# Functor From Category of Covering Spaces to Category of Sets Equipped With An Action By The Fundamental Groupoid

I have some problems with the understanding of the connection between covering spaces and the fundamental groupoid.

Let $$X$$ be a topological space and let $$\Pi_1(X)$$ denote the fundamental groupoid of $$X$$. We can then construct the category $$\Pi_1(X)-\mathbf{Sets}$$, which has functors $$F:\Pi_1(X)\to\mathbf{Sets}$$ as its objects and natural transformations $$\eta:F\Longrightarrow G$$ as its morphisms ($$F,G\in\Pi_1(X)-\mathbf{Sets}$$). Let furthermore $$\mathbf{Cov}(X)$$ be the category of covering spaces over $$X$$. I would like to construct a functor $$\Phi:\mathbf{Cov}(X)\to\Pi_1(X)-\mathbf{Sets}.$$ I think I have managed to construct the functors correctly, but not the morphisms.

## Construction of Functors

Let $$p:Y\to X$$ be a covering space, we define a functor $$F:\Pi_1(X)\to\mathbf{Sets}$$ as follows.

Objects of the fundamental groupoid are points of $$X$$, so we simply define $$F(x)=p^{-1}(x),$$ where $$x\in X$$.

A morphism in $$\Pi_1(X)$$, between two points $$x_0$$ and $$x_1$$, is a path $$\alpha:x_0\to x_1$$ modulo homotopy equivalence. Given such a path, we construct a set-theoretical map $$\alpha_*:p^{-1}(x_0)\to p^{-1}(x_1)$$ (from a point $$y_0\in p^{-1}(x_0)$$ to a point $$y_1\in p^{-1}(x_1)$$) as follows:

We have a covering space $$p:Y\to X$$, which makes it possible to apply the path-lifting property. That is, we lift $$\alpha:x_0\to x_1$$ to a path $$\widetilde{\alpha}:y_0\to y_1$$, such that $$\beta(0)=y_0$$ and $$p\widetilde{\alpha}=\alpha$$. Letting $$\widetilde{\alpha}(1)=y_1$$, one can check that $$p\widetilde{\alpha}(1)=p(y_1)=\alpha(1)=x_1$$, which gives a well-defined map $$\alpha_*:p^{-1}(x_0)\to p^{-1}(x_1).$$ One can then check that this satisfies the functorial properties.

## Construction of Morphisms

So, I think I know how to do this step without any reference to functors (it feels like it should be an easy translation to functors, but somehow I cannot figure it out). Let us begin to do this without any functors.

### A Construction Without Functors

Let $$X$$ be a topological space. Without the category-theoretical language, I think this is how the argument goes.

Let $$(Y_0,p_0)$$ and $$(Y_1,p_1)$$ be covering spaces of a topological space $$X$$. A morphism between the two covering spaces is then a map $$\varphi:Y_0\to Y_1$$ such that $$p_1\circ \varphi=p_0$$.

We want to use the covering space morphism to construct a $$\pi_1(X,x)-\text{Set}$$ morphism; That is, maps between sets which preserves the action by the paths. What one can do is to simply combine the covering space morphism and the path-lifting diagram to show this.

Let $$\alpha:I\to X$$ be a path, by the path lifting property, we can lift $$\alpha$$ to a path $$\widetilde{\alpha}:I\to Y$$ such that $${p_0}_*(\widetilde{\alpha})=\alpha$$ and $$\widetilde{\alpha}(0)=y$$, with $$y\in p_0^{-1}(x)$$. Thus $$\alpha_*(y)=\widetilde{\alpha}(1).$$ We get another path through $$\varphi_*(\widetilde{\alpha})$$ in $$Y$$, with initial point $$\varphi_*(\widetilde{\alpha})(0)=\varphi(y)$$ and terminal point $$\varphi_*(\widetilde{\alpha})(1)=\varphi(\alpha_*)(y)$$.

The following extended commutative diagram shows that $$p_*[\varphi_*(\widetilde{\alpha})]=\alpha$$ (with an extra map $$p_0:Y_0\to X$$, but I didn't knew how to include it, without making a total mess of the diagram). $$\require{AMScd}$$ $$\begin{CD} Y_0 @>\varphi>> Y_1\\ @AA\widetilde{\alpha}A @VVp_1V\\ I @>\alpha>> X \end{CD}$$ Hence $$\varphi_*(\widetilde{\alpha})$$ is a lifting of $$\alpha$$. So we get that the terminal point of $$\alpha_*(\varphi y)$$ is the terminal point of $$\varphi_*(\widetilde{\alpha})$$, also. So $$\alpha_*(\varphi(y))=\varphi(\alpha_*(y)).$$ Thus, we have a morphism between path-actions.

### A Construction With Functors

Given a covering space homomorphism $$\varphi:(Y_0,p_0)\to (Y_1,p_1)$$, we want to map it to a morphism in $$\Pi_1(X)-\mathbf{Sets}$$, by applying $$\Phi$$. In a category of functors a morphism is a natural transformation.

A natural transformation is constructed as follows. Given two functors $$\Gamma,\Delta:\mathcal{A}\to\mathcal{B}$$ between two categories $$\mathcal{A},\mathcal{B}$$, we have morphism $$\mu_X:\Gamma(X)\to\Delta(X)$$, and a commutative diagram $$\require{AMScd}$$ $$\begin{CD} \Gamma(X) @>\mu_X>> \Delta(X)\\ @VV\Gamma(f)V @VV\Delta(f)V\\ \Gamma(Y) @>\mu_Y>> \Delta(Y), \end{CD}$$ where $$f:X\to Y$$ is a morphism in $$\mathcal{A}$$.

In our case, we have two functor $$F,G:\Pi_1(X)\to\mathbf{Sets}$$, morphisms $$\eta_x:F(x)\to G(x)$$ and a commutative diagram $$\require{AMScd}$$ $$\begin{CD} F(x_0) @>\eta_{x_0}>> G(x_0)\\ @VVF(\alpha)V @VVG(\alpha)V\\ F(x_1) @>\eta_{x_1}>> G(x_1), \end{CD}$$ with $$\alpha:x_0\to x_1$$ a path.

Question 1. What does the natural transformation diagram look like? Is it correct to rewrite it as $$\require{AMScd}$$ $$\begin{CD} p_0^{-1}(x_0) @>\eta_{x_0}>> p_1^{-1}(x_0)\\ @VVF(\alpha)V @VVG(\alpha)V\\ p_0^{-1}(x_1) @>\eta_{x_1}>> p_1^{-1}(x_1), \end{CD}$$ or do we get something like $$\require{AMScd}$$ $$\begin{CD} p_0^{-1}(x_0) @>\eta_{x_0}>> p_0^{-1}(x_0)\\ @VVF(\alpha)V @VVG(\alpha)V\\ p_1^{-1}(x_1) @>\eta_{x_1}>> p_1^{-1}(x_1)? \end{CD}$$ Question 2. How do we define $$\eta_x:F(x)\to G(x)$$? If $$F(x)=p_0^{-1}(x)$$ and $$G(x)=p_1^{-1}(x)$$, then we want to construct a map $$\eta_x:p_0^{-1}(x)\to p_1^{-1}(x)$$. I would like to construct this map through $$\varphi$$, I guess. But I'm not really sure how to do it.

Question 3. If $$F(x_0)=G(x_0)=p_0^{-1}(x_0)$$. Don't we have the following equality then $$F(\alpha)=G(\alpha)$$? Which I think maybe talks against on of my suggestions in Question 1.

I think all I should do, is to try to mimic what we did in the previous subsection. Or well, if not, more importantly just how to construct the natural transformation. But I don't really know what I am doing at all, there are just so much abstract things happening at the same time right now - which makes me confused.

I would be really happy if someone could help me out with the expression of the natural transformation.

Best wishes,

Joel

For question 2, the map $$\eta_x: p_0^{-1}(x) \to p_1^{-1}(x)$$ is just the restriction of the covering space homomorphism $$\phi: Y_0 \to Y_1$$ to $$p_0^{-1}(x)$$. The image lands in $$p_1^{-1}(x)$$ since $$\phi$$ preserves the projections.
To show that the naturality square commutes for $$\eta$$ constructed this way, use the uniqueness of path-lifting for covering spaces.