TITLE: Does anyone know a basepoint-free construction of universal covers?
QUESTION [23 upvotes]: Let $X$ be a real manifold (for simplicity). The standard construction of the universal cover $\varphi: \widetilde{X} \longrightarrow X$ involves fixing a basepoint $p \in X$ and considering homotopy classes of paths from $p$ to $x \in X$.
Is there an alternative construction of $\varphi$ that avoids choosing a basepoint?

REPLY [4 votes]: Here is another attempt at pinning down the meaning of "canonical" in reference to  Tom's answer. 

Let $X$ be a nice space (connected, locally path-connected and semi-locally simply connected).
Let $\pi_X$ be the fundamental groupoid of $X$: this is a category whose objects
are points $x\in X$, where a morphism $x\to y$ is a homotopy class of path fixing the endpoints. 
Let $U_X$ be the groupoid of universal covers: an object is a universal cover $X_1 \to X$ and a morphism $X_1 \to X_2$ is an isomorphism of covers over the identity map of $X$.  

There is a functor $$f:\pi_X\to U_X$$ (i.e., a homomorphism of groupoids)  given by the usual construction of a universal cover. Then $f$ is an equivalence of categories (covering space theory).  
Let $$g: U_X \to \pi_X$$ be its adjoint (which is defined up to unique isomorphism).
This means for any $\tilde X\in U_X$, with $g(\tilde X) = x\in X$ we have a preferred
isomorphism 
$$
f(x) \cong \tilde X\, .
$$
In other words, a universal cover determines a basepoint and a basepoint determines a universal cover.