nLab category of covering spaces

Definition

For $X$ a topological space, the category of covering spaces $Cov(X)$ is the category whose

• objects are covering spaces $E \overset{p}{\to} X$ over $X$;

• morphisms are homomorphisms of covering spaces, hence continuous functions $f \colon E_1 \longrightarrow E_2$ such that we have a commuting diagram

  $$\array{ E_1 && \overset{f}{\longrightarrow} && E_2 \\ & {}_{\mathllap{p_1}}\searrow && \swarrow_{\mathrlap{p_2}} \\ && X }$$

This is equivalently the full subcategory of the slice category $Top_{/X}$ of the category of topological spaces over $X$ on those objects which are covering spaces.

Properties

Proposition

(fundamental theorem of covering spaces)

For $X$ a topological space then forming monodromy is a functor from the category of covering spaces over $X$ to that of permutation groupoid representations of the fundamental groupoid of $X$:

$$Fib \;\colon\; Cov(X) \longrightarrow Set^{\Pi_1(X)} \,.$$

If $X$ is locally path connected and semi-locally simply connected, then this is an equivalence of categories.

See at fundamental theorem of covering spaces for details.

Related concepts

• category of vector bundles