Covering space

Since a covering ${\displaystyle \pi :E\rightarrow X}$ maps each of the disjoint open sets of ${\displaystyle \pi ^{-1}(U)}$ homeomorphically onto ${\displaystyle U}$ it is a local homeomorphism, i.e. ${\displaystyle \pi }$ is a continuous map and for every ${\displaystyle e\in E}$ there exists an open neighborhood ${\displaystyle V\subset E}$ of ${\displaystyle e}$, such that ${\displaystyle \pi |_{V}:V\rightarrow \pi (V)}$ is a homeomorphism.

It follows that the covering space ${\displaystyle E}$ and the base space ${\displaystyle X}$ locally share the same properties.

• If ${\displaystyle X}$ is a connected and non-orientable manifold, then there is a covering ${\displaystyle \pi$ :{\tilde {X}}\rightarrow X} of degree ${\displaystyle 2}$, whereby ${\displaystyle {\tilde {X}}}$ is a connected and orientable manifold.[1]^: 234
• If ${\displaystyle X}$ is a connected Lie group, then there is a covering ${\displaystyle \pi$ :{\tilde {X}}\rightarrow X} which is also a Lie group homomorphism and ${\displaystyle {\tilde {X}}:=\{\gamma$ :\gamma {\text{ is a path in X with }}\gamma (0)={\boldsymbol {1_{X}}}{\text{ modulo homotopy with fixed ends}}\}} is a Lie group.[2]^: 174
• If ${\displaystyle X}$ is a graph, then it follows for a covering ${\displaystyle \pi :E\rightarrow X}$ that ${\displaystyle E}$ is also a graph.[1]^: 85
• If ${\displaystyle X}$ is a connected manifold, then there is a covering ${\displaystyle \pi$ :{\tilde {X}}\rightarrow X} , whereby ${\displaystyle {\tilde {X}}}$ is a connected and simply connected manifold.[3]^: 32
• If ${\displaystyle X}$ is a connected Riemann surface, then there is a covering ${\displaystyle \pi$ :{\tilde {X}}\rightarrow X} which is also a holomorphic map[3]^: 22 and ${\displaystyle {\tilde {X}}}$ is a connected and simply connected Riemann surface.[3]^: 32

Let ${\displaystyle p,q}$ and ${\displaystyle r}$ be continuous maps, such that the diagram

commutes.

• If ${\displaystyle p}$ and ${\displaystyle q}$ are coverings, so is ${\displaystyle r}$.
• If ${\displaystyle p}$ and ${\displaystyle r}$ are coverings, so is ${\displaystyle q}$.[4]^: 485

Let ${\displaystyle X}$ and ${\displaystyle X'}$ be topological spaces and ${\displaystyle p:E\rightarrow X}$ and ${\displaystyle p':E'\rightarrow X'}$ be coverings, then ${\displaystyle p\times p':E\times E'\rightarrow X\times X'}$ with ${\displaystyle (p\times p')(e,e')=(p(e),p'(e'))}$ is a covering.[4]^: 339

Let ${\displaystyle X}$ be a topological space and ${\displaystyle p:E\rightarrow X}$ and ${\displaystyle p':E'\rightarrow X}$ be coverings. Both coverings are called equivalent, if there exists a homeomorphism ${\displaystyle h:E\rightarrow E'}$, such that the diagram

commutes. If such a homeomorphism exists, then one calls the covering spaces ${\displaystyle E}$ and ${\displaystyle E'}$ isomorphic.

An important property of the covering is, that it satisfies the lifting property, i.e.:

Let ${\displaystyle I}$ be the unit interval and ${\displaystyle p:E\rightarrow X}$ be a covering. Let ${\displaystyle F:Y\times I\rightarrow X}$ be a continuous map and ${\displaystyle {\tilde {F}}_{0}:Y\times \{0\}\rightarrow E}$ be a lift of ${\displaystyle F|_{Y\times \{0\}}}$, i.e. a continuous map such that ${\displaystyle p\circ {\tilde {F}}_{0}=F|_{Y\times \{0\}}}$. Then there is a uniquely determined, continuous map ${\displaystyle {\tilde {F}}:Y\times I\rightarrow E}$, which is a lift of ${\displaystyle F}$, i.e. ${\displaystyle p\circ {\tilde {F}}=F}$.[1]^: 60

If ${\displaystyle X}$ is a path-connected space, then for ${\displaystyle Y=\{0\}}$ it follows that the map ${\displaystyle {\tilde {F}}}$ is a lift of a path in ${\displaystyle X}$ and for ${\displaystyle Y=I}$ it is a lift of a homotopy of paths in ${\displaystyle X}$.

Because of that property one can show, that the fundamental group ${\displaystyle \pi _{1}(S^{1})}$ of the unit circle is an infinite cyclic group, which is generated by the homotopy classes of the loop ${\displaystyle \gamma :I\rightarrow S^{1}}$ with ${\displaystyle \gamma (t)=(\cos(2\pi t),\sin(2\pi t))}$.[1]^: 29

Let ${\displaystyle X}$ be a path-connected space and ${\displaystyle p:E\rightarrow X}$ be a connected covering. Let ${\displaystyle x,y\in X}$ be any two points, which are connected by a path ${\displaystyle \gamma }$, i.e. ${\displaystyle \gamma (0)=x}$ and ${\displaystyle \gamma (1)=y}$. Let ${\displaystyle {\tilde {\gamma }}}$ be the unique lift of ${\displaystyle \gamma }$, then the map

${\displaystyle L_{\gamma }:p^{-1}(x)\rightarrow p^{-1}(y)}$ with ${\displaystyle L_{\gamma }({\tilde {\gamma }}(0))={\tilde {\gamma }}(1)}$

is bijective.[1]^: 69

If ${\displaystyle X}$ is a path-connected space and ${\displaystyle p:E\rightarrow X}$ a connected covering, then the induced group homomorphism

${\displaystyle p_{\#}:\pi _{1}(E)\rightarrow \pi _{1}(X)}$ with ${\displaystyle p_{\#}([\gamma ])=[p\circ \gamma ]}$,

is injective and the subgroup ${\displaystyle p_{\#}(\pi _{1}(E))}$ of ${\displaystyle \pi _{1}(X)}$ consists of the homotopy classes of loops in ${\displaystyle X}$, whose lifts are loops in ${\displaystyle E}$.[1]^: 61

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be Riemann surfaces, i.e. one dimensional complex manifolds, and let ${\displaystyle f:X\rightarrow Y}$ be a continuous map. ${\displaystyle f}$ is holomorphic in a point ${\displaystyle x\in X}$, if for any charts ${\displaystyle \phi _{x}:U_{1}\rightarrow V_{1}}$ of ${\displaystyle x}$ and ${\displaystyle \phi _{f(x)}:U_{2}\rightarrow V_{2}}$ of ${\displaystyle f(x)}$, with ${\displaystyle \phi _{x}(U_{1})\subset U_{2}}$, the map ${\displaystyle \phi _{f(x)}\circ f\circ \phi _{x}^{-1}:\mathbb {C} \rightarrow \mathbb {C} }$ is holomorphic.

If ${\displaystyle f}$ is holomorphic at all ${\displaystyle x\in X}$, we say ${\displaystyle f}$ is holomorphic.

The map ${\displaystyle F=\phi _{f(x)}\circ f\circ \phi _{x}^{-1}}$ is called the local expression of ${\displaystyle f}$ in ${\displaystyle x\in X}$.

If ${\displaystyle f:X\rightarrow Y}$ is a non-constant, holomorphic map between compact Riemann surfaces, then ${\displaystyle f}$ is surjective and an open map,[3]^: 11 i.e. for every open set ${\displaystyle U\subset X}$ the image ${\displaystyle f(U)\subset Y}$ is also open.

Let ${\displaystyle f:X\rightarrow Y}$ be a non-constant, holomorphic map between compact Riemann surfaces. For every ${\displaystyle x\in X}$ there exist charts for ${\displaystyle x}$ and ${\displaystyle f(x)}$ and there exists a uniquely determined ${\displaystyle k_{x}\in \mathbb {N_{>0}} }$, such that the local expression ${\displaystyle F}$ of ${\displaystyle f}$ in ${\displaystyle x}$ is of the form ${\displaystyle z\mapsto z^{k_{x}}}$.[3]^: 10 The number ${\displaystyle k_{x}}$ is called the ramification index of ${\displaystyle f}$ in ${\displaystyle x}$ and the point ${\displaystyle x\in X}$ is called a ramification point if ${\displaystyle k_{x}\geq 2}$. If ${\displaystyle k_{x}=1}$ for an ${\displaystyle x\in X}$, then ${\displaystyle x}$ is unramified. The image point ${\displaystyle y=f(x)\in Y}$ of a ramification point is called a branch point.

Let ${\displaystyle f:X\rightarrow Y}$ be a non-constant, holomorphic map between compact Riemann surfaces. The degree ${\displaystyle \operatorname {deg} (f)}$ of ${\displaystyle f}$ is the cardinality of the fiber of an unramified point ${\displaystyle y=f(x)\in Y}$, i.e. ${\displaystyle \operatorname {deg} (f):=|f^{-1}(y)|}$.

This number is well-defined, since for every ${\displaystyle y\in Y}$ the fiber ${\displaystyle f^{-1}(y)}$ is discrete[3]^: 20 and for any two unramified points ${\displaystyle y_{1},y_{2}\in Y}$, it is: ${\displaystyle |f^{-1}(y_{1})|=|f^{-1}(y_{2})|.}$

It can be calculated by:

${\displaystyle \sum _{x\in f^{-1}(y)}k_{x}=\operatorname {deg} (f)}$ [3]^: 29

A continuous map ${\displaystyle f:X\rightarrow Y}$ is called a branched covering, if there exists a closed set with dense complement ${\displaystyle E\subset Y}$, such that ${\displaystyle f_{|X\smallsetminus f^{-1}(E)}:X\smallsetminus f^{-1}(E)\rightarrow Y\smallsetminus E}$ is a covering.

• Let ${\displaystyle n\in \mathbb {N} }$ and ${\displaystyle n\geq 2}$, then ${\displaystyle f:\mathbb {C} \rightarrow \mathbb {C} }$ with ${\displaystyle f(z)=z^{n}}$ is branched covering of degree ${\displaystyle n}$, whereby ${\displaystyle z=0}$ is a branch point.
• Every non-constant, holomorphic map between compact Riemann surfaces ${\displaystyle f:X\rightarrow Y}$ of degree ${\displaystyle d}$ is a branched covering of degree ${\displaystyle d}$.

Let ${\displaystyle p:{\tilde {X}}\rightarrow X}$ be a simply connected covering. If ${\displaystyle \beta :E\rightarrow X}$ is another simply connected covering, then there exists a uniquely determined homeomorphism ${\displaystyle \alpha$ :{\tilde {X}}\rightarrow E} , such that the diagram

commutes.[4]^: 482

This means that ${\displaystyle p}$ is, up to equivalence, uniquely determined and because of that universal property denoted as the universal covering of the space ${\displaystyle X}$.

A universal covering does not always exist, but the following properties guarantee its existence:

Let ${\displaystyle X}$ be a connected, locally simply connected, then there exists a universal covering ${\displaystyle p:{\tilde {X}}\rightarrow X}$.

${\displaystyle {\tilde {X}}}$ is defined as ${\displaystyle {\tilde {X}}:=\{\gamma$ :\gamma {\text{ is a path in }}X{\text{ with }}\gamma (0)=x_{0}\}/{\text{ homotopy with fixed ends}}} and ${\displaystyle p:{\tilde {X}}\rightarrow X}$ by ${\displaystyle p([\gamma ]):=\gamma (1)}$.[1]^: 64

The topology on ${\displaystyle {\tilde {X}}}$ is constructed as follows: Let ${\displaystyle \gamma :I\rightarrow X}$ be a path with ${\displaystyle \gamma (0)=x_{0}}$. Let ${\displaystyle U}$ be a simply connected neighborhood of the endpoint ${\displaystyle x=\gamma (1)}$, then for every ${\displaystyle y\in U}$ the paths ${\displaystyle \sigma _{y}}$ inside ${\displaystyle U}$ from ${\displaystyle x}$ to ${\displaystyle y}$ are uniquely determined up to homotopy. Now consider ${\displaystyle {\tilde {U}}:=\{\gamma .\sigma _{y}:y\in U\}/{\text{ homotopy with fixed ends}}}$, then ${\displaystyle p_{|{\tilde {U}}}:{\tilde {U}}\rightarrow U}$ with ${\displaystyle p([\gamma .\sigma _{y}])=\gamma .\sigma _{y}(1)=y}$ is a bijection and ${\displaystyle {\tilde {U}}}$ can be equipped with the final topology of ${\displaystyle p_{|{\tilde {U}}}}$.

The fundamental group ${\displaystyle \pi _{1}(X,x_{0})=\Gamma }$ acts freely through ${\displaystyle ([\gamma ],[{\tilde {x}}])\mapsto [\gamma .{\tilde {x}}]}$ on ${\displaystyle {\tilde {X}}}$ and ${\displaystyle \psi$ :\Gamma \backslash {\tilde {X}}\rightarrow X} with ${\displaystyle \psi ([\Gamma {\tilde {x}}])={\tilde {x}}(1)}$ is a homeomorphism, i.e. ${\displaystyle \Gamma \backslash {\tilde {X}}\cong X}$.

• ${\displaystyle r\colon \mathbb {R} \to S^{1}}$ with ${\displaystyle r(t)=(\cos(2\pi t),\sin(2\pi t))}$ is the universal covering of the unit circle ${\displaystyle S^{1}}$.
• ${\displaystyle p\colon S^{n}\to \mathbb {R} P^{n}\cong \{+1,-1\}\backslash S^{n}}$ with ${\displaystyle p(x)=[x]}$ is the universal covering of the projective space ${\displaystyle \mathbb {R} P^{n}}$ for ${\displaystyle n>1}$.
• ${\displaystyle q\colon SU(n)\ltimes \mathbb {R} \to U(n)}$ with ${\displaystyle q(A,t)={\begin{bmatrix}\exp(2\pi it)&0\\0&I_{n-1}\end{bmatrix}}A}$ is the universal covering of the unitary group ${\displaystyle U(n)}$.[5]^{: 5, Theorem 1}
• Since ${\displaystyle SU(2)\cong S^{3}}$, it follows that the quotient map ${\displaystyle f:SU(2)\rightarrow \mathbb {Z_{2}} \backslash SU(2)\cong SO(3)}$ is the universal covering of the ${\displaystyle SO(3)}$.
  

  The Hawaiian earring. Only the ten largest circles are shown.
• A topological space which has no universal covering is the Hawaiian earring:

${\displaystyle X=\bigcup _{n\in \mathbb {N} }\left\{(x_{1},x_{2})\in \mathbb {R} ^{2}:{\Bigl (}x_{1}-{\frac {1}{n}}{\Bigr )}^{2}+x_{2}^{2}={\frac {1}{n^{2}}}\right\}}$

One can show that no neighborhood of the origin ${\displaystyle (0,0)}$ is simply connected.[4]^{: 487, Example 1}

Let ${\displaystyle p:E\rightarrow X}$ be a covering. A deck transformation is a homeomorphism ${\displaystyle d:E\rightarrow E}$, such that the diagram of continuous maps

commutes. Together with the composition of maps, the set of deck transformation forms a group ${\displaystyle \operatorname {Deck} (p)}$, which is the same as ${\displaystyle \operatorname {Aut} (p)}$.

• Let ${\displaystyle q\colon S^{1}\to S^{1}}$ be the covering ${\displaystyle q(z)=z^{n}}$ for some ${\displaystyle n\in \mathbb {N} }$, then the map ${\displaystyle d:S^{1}\rightarrow S^{1}:z\mapsto z\,e^{2\pi i/n}}$ is a deck transformation and ${\displaystyle \operatorname {Deck} (q)\cong \mathbb {Z} /\mathbb {nZ} }$.
• Let ${\displaystyle r\colon \mathbb {R} \to S^{1}}$ be the covering ${\displaystyle r(t)=(\cos(2\pi t),\sin(2\pi t))}$, then the map ${\displaystyle d_{k}:\mathbb {R} \rightarrow \mathbb {R} :t\mapsto t+k}$ with ${\displaystyle k\in \mathbb {Z} }$ is a deck transformation and ${\displaystyle \operatorname {Deck} (r)\cong \mathbb {Z} }$.

Let ${\displaystyle X}$ be a path-connected space and ${\displaystyle p:E\rightarrow X}$ be a connected covering. Since a deck transformation ${\displaystyle d:E\rightarrow E}$ is bijective, it permutes the elements of a fiber ${\displaystyle p^{-1}(x)}$ with ${\displaystyle x\in X}$ and is uniquely determined by where it sends a single point. In particular, only the identity map fixes a point in the fiber.[1]^: 70 Because of this property every deck transformation defines a group action on ${\displaystyle E}$, i.e. let ${\displaystyle U\subset X}$ be an open neighborhood of a ${\displaystyle x\in X}$ and ${\displaystyle {\tilde {U}}\subset E}$ an open neighborhood of an ${\displaystyle e\in p^{-1}(x)}$, then ${\displaystyle \operatorname {Deck} (p)\times E\rightarrow E:(d,{\tilde {U}})\mapsto d({\tilde {U}})}$ is a group action.

A covering ${\displaystyle p:E\rightarrow X}$ is called normal, if ${\displaystyle \operatorname {Deck} (p)\backslash X\cong E}$. This means, that for every ${\displaystyle x\in X}$ and any two ${\displaystyle e_{0},e_{1}\in p^{-1}(x)}$ there exists a deck transformation ${\displaystyle d:E\rightarrow E}$, such that ${\displaystyle d(e_{0})=e_{1}}$.

Let ${\displaystyle X}$ be a path-connected space and ${\displaystyle p:E\rightarrow X}$ be a connected covering. Let ${\displaystyle H=p_{\#}(\pi _{1}(E))}$ be a subgroup of ${\displaystyle \pi _{1}(X)}$, then ${\displaystyle p}$ is a normal covering iff ${\displaystyle H}$ is a normal subgroup of ${\displaystyle \pi _{1}(X)}$.

If ${\displaystyle p:E\rightarrow X}$ is a normal covering and ${\displaystyle H=p_{\#}(\pi _{1}(E))}$, then ${\displaystyle \operatorname {Deck} (p)\cong \pi _{1}(X)/H}$.

If ${\displaystyle p:E\rightarrow X}$ is a path-connected covering and ${\displaystyle H=p_{\#}(\pi _{1}(E))}$, then ${\displaystyle \operatorname {Deck} (p)\cong N(H)/H}$, whereby ${\displaystyle N(H)}$ is the normaliser of ${\displaystyle H}$.[1]^: 71

Let ${\displaystyle E}$ be a topological space. A group ${\displaystyle \Gamma }$ acts discontinuously on ${\displaystyle E}$, if every ${\displaystyle e\in E}$ has an open neighborhood ${\displaystyle V\subset E}$ with ${\displaystyle V\neq \emptyset }$, such that for every ${\displaystyle \gamma \in \Gamma }$ with ${\displaystyle \gamma V\cap V\neq \emptyset }$ one has ${\displaystyle d_{1}=d_{2}}$.

If a group ${\displaystyle \Gamma }$ acts discontinuously on a topological space ${\displaystyle E}$, then the quotient map ${\displaystyle q:E\rightarrow \Gamma \backslash E}$ with ${\displaystyle q(e)=\Gamma e}$ is a normal covering.[1]^: 72 Hereby ${\displaystyle \Gamma \backslash E=\{\Gamma e:e\in E\}}$ is the quotient space and ${\displaystyle \Gamma e=\{\gamma (e):\gamma \in \Gamma \}}$ is the orbit of the group action.

• The covering ${\displaystyle q\colon S^{1}\to S^{1}}$ with ${\displaystyle q(z)=z^{n}}$ is a normal coverings for every ${\displaystyle n\in \mathbb {N} }$.
• Every simply connected covering is a normal covering.

Let ${\displaystyle \Gamma }$ be a group, which acts discontinuously on a topological space ${\displaystyle E}$ and let ${\displaystyle q:E\rightarrow \Gamma \backslash E}$ be the normal covering.

• If ${\displaystyle E}$ is path-connected, then ${\displaystyle \operatorname {Deck} (q)\cong \Gamma }$.[1]^: 72

• If ${\displaystyle E}$ is simply connected, then ${\displaystyle \operatorname {Deck} (q)\cong \pi _{1}(X)}$.[1]^: 71

• Let ${\displaystyle n\in \mathbb {N} }$. The antipodal map ${\displaystyle g:S^{n}\rightarrow S^{n}}$ with ${\displaystyle g(x)=-x}$ generates, together with the composition of maps, a group ${\displaystyle D(g)\cong \mathbb {Z/2Z} }$ and induces a group action ${\displaystyle D(g)\times S^{n}\rightarrow S^{n},(g,x)\mapsto g(x)}$, which acts discontinuously on ${\displaystyle S^{n}}$. Because of ${\displaystyle \mathbb {Z_{2}} \backslash S^{n}\cong \mathbb {R} P^{n}}$ it follows, that the quotient map ${\displaystyle q\colon S^{n}\rightarrow \mathbb {Z_{2}} \backslash S^{n}\cong \mathbb {R} P^{n}}$ is a normal covering and for ${\displaystyle n>1}$ a universal covering, hence ${\displaystyle \operatorname {Deck} (q)\cong \mathbb {Z/2Z} \cong \pi _{1}({\mathbb {R} P^{n}})}$ for ${\displaystyle n>1}$.
• Let ${\displaystyle SO(3)}$ be the special orthogonal group, then the map ${\displaystyle f:SU(2)\rightarrow SO(3)\cong \mathbb {Z_{2}} \backslash SU(2)}$ is a normal covering and because of ${\displaystyle SU(2)\cong S^{3}}$, it is the universal covering, hence ${\displaystyle \operatorname {Deck} (f)\cong \mathbb {Z/2Z} \cong \pi _{1}(SO(3))}$.
• With the group action ${\displaystyle (z_{1},z_{2})*(x,y)=(z_{1}+(-1)^{z_{2}}x,z_{2}+y)}$ of ${\displaystyle \mathbb {Z^{2}} }$ on ${\displaystyle \mathbb {R^{2}} }$, whereby ${\displaystyle (\mathbb {Z^{2}} ,*)}$ is the semidirect product ${\displaystyle \mathbb {Z} \rtimes \mathbb {Z} }$, one gets the universal covering ${\displaystyle f:\mathbb {R^{2}} \rightarrow (\mathbb {Z} \rtimes \mathbb {Z} )\backslash \mathbb {R^{2}} \cong K}$ of the klein bottle ${\displaystyle K}$, hence ${\displaystyle \operatorname {Deck} (f)\cong \mathbb {Z} \rtimes \mathbb {Z} \cong \pi _{1}(K)}$.
• Let ${\displaystyle T=S^{1}\times S^{1}}$ be the torus which is embedded in the ${\displaystyle \mathbb {C^{2}} }$. Then one gets a homeomorphism ${\displaystyle \alpha :T\rightarrow T:(e^{ix},e^{iy})\mapsto (e^{i(x+\pi )},e^{-iy})}$, which induces a discontinuous group action ${\displaystyle G_{\alpha }\times T\rightarrow T}$, whereby ${\displaystyle G_{\alpha }\cong \mathbb {Z/2Z} }$. It follows, that the map ${\displaystyle f:T\rightarrow G_{\alpha }\backslash T\cong K}$ is a normal covering of the klein bottle, hence ${\displaystyle \operatorname {Deck} (f)\cong \mathbb {Z/2Z} }$.
• Let ${\displaystyle S^{3}}$ be embedded in the ${\displaystyle \mathbb {C^{2}} }$. Since the group action ${\displaystyle S^{3}\times \mathbb {Z/pZ} \rightarrow S^{3}:((z_{1},z_{2}),[k])\mapsto (e^{2\pi ik/p}z_{1},e^{2\pi ikq/p}z_{2})}$ is discontinuously, whereby ${\displaystyle p,q\in \mathbb {N} }$ are coprime, the map ${\displaystyle f:S^{3}\rightarrow \mathbb {Z_{p}} \backslash S^{3}=:L_{p,q}}$ is the universal covering of the lens space ${\displaystyle L_{p,q}}$, hence ${\displaystyle \operatorname {Deck} (f)\cong \mathbb {Z/pZ} \cong \pi _{1}(L_{p,q})}$.

Let ${\displaystyle X}$ be a topological space. The objects of the category ${\displaystyle {\boldsymbol {Cov(X)}}}$ are the coverings ${\displaystyle p:E\rightarrow X}$ of ${\displaystyle X}$ and the morphisms between two coverings ${\displaystyle p:E\rightarrow X}$ and ${\displaystyle q:F\rightarrow X}$ are continuous maps ${\displaystyle f:E\rightarrow F}$, such that the diagram

commutes.

Let ${\displaystyle G}$ be a topological group. The category ${\displaystyle {\boldsymbol {G-Set}}}$ is the category of sets which are G-sets. The morphisms are G-maps ${\displaystyle \phi :X\rightarrow Y}$ between G-sets. They satisfy the condition ${\displaystyle \phi (gx)=g\,\phi (x)}$ for every ${\displaystyle g\in G}$.

Let ${\displaystyle X}$ be a connected and locally simply connected space, ${\displaystyle x\in X}$ and ${\displaystyle G=\pi _{1}(X,x)}$ be the fundamental group of ${\displaystyle X}$. Since ${\displaystyle G}$ defines, by lifting of paths and evaluating at the endpoint of the lift, a group action on the fiber of a covering, the functor ${\displaystyle F:{\boldsymbol {Cov(X)}}\longrightarrow {\boldsymbol {G-Set}}:p\mapsto p^{-1}(x)}$ is an equivalence of categories.[1]^: 68–70