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   \Blatt{2}
   \Abgabe{02/05/2002}


   \textbf{Defintion:\hspace{0.2cm}}
   Let $X$ be a topological space.
   \begin{enumerate}
   \item 
     A tuple $(E,\pi)$ is a \emph{covering space} if
     $\pi:E\rightarrow X$ is a local homeomorphism, i.\ e.\ for
     each $e\in E$ there are open neighbourhoods $U(e)\subset E$
     and $U\big(\pi(e)\big)\subset X$ such that
     $\pi_|:U(e)\rightarrow U\big(\pi(e)\big)$ is a homeomorphism.
     
     If $(E,\pi)$ and $\big(E',\pi')$ are covering spaces,
     then a continous map $\phi:E\rightarrow E'$ is called a
     \emph{morpism of covering spaces} if the following diagram
     commutes:
     \begin{displaymath}
       \xymatrix{
         E\ar[rr]^\phi\ar[dr]_\pi && E'\ar[dl]^{\pi'}\\ & X
         }
     \end{displaymath}
     By $\kc\ks(X)$ we denote the category of covering spaces of $X$. 
   \item      
     If $(E,\pi)$ is a covering space, we define the sheaf $\LL
     E:=\kf_E$ as the sheaf of sections of $\pi$, i.\ e.\ $\LL
     E(U)=\{s:U\rightarrow E\;|\; s \mbox{ continous},\pi\circ
     s=\id_U\}$ for $U\subseteq X$ open with the obvious restriction maps.

     If $\phi:(E,\pi)\rightarrow \big(E',\pi')$ is a morphism of
     covering spaces we define a sheaf morphism $\LL\phi=\widetilde{\phi}:\LL
     E\rightarrow \LL E'$ by  
     \begin{displaymath}
       \LL\phi(U):\LL E(U)=\kf_E(U)\rightarrow \LL E'(U)=\kf_{E'}(U):
       s\mapsto \phi\circ s
     \end{displaymath}
     for $U\subseteq X$ open.
   \item If $\kf$ is a presheaf of sets on $X$ we define
     $\Gamma\kf=\coprod_{x\in X}\kf_x$ and
     $\pr:\Gamma\kf\rightarrow X$ the canonical projection. 

     For $s\in\kf(U)$ with $U\subseteq X$ open, we define
     $\tilde{s}:U\rightarrow \Gamma\kf:x\rightarrow s_x$, and we use
     these maps to define a topology on $\Gamma\kf$. Namely, 
     $V\subseteq \Gamma\kf$ is open in $\Gamma\kf$ if and only if for any $U\subseteq
     X$ open and for any $s\in\kf(U)$ the preimage
     $\tilde{s}^{-1}(V)\subseteq U$ is open in $U$.

     If $\varphi:\kf\rightarrow\kf'$ is a morphism of presheaves we
     then define 
     \begin{displaymath}
       \Gamma\varphi=\widehat{\varphi}:\Gamma\kf\rightarrow\Gamma\kf':s_x\rightarrow\varphi_x(s_x).
     \end{displaymath}

     By $\Preshv(X)$ we denote the category of
     presheaves on $X$, and by $\Shv(X)$ the subcategory of sheaves.
   \end{enumerate}

   \Nr
   Show that $\Gamma:\kc\ks(X)\rightarrow \Shv(X)$ is functor.

   \Nr 
   Show that $\LL:\Preshv(X)\rightarrow \kc\ks(X)$ is functor.
   
   \Nr
   Let $\kf$ be a sheaf on $X$ and $(E,\pi)$ a covering space. Show:
   \begin{enumerate}
   \item $\LL\Gamma\kf$ is naturally\footnote{I.\ e.\ if $\varphi:\kf
       \rightarrow \kf'$ is a morphism of sheaves, then the following
       diagram commutes 
       \begin{displaymath}
         \xymatrix@C1.3cm{
           \kf\ar[r]^\varphi\ar[d]_\cong&\kf'\ar[d]^\cong\\
           \LL\Gamma\kf\ar[r]^{\LL\Gamma\varphi}&\LL\Gamma\kf'.
           }
       \end{displaymath}
       }
     isomorphic to $\kf$.
   \item $(\Gamma\LL E,\pr)$ is naturally\footnote{I.\ e.\ if $\phi:(E,\pi)
       \rightarrow \big(E'\pi'\big)$ is a morphism of covering spaces, then the following
       diagram commutes 
       \begin{displaymath}
         \xymatrix@C1.3cm{
           (E,\pi)\ar[r]^\varphi\ar[d]_\cong&\big(E',\pi'\big)\ar[d]^\cong\\
           (\Gamma\LL E,\pr)\ar[r]^{\Gamma\LL\varphi}&\big(\Gamma\LL E',\pr\big).
           }
       \end{displaymath}
       } isomorphic to $(E,\pi)$.
   \end{enumerate}
   
\end{document}


