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\begin{document}
\title[]{Stacks HW7 - final assignment}\maketitle
\begin{enumerate}

% \item What is the correct notion of monomorphism in a 2-category? Let
%   $F\colon C \to D$ be a functor. Say that $F$ is a monomorphism if any pair of morphisms
% \begin{equation}
%    \xymatrix@R-1.3pc{
%     X \ar@<.5ex>[r]\ar@<-.5ex>[r] & C
%    }\end{equation}
% whose compositions 
% \begin{equation}
%    \xymatrix@R-1.3pc{
%     X \ar@<.5ex>[r]\ar@<-.5ex>[r] & D
%    }\end{equation}
% are \emph{isomorphic} are themselves isomorphic. Show that $F$ is a
% monomorphism iff $F$ is fully faithful.

% \item Do the analgous problem for epimorphisms.
\item Let $C$ and $D$ be categories and calculate very explicitly the 2-limit of the diagram
\begin{equation}
   \xymatrix@R-1.3pc{
    C \ar@<-1 ex>[r]\ar[r]\ar@<+1 ex>[r]  & D
   }\end{equation}

\item Show that a morphism $\calX \to \calY$ of categories is a
  monomorphism (i.e., fully faithful) iff the diagonal is an equivalence.

\item Let $C$ be a site and let $X' \to X$ be a covering in $C$. Show
  that the category $Sh(X' \to X)$ is equivalent to the 2-limit of the diagram
\begin{equation}
   \xymatrix@R-1.3pc{
    \widetilde{X'} \ar@<-.5 ex>[r]\ar@<+.5 ex>[r]  & \widetilde{X''} \ar@<-1 ex>[r]\ar[r]\ar@<+1 ex>[r]  & \widetilde{X'''}
   }\end{equation}

\item Let $D \to C$ be a fibred category. Show that the maps $D(V) \to
  D(U)$ defined in class are functors, and that, for a pair of maps
  $U \to V \to W$, the composition of the functors $D(W) \to D(V) \to
  D(U)$ is isomorphic to $D(W) \to D(U)$.
% \item Let $C$ be a site and let $F$ be a functor. Show that $C_{/F}
%   \to C$ is a stack iff $F$ is a sheaf.
\item Prove the 2-Yoneda lemma.
\item Let $\calX \to \Sch$ be a fibered category. Show that if the
  fibers are setoids, then $\calX$ is equivalent to $\Sch_{/F}$ for
  some functor $F$. Show that in this case  $F$ is a sheaf iff $\calX
  \to \Sch$ is a stack.
\end{enumerate}
\end{document}

