$-structured fibre bundle with typical fibre a discrete set of $n$ points, and so their isomorphism classes naturally correspond to isomorphism classes of principal $S_n$-bundles on $X$, which are in turn classified by an Eilenberg–MacLane space $\mathrm{B} S_n = K(S_n, 1)$.

Question 1. Is there a universal $n$-fold covering map of $\mathrm{B} S_n$, i.e. a $n$-fold covering map $T_n \to \mathrm{B} S_n$ such that every $n$-fold covering map of $X$ is obtained (up to isomorphism) as a pullback of $T_n \to \mathrm{B} S_n$ along the classifying map?

It seems to me that once this is done, we can improve the situation slightly and get a classifying space for all finite covering maps by considering $\coprod_{n \in \mathbb{N}} \mathrm{B} S_n$.

Question 2. Does the obvious generalisation work, i.e. does $\mathrm{B} S_{\kappa}$ classify $\kappa$-fold covering maps for each cardinal $\kappa$?

Answer 1

The answer to both your questions is yes, and Qiaochu gave the basic idea. The base space is $BS_n$ and the fiber is $ES_n$. You can make this concrete (very analogous to Grassmannians) by using the model $BS_n \equiv C_n(\mathbb R^\infty) / S_n$ and $ES_n = C_n(\mathbb R^\infty)$ where $C_n$ indicates the configuration space of $n$ labelled points in $\mathbb R^\infty$. i.e. $C_n (\mathbb R^\infty) = Emb(\{ 1,2,\cdots, n\}, \mathbb R^\infty)$.

edit: this is a response to Zhen Lin's 2nd comment:

The theory of classifying spaces (or looking at it another way, obstruction theory). For simplicity, assume $X$ is connected. Give $X$ a CW-structure with one $0$-cell, then a map $X \to BS_n$ when restricted to the $1$-skeleton gives a homomorphism $\pi_1 X \to S_n$, this is the action of $\pi_1$ on $S_n$ described in most intro algebraic topology courses. Now ask, can you extend the map on the $1$-skeleton $X^1 \to BS_n$ to the $2$-skeleton $X^2 \to BS_n$ ? The obstructions (if any) would be elements of $\pi_1 BS_n$, corresponding to the action on the fiber along a $2$-cell attachment. But these are trivial since the covering space pulls-back to a cover of $D^2$, and covering spaces over discs are trivial. Similarly, the obstruction to extending to $X^3$ are elements of $\pi_2 BS_n = *$.

Answer 2

I can see why this question is asked, but feel the better way of looking at covering maps of $X$, see here, is to say that if $X$ is "locally nice" then the fundamental groupoid functor $\pi_1$ determines an equivalence of categories

$$TopCov(X) \to GpdCov(\pi_1 X), $$

from covering maps of $X$ to covering morphisms of $\pi_1 X$. A covering morphism $p: Q \to G$ of groupoids satisfies for $x \in Ob Q$ and $g \in G$ starting at $px$ there is a unique $h \in Q$ starting at $x$ and such that $p(h)=g$.

One then shows that the category $GpdCov(G)$ is equivalent to the category of actions of $G$ on sets. If $G$ acts on a set $S$ on the left then there is an action groupoid which one can write $S \rtimes G$ and the projection $S \rtimes G \to G$ is a covering morphism.