This is one of exercises I have after a chapter about the fundamental theorem of covering spaces (see its statement). Category $\mathrm{Set}^{\Pi_1(X)}$ there denotes the category of functors from the fundamental groupoid of $X$ to the category $\mathrm{Set}$. Note that a similar question came up here (e.g. this question and this one), but they ask for the explicit computation of covering spaces that I don't need here and use results unknown to me. In particular, I don't have the result relating the number of subgroups of the fundamental group to the number covering spaces at my disposal. Moreover, I want a proof that uses the fundamental theorem.
I will now describe my thoughts. Since the equivalence of categories requires a functor to be fully faithful as well as there to be a bijection between isomorphism classes of each category (essential surjectivity), I think I need to prove that there is a finite number of isomorphism classes of $\mathrm{Set}^{\Pi_1(X)}$ and describe morphisms between them, which also should be a finite number. However, I don't know how to do it. Below are quick observations.
- Path-connectivity implies that the functor maps points of $\mathbb{R}P^2 \times \mathbb{R}P^2$ to bijective sets;
- morphisms in $\mathrm{Set}^{\Pi_1(X)}$ are natural transformations;
- there is a trivial functor in $\mathrm{Set}^{\Pi_1(X)}$ that maps each object to a singleton set;
- since there is an identity loop and both $F_1$ and $F_2$ are functors, natural transformation $\Phi: F_1(b) \to F_2(b)$ is required to be an identity function for some object $b$; then it follows that two functors are not isomorphic unless loops are mapped to same permutations for a fixed object;
- two functors are also not isomorphic if they map objects to sets of different cardinality since $\Phi$ is required to be a bijection for a natural transformation to be a natural isomorphism;
Then we could look at functors that map each object to a two-element set. Let's look at how they permute two-element sets with images of loops. I then see four possible functors. If we denote loops with elements of $\mathbb{Z}_2 \times \mathbb{Z}_2$, then they are a functor that maps all loops to identity permutations, a functor that maps $(0, 1)$ and $(1, 0)$ to a transposition of two elements and $(0, 0), (1, 1)$ to an identity permutation, and a functor that maps $(1,1)$ and exactly one of $(0, 1), (1, 0)$ to a transposition. Otherwise, there is inconsistency with group operation.
But then I think we can make infinitely many isomorphism classes just by mapping an object $b$ to a (possibly infinite) set and sending some loops to a transposition of two fixed elements, which reduces it to the case above, and this doesn't even list all of them. What is more, I only considered how a functor works for a fixed object $b$, but it can also be the case that loops at two objects $b, b^{\prime}$ define different permutations of underlying sets for respective loops, which results in infinitely many combinations. I also don't know how to describe natural transformations between sets from different objects, so I don't know how to approach the second question in my title.
And another point that really confuses me is the fact that the question asks to count connected covering spaces. The fundamental theorem works for disconnected spaces, and I don't see how this can help here at all.
Best Answer
If $X$ is path connected, then for any choice of $x\in X$ we have an equivalence $\pi_1(X,x)\simeq \Pi(X)$, therefore also an equivalence $\mathrm{Set}^{\pi_1(X,x)}\simeq\mathrm{Set}^{\Pi(X)}$. The left hand side is isomorphic to the category of sets with a (left) $\pi_1(X,x)$-action and equivariant maps. Denote by $\mathrm{Or}_{\pi_1(X,x)}$ the subcategory of sets with a transitive $\pi_1(X,x)$-action. The fundamental theorem tells you a bit more, namely that there is an equivalence of categories between $\mathrm{Or}_{\pi_1(X,x)}$ and the category of connected coverings of $X$. Note that any $G$-set decomposes as the disjoint union of transitive $G$-sets and under the equivalence of the fundamental theorem this corresponds to decomposing a covering space along its connected components.
In the case at hand, $\pi_1(X)\cong \mathbb{Z}/2\times\mathbb{Z}/2$ and it is easy to check that there are only 5 transitive $\mathbb{Z}/2\times\mathbb{Z}/2$-sets (up to isomorphism), therefore only 5 connected coverings (up to isomorphism).