Categorification of covering morphisms

Question

Given a category $\mathsf{A}$, let $\mathsf{Fam}(\mathsf{A})$ be its free coproduct cocompletion (which is always extensive). This means every object has a unique up to iso presentation as a coproduct of connected objects. This category is really part of the data of a fibration $\Pi_0:\mathsf{Fam}(\mathsf{A})\longrightarrow \mathsf{Set}$ assigning to each object its set of connected components. There's also an adjunction $\Pi_0\dashv H$ where $H$ is the "discrete" functor taking a set $A$ to $A\cdot \mathbf{1}=\coprod_A\mathbf{1}$.

In the book Galois Theories by Borceux and Janelidze, a neat process of abstraction leads to the following definition.

Definition 6.5.9. An arrow $\alpha:A\longrightarrow B$ in $\mathsf{C}=\mathsf{Fam}(\mathsf A)$ is said to be a covering morphism if there exists an effective descent morphism $p:E\longrightarrow B$ such that the square below is a pullback. $$\require{AMScd} \begin{CD} E\times_BA @>{\eta_{E\times_BA}}>> H\Pi_0(E\times_BA)\\ @V{p^\ast\alpha}VV @VV{H\Pi_0(p^\ast\alpha)}V\\ E @>>{\eta_E}> H\Pi_0(E) \end{CD}$$

Concretely, the unit $\eta$ takes a point to its connected component.

Since I'm having a hard time visualizing this definition as it's presented, I thought of abstracting the definition of a fiber bundle, and the trying to abstract the definition of a covering space as a fiber bundle with discrete fibers.

Definition 1. A trivial fiber bundle with fiber $F$ is an arrow which is isomorphic to $\pi_1:B\times F\longrightarrow B$ in $\mathsf{C}/B$.

Definition 2. Let $\mathcal M$ be a class of arrows in a site $\mathsf{C}$. An arrow $\pi:E\longrightarrow B$ is said to be locally in $\mathcal M$ if there's a covering $\left\{u_i:U_i\rightarrow B \right\}$ such that $u_i^\ast\pi$ are all in $\mathcal M$.

Definition 3. A fiber bundle with fiber $F$ a locally trivial fiber bundle with fiber $F$, i.e it is locally in the class of projections $X\times F\longrightarrow X$.

If $\mathsf{C}$ is complete, trivial fiber bundles are stable under base change (with invariant fiber). If $\mathsf{C}$ is a complete superextensive site, fiber bundles are stable under base change, also with invariant fiber. So complete superextensive sites look like especially good settings for working with fiber bundles.

Definition 4. A covering morphism is a fiber bundle such that $F$ is in the essential image of $H$.

Why is this definition poor, and why does it not capture what we want a covering morphism to capture? How does it compare to definition 6.5.9 (e.g if we take the extensive topology on $\mathsf{C}$)?

What do we want a covering morphism to capture?

Answer 1

First, definition 3 must be mended to allow varying fibers, so a fiber bundle is locally some product projection. Then, definition 4 also allows for varying discrete fibers. Whenever both definitions (this and 6.5.9) are applicable, they coincide.

We'll work our way into increasing generality starting with spaces. $\mathsf{Top}$ with its usual topology is a superextensive site. This allows to replace every covering family with a single arrow - its associated cover (see below). Being an extensive category, $\mathsf{Top}$ has universal coproducts, which ensures the the squares on the left below are pullbacks iff the right one is.

$$\require{AMScd} \begin{CD} U_i\times F_i @>>> A\\ @VVV @VV{\alpha}V\\ U_i @>>> B \end{CD}\forall i\iff\require{AMScd} \begin{CD} \coprod_iU_i\times F_i @>>> A\\ @V{p^\ast\alpha}VV @VV{\alpha}V\\ \coprod_iU_i @>>{p}> B \end{CD}$$

This allows us to encapsulate the local triviality definition for fiber bundles in terms of a single base change. Next, for spaces, note the associated cover $p$ is an étale surjection. Étale surjections are effective descent morphsims of spaces, which suggests generalizing to such morphisms in general contexts.

The problem is that we can't even write a pullback square with varying fibers if we replace $\coprod_iU_i$ by $E$. However, if we assume $\mathsf{C}$ is actually a free coproduct cocompletion, we may write $E\cong \coprod_iE_i$ where $E_i$ are the connected components of $E$.

This yields a notion of a "generalized fiber bundle" - an arrow $\alpha$ for which there exists an effective descent morphism making the square below a pullback.

$$\begin{CD} \coprod_iE_i\times F_i @>>> A\\ @VVV @VV{\alpha}V\\ \coprod_iE_i @>>{p}> B \end{CD}$$

Following the setting of spaces, say an arrow is a trivial covering morphism if it's a trivial fiber bundle with discrete fiber (in the essential image of the copower functor $H:A\mapsto \coprod_A\mathbf 1=A\cdot \bf 1$. Since $\mathsf{C}$ is extensive and has products, it is distributive, hence $E_i\times F_i\cong |F_i|\cdot E_i$. This shows the connected component decomposition of the total space of a trivial covering morphism simply has duplicates of the connected compnents of the base space, with multiplicity equal to the cardinality of the fiber (which is always given by $|\Pi_0(F_i)|$).

Proposition. Suppose $\mathsf C$ is a free coproduct cocompletion. Then $\alpha$ is a trivial covering morphism if and only if the square below is a pullback.

$$\require{AMScd} \begin{CD} A @>{\eta_A}>> H\Pi_0(A)\\ @V{\alpha}VV @VV{H\Pi_0(\alpha)}V\\ B @>>{\eta_B}> H\Pi_0(B) \end{CD}$$

Proof. Let $B\cong\coprod_iB_i$ be the connected components decompositio of $B$. The pullback $B\times_{H\Pi_0(B)}H\Pi_0(A)$ is $\coprod_i | (\Pi_0\alpha)^\ast \left\{ i \right\}|\cdot B_i$ with projection on $B$ given componentwise by identities. The description for $\alpha$ as a trivial fiber bundle with discrete fibers is equivalent, since the components $\alpha:\coprod_i | (\Pi_0\alpha)^\ast \left\{ i \right\}|\cdot B_i\rightarrow \coprod_iB_i$ correspond by connectedness of $B_i$ to the identity. $\square$