\documentclass[12pt, reqno]{amsart}

\input{frontmatter/packages.tex}
\input{frontmatter/standardMacros.tex}
\input{frontmatter/extraMacros.tex}
\input{frontmatter/theoremStyle.tex}



\begin{document}
\title[]{Stacks HW1 - Functors}\maketitle




\begin{enumerate}
% \item Prove Yoneda's Lemma: for any category $C$, the functor $C\to Func(C^{op},(Set))$
% which takes an object $X\in C$ to the functor $h_X\colon C^{op}\to
% (Set)$ (defined by $h_X(T)=Hom_C(T,X)$) is fully faithful. 
% \vspace{5pt}


\item Prove a slightly more general version of Yoneda's Lemma -- let $C$ be a category, $X\in C$ an object, $h_X$ the functor $h_X(T)=Hom_C(T,X)$, and 
$F\colon C^{op}\to \Sets$ a functor. Then $Hom(h_X,F)\cong F(X)$.
\vspace{5pt}





\item  Play the game ``find the representing object'' whenever you get the chance. Determine if the following functors are representable. If they are, find the representing object.
  \begin{enumerate}
 
\item The functor $\Top^{\op}\to \Sets$ taking a topological space $X$
    to the set of open subsets of $X$.  
\item The functor $\Top^{\op}\to \Sets$
    taking a topological space $X$ to the set of closed subsets of
    $X$.  
\item The functor $\Top^{\op}\to \Sets$ taking a topological space
    $X$ to the open subsets of $X$ whose complement is also open.  
\item The functor $\mathbf{HausTop}^{\op}\to \Sets$ taking a topological space $X$
    to the set of open subsets of $X$.  ($\mathbf{HausTop}$ is the category of Hausdorff topological spaces.)
\item The functor $\A^n-\{(0,\ldots,0)\}\colon(\Sch^{\op})\to (\Sets)$ taking a
    scheme $T$ to $\{(f_1,\dots, f_n)\in \calO_T(T)^n|$the $f_i$ do not
    all simultaneously vanish at the origin$\}$.  
\item The functor
    $(\A^n-\{(0,\ldots,0)\})/\G_m\colon \Sch^{\op} \to \Sets$ taking a scheme $T$ to
    $(\A^n-\{(0,\ldots,0)\})(T)/\sim$, where $\sim$ is the equivalence relation
    $(f_1,\dots, f_n)\sim (f_1',\dots, f_n')$ if there is a unit $u\in
    \calO_T(T)$ such that $f_i'=uf_i$ for each $i$.
  \end{enumerate}
\vspace{5pt}

\item Give 3 examples of equivalences of categories that are not isomorphisms of categories.

\item Are the categories \vspace{1cm}
\[
\xymatrix{
\bullet \ar@(ul,ur)
} \text{ and }
\xymatrix{
\bullet 
} 
\]
 equivalent?

\item Are the categories \vspace{1cm}
\[
\xymatrix{
\bullet \ar@/{}_{.5pc}/[r] \ar@(ul,ur)
& \bullet \ar@/{}_{.5pc}/[l] \ar@(ul,ur)
} 
\text{ and }
\xymatrix{
\bullet \ar@(ul,ur)
} 
\]
 equivalent?

\end{enumerate}
\end{document}


  \begin{center}
    \begin{Large}
\xymatrix{                                                           
& \uncover<4->{M = H^1_{\text{cris}}(\Jac_C, \Z_p) 
         \only<6->{\ar[dr] }  }
& 
\\ \uncover<3->{\Jac_C  
          \only<4->{\ar@{-->}[ru] } } 
          \only<5->{\ar@{-->}[rd] }
&                                                                    
&  \uncover<6->{ H^1_{\text{dR}}(\Jac_C, \F_p) }
\\                                                       
& \uncover<5->{\{\Jac_C[p^n]\}_n 
         \only<5->{\ar@{-->}[uu] } }
& \uncover<7->{V^{-1}\colon df \mapsto\frac{\text{``}d(f^p)\text{"}}{p}
         \only<7->{\ar@{}[u]^{\scalebox{1.35}{$\circlearrowleft$}} } }
}
    \end{Large}
  \end{center}
