----
> [!definition] Definition. ([[universal cover]])
> We say that a [[covering space|covering map]] $p:\widetilde{X} \to X$ is a **universal cover** if $\widetilde{X}$ is [[simply connected]].
> \
> Universal covering is unique up to [[isomorphism of covering spaces|covering space equivalence]], as a corollary of [[The Galois Correspondence for Covering Spaces]]. Thus it makes sense to refer to 'the' [[universal cover]] of a [[topological space|space]].
>
> See also [[constructing universal covers]].
^definition
In light of the lifting property for covering spaces, the [[universal cover]] satisfies the [[universal property]] that $p$ lifts through any other [[covering space|covering map]] $\pi: (E, e_{0}) \to (X, x_{0})$ to a map $\underline{\pi}:\widetilde{X} \to E$ that is a covering map onto the [[connected component]] of $E$ containing $e_{0}$.
```tikz
\usepackage{tikz-cd}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage[mathscr]{euscript}
\usepackage{tikzit}
\usetikzlibrary{shapes.geometric}
\begin{document}
% TiKZ style file generated by TikZiT. You may edit this file manually,
% but some things (e.g. comments) may be overwritten. To be readable in
% TikZiT, the only non-comment lines must be of the form:
% \tikzstyle{NAME}=[PROPERTY LIST]
% Node styles
\tikzstyle{red dot}=[fill={rgb,255: red,191; green,0; blue,64}, draw=black, shape=circle]
\tikzstyle{green dot}=[fill={rgb,255: red,0; green,128; blue,128}, draw=black, shape=circle]
\tikzstyle{rectangle}=[fill=white, draw=black, shape=rectangle]
\tikzstyle{triangle}=[fill=white, draw=black, regular polygon sides=3, regular polygon, shape=regular polygon, tikzit shape=regular polygon, shape border rotate=180, minimum size=5mm, inner sep=0pt]
\tikzstyle{bullet}=[fill=black, draw=black, shape=circle, minimum size=2mm, inner sep=0pt]
\tikzstyle{noborder}=[fill=none, draw=none, shape=circle]
\tikzstyle{widerectangle}=[fill=white, draw=black, shape=rectangle, minimum width=16mm, minimum height=4mm, inner sep=1pt, tikzit shape=rectangle]
% Edge styles
\tikzstyle{new edge style 0}=[-]
%% paste tikzpicture here
\begin{tikzcd} (\widetilde X,\widetilde x_0) \arrow[d, "p"'] \arrow[r, "\exists !\,\underline{\pi}", dashed] & (E,e_0) \arrow[ld, "\pi"] \\ (X,x_0) & \end{tikzcd}
%% end paste
\end{document}
```
when it exists. That is, the universal cover is the [[terminal object|initial object]] in the [[category]] of based covering spaces over $X$.
----
####
----
#### References
> [!backlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM [[]]
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
> [!frontlink]
> ```dataview
> TABLE rows.file.link as "Further Reading"
> FROM outgoing([[]])
> FLATTEN file.tags as Tag
> WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom"
> GROUP BY Tag
> ```
#reformatrevisebatch01