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\begin{document}
\title[]{Stacks HW4 - sheaves and topoi}\maketitle


\begin{enumerate}

\item Let $T$ be a topos. Define a topology $\tau$ on $T$ by
    declaring a morphism $F' \to F$ to be a covering if it is a
    surjection of sheaves. (This is called the canonical topology.)

 
  \begin{enumerate}

  \item Show that the canonical topology is a topology.
  \item Show that the associated topos is equivalent to $T$.

  \item Show that the canonical topology is the largest topology preserving the
    topos; i.e., prove that if $p\colon X' \to X$ is a morhpism and if every sheaf $F$ in $T$ satisfies the sheaf
    axiom with respect to $p$, then $p$ is an epimorphism in $T$.
  \end{enumerate}

\item Define a site $C$ to be \emph{subcanonical} if for every object
  $X \in C$, $h_X$ is a sheaf. (So, for instance, the etale site of a
  scheme is subcanonical.) Give an example of a site $C$ which is not subcanonical.


\item Let $X' \to X$ be an \'etale surjection of affine schemes and
  let $Y$ be a scheme. Show that the `first sheaf axiom' is satisfied,
  i.e., that the map 
\[
h_Y(X) \to h_Y(X') 
\]
is injective. (Assume that we already know this for $Y$ affine.)


\item Let $F\colon C \to D$ and $G \colon D \to C$ be a pair of
  functors. We say that $F$ is 
  \emph{left adjoint} to $G$ if there is an isomorphism of (bi)-functors 
\[
\Hom(F(-),-) \cong \Hom(-, G(-)).
\]
Let $C'$ be a category and let $C$ be a subcategory. Denote by $i$ the
inclusion $C \to C'$, and suppose that $i$ has a left adjoint $a\colon
C' \to C$. (For instance, $a$ could be sheafification.) Let $D\colon I
\to C$ be a diagram.

Prove that $a(\varinjlim i \circ D) = \varinjlim D$



\end{enumerate}
\end{document}

