If you take a category connCov_*(X) of _pointed_ connected covering spaces of a pointed well-behaved space X, then this is a preorder (at most one arrow between any two objects), and this is equivalent to the poset Sub(G) of subgroups of G = pi_1(X). However, Sub(G) is isomorphic to the reflection of connCov_*(X) into the category of posets. Without the presence of basepoints, it is, as Ronnie Brown would say, more natural to use groupoids. David
No-no, that is in fact the very next exercise:
Covering space theory: Requiring covering spaces of a (well-behaved) connected topological space B to be connected, let \sC ov(B) be the category of covering spaces of B and maps over B. If G is the fundamental group of B, then the orbit category of G is {\em equivalent}, not {\em isomorphic}, to \sC ov(B). Sketch the proof. (Hint: use a universal cover of B to construct a skeleton of the category \sC ov(B).)
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