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 $\kappa$?