\documentclass{amsart}
\title{Algebra 2. Problem Set 2}
\author{}

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\usepackage[all]{xy}
\xyoption{arc}
\usepackage[margin=1.25in]{geometry}

\newcommand{\csets}{\mathbf{Sets}}
\newcommand{\contra}[2]{\mathbf{Func}\left(#1^{\scriptstyle{\operatorname{op}}}, #2 \right)}

\newcommand{\C}{\mathcal{C}}
\newcommand{\D}{\mathcal{D}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\h}{\mathsf{h}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Aut}{\operatorname{Aut}}
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\newcommand{\Ker}{\operatorname{Ker}}

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\newcommand{\wt}[1]{\widetilde{#1}}

\newcommand{\problem}[1]{\noindent {\bf Problem #1.}}

\begin{document}
\maketitle

\begin{center}
{\em Unless otherwise stated, the functors below are covariant.}
\end{center}

\problem{1} State the Frobenius reciprocity as adjointness of induction
and restriction functors. See Lecture 21 of Algebra 1 at the following link,
for the statement and proof of Frobenius reciprocity (Thm. 21.4 page 5).

\url{https://people.math.osu.edu/gautam.42/F17/notes.html}\\

\problem{2} Let $\csets$ be the category of sets. Given two sets
$X,Y\in \csets$, define a functor $F : \csets \to \csets$ as follows:
$F(Z) = \Hom_{\csets}(X,Z)\times \Hom_{\csets}(Y,Z)$ for every $Z\in \csets$.
For every morphism $f : Z\to Z'$ in $\csets$, $F(f)$ is componentwise
composition with $f$. That is, $F(f)\! : (g_1,g_2) \mapsto  
(f\circ g_1, f\circ g_2)$. Is this functor representable?\\

\problem{3} With the notations of problem 2 above, let $G$ be the
contravariant analogue of $F$. That is, $G(Z) = \Hom_{\csets}(Z,X)\times \Hom_{\csets}(Z,Y)$.
Determine whether $G$ is representable.\\

\problem{4} For a set $X$, let $\mathfrak{P}(X)$ be the set of 
all subsets of $X$. Consider the {\em covariant} functor $X\mapsto \mathfrak{P}(X)$
which sends $f:X\to Y$ to $\wt{f} : \mathfrak{P}(X)\to \mathfrak{P}(Y)$
given by: $\wt{f}(X') = f(X')\subset Y$, for every $X'\subset X$.
Prove that this functor is {\bf not} representable.\\

\problem{5} Let $\mathbf{Ab}$ be the category of abelian groups,
and $\mathbf{Gps}$ be the category of all groups. Let $F : \mathbf{Ab}\to \mathbf{Gps}$
be the inclusion of subcategory, and let $F' : \mathbf{Gps} \to \mathbf{Ab}$
be the functor which sends a group $G$ to $G/[G,G]$. Recall that
$[G,G]$ is the normal subgroup of $G$ generated by elements of the 
form $aba^{-1}b^{-1}$ for $a,b\in G$ (called the commutator subgroup).
Do these functors form an adjoint pair (in some order)?\\

\problem{6} Let $\C$ be a category and let $\contra{\C}{\csets}$ be the
category of contravariant functors from $\C$ to $\csets$. Recall that for $X\in \C$,
we defined $\h_X := \Hom_{\C}(-, X) \in \contra{\C}{\csets}$.
Prove that for every $F\in \contra{\C}{\csets}$ we have a bijection between
$\Hom_{\contra{\C}{\csets}}(\h_X,F)$ and $F(X)$.\\

\problem{7} Let $F : \C\to \D$ be an equivalence of categories. That is,
$F$ is faithful, full and essentially surjective. Prove that $F$ admits
both a left and a right adjoint.\\

\problem{8} Consider the functors given in the figure below. Assume that
$(F_1,F_2)$ and $(F_1,\wt{F_2})$ are both adjoint pairs. Prove that there
is a natural isomorphism (of functors) between $F_2$ and $\wt{F_2}$. 

\[
\xy
(0,0)*{\C}; (50,0)*{\D};
(2,2)*{}; (48,2)*{} **\crv{(25,15)}?(1)*\dir{>};
(48,-2)*{}; (2,-2)*{} **\crv{(25,-15)}?(1)*\dir{>};
(48,-5)*{}; (2,-5)*{} **\crv{(25,-18)}?(1)*\dir{>};
(25,10)*{F_1}; (25,-15)*{F_2};
(25,-5)*{\wt{F_2}};
\endxy
\]



\end{document}
