Universal covering and double cover functors

Question

Initially posted on MSE

Let $\mathsf{CW}$ be the category of CW-complexes and $\mathsf{CW}_*$ that of pointed CW-complexes (possibly disconnected, one basepoint in each component). I would like to know whether there is a functor $\mathsf{CW} \to \mathsf{CW}$ mapping a CW-complex to its universal cover. In the pointed case this works, i.e. there is a functor $\mathsf{CW}_* \to \mathsf{CW}_*$ mapping a pointed CW-complex $(X,x_0)$ to its universal cover $$ \widetilde X(x_0) \colon = \{[\alpha \colon I \to X] \mid \alpha(0)= x_0 \} $$ where square brackets denote homotopy relative endpoints, and the covering projection is $[\alpha] \mapsto \alpha(1)$. The basepoint of $\widetilde X(x_0)$ is the class of the constant map, and induced maps are given by requiring that basepoints map to basepoints, using lift uniqueness. But without the basepoint I am unable to define this as a functor.

A related question (one that I am more interested in) is: define the category $\mathsf{CW}^\text{tw}$ to be the category of pairs $(X,w)$ with $X$ a CW-complex and $w \in H^1(X;\mathbb{Z}_2)$, and maps preserving cohomology classes. Is there a functor $\mathsf{CW}^\text{tw} \to \mathsf{CW}$ which maps a $(X,w)$ to a two sheeted covering space $\overline X \to X$ whose first Stiefel-Whitney class is $w$? Again, this is possible in the pointed setting, and a positive answer to the first question would imply a positive answer here as well. The second functor actually exists on the subcategory $\mathsf{MFD} < \mathsf{CW}^\text{tw}$ of smooth (actually also topological) manifolds with their first Stiefel-Whitney classes: take the sphere bundle of the top exterior algebra of the tangent bundle. This makes me think it would not be an easy task to show that they don't exist over all CW-complexes, i.e. the statement is not that implausible after all.

I would be very grateful for any help, including references and hints. Thank you for reading :)

Answer 1

I am supposing that you want to take all continuous maps between CW-complexes in $\mathsf{CW}$, and I am also supposing for simplicity that you want the purported universal cover functor $U : \mathsf{CW} \to \mathsf{CW}$ to be continuous, and that you want it to be augmented $\eta: U \Rightarrow Id_{\mathsf{CW}}$ so that $\eta_X : U(X) \to X$ is the universal covering map.

Then this is impossible. This is because for each CW-complex $X$ one can then form the continuous map $$X = \mathrm{Hom}_{\mathsf{CW}}(*, X) \to \mathrm{Hom}_{\mathsf{CW}}(U(*), U(X)) = \mathrm{Hom}_{\mathsf{CW}}(*, U(X)) = U(X)$$ a canonical map from each space to its universal cover, and the natural transformation $\eta$ shows that this is a section of the covering map $\eta_X : U(X) \to X$.