I have been reading Hain's notes Lectures on Moduli Spaces of Elliptic Curves, and would like some "philosophical" intuition on the definition of orbifold basepoints.
Let $X$ be a simply connected topological space, and let $\Gamma$ be a discrete group acting on $X$. We denote by $\Gamma\backslash\backslash X$ the corresponding (basic) orbifold. The space $X$ can be regarded as the orbifold $1\backslash\backslash X$ under the action of the trivial group.
On p.18 Hain says that the natural orbifold "universal covering" map $p:X\to\Gamma\backslash\backslash X$ should be regarded as a base point of the orbifold $\Gamma\backslash\backslash X$. I understand how pointed morphims of orbifolds (Defintion 3.1) preserve such "base points", but I was wondering: is there an intuitive reason for calling such a map a base point of the orbifold?
For example, we can look at the analogy with geometric base points in algebraic geometry. By definition a geometric base point of a scheme $X$ is a morphism $\text{Spec}(\bar{k})\to X$ where $\bar{k}$ is an algebraically closed field. Now in the theory of the etale fundamental group, the universal cover of $\text{Spec}(k)$ (for $k$ perfect, at least) is $$\text{Spec}(\bar{k})\to\text{Spec}(k),$$ induced by the choice of an embedding $k\hookrightarrow\bar{k}$. So in this case a base point of $\text{Spec}(k)$ is the same as a universal cover. But this doesn't generalise to schemes of higher dimension.
To motivate the above idea for orbifolds, therefore, I am wondering: are there any other analogies/results in geometry or topology where regarding the universal cover of a space as an abstract base point makes sense? Perhaps such a thing comes from the worlds of stacks/topoi? Or is this just a convenient notation?
Exercise: Suppose $X$ is a topological space. When defining the fundamental group of $X$, you can take any map $f : Z \to X$ from a simply connected space $Z$—i.e. $Z$ contractible—to $X$ to be a basepoint. The point is that there is a unique homotopy class of paths between any two points $z$, $z'$ of $Z$. This gives a distinguished homotopy class of path in $X$ between $f(z)$ and $f(z')$.
If you take the Grothendieck point of view, then:
$\text{}$1. The category of coverings $\text{cov}(Z)$ of $Z$ is equivalent to the category of sets.
$\text{}$2. The map $\text{cov}(X) \to \text{cov}(Z)$ induced by $f$ is a fiber functor.
$\text{}$3. $\pi_1(Z,f)$ is the automorphism group of this fiber functor.
$\text{}$4. If $f : Z \to X$ is a universal covering, then $\pi_1(X,f)$ is just the group of deck transformations $\text{Aut}(Z/X)$.
The analogy with arithmetic theory:
$\text{}$1. $Z$ corresponds to an algebraically closed field $K$.
$\text{}$2. $f : Z \to X$ corresponds to a map $\text{Spec}\,K \to X$.