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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 $\kappa$?

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    Incidentally, if you want to pass to arbitrary CW-complexes, you can instead talk about functors from the fundamental groupoid of X to the various groupoids $(G \rightrightarrows *)$. In fact, the realization of this target is nothing more than the bar construction of $BG$!2012-12-02

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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 $\pi_2 BS_n = *$.

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    My response is a little too long for a comment so it appears below the "edit" line in my answer.2012-12-31
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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.

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    Yes, I was already aware of this. Unfortunately by replacing spaces by groupoids in this question, things take the flavour of "make the homotopy hypothesis true by defining spaces to be Kan complexes and defining $\infty$-groupoids to be Kan complexes".2012-12-31