[[Covering]]
# Universal covering

The **universal covering** $\hat{p} : \hat{X}\to X$ is a [[covering]] with a [[Simple connectedness|simply connected]] covering space $\hat{X}$. #m/def/homotopy 
It follows immediately that the [[Characteristic subgroup of a covering|characteristic subgroup]] of $\hat{p}$ is trivial, 
and by [[deck transformation group of a regular covering as quotient]]
$$
\begin{align*}
\Aut_{\Cov_{X}}(p) \cong \pi_{1}(X,x_{0})
\end{align*}
$$
for any $x_{0} \in X$.
The universal covering is universal in the sense that $\hat{p} : (\hat{X},\hat{x}_{0}) \twoheadrightarrow (X,x_{0})$ is the [[Initial and terminal objects|initial object]] of the [[Category of coverings|category of pointed connected coverings]] $\cat{Cov}_{(X,x_{0})}$,
assuming $X$ is locally path-connected.

## Properties

- The universal covering is [[Regular covering|regular]].
- [[Main theorem of coverings]]
- [[A locally path-connected, connected space is semilocally simply connected iff it has a universal covering]].

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#state/develop | #lang/en | #SemBr