\documentclass[12pt]{amsart}
\title{Algebra 2. Problem Set 1}
\author{}

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\usepackage[all]{xy}
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\newcommand{\CalC}{\mathcal{C}}
\newcommand{\CalD}{\mathcal{D}}

\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\End}{\operatorname{End}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Id}{\operatorname{Id}}
\newcommand{\Ker}{\operatorname{Ker}}

\newcommand{\fd}{\scriptstyle{\operatorname{fd}}}

\newcommand{\ul}[1]{\underline{#1}}

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

\begin{document}
\maketitle

\begin{center}
{\em Unless otherwise stated, the functors below are covariant.}\\
References are made here to the lecture notes. For example,
\S 2.5 refers to Lecture 2, paragraph $5^{th}$ (also labelled
as (2.5) in Lecture 2).\\
\end{center}


\problem{1} Let $F:\CalC\to\CalD$ be a functor from a category $\CalC$ to another
category $\CalD$. Assume that $F$ is faithful. For a morphism $f
\in \Hom_{\CalC}(A,B)$, prove that if $F(f)$ is injective (resp. surjective)
then so is $f$.\\

\problem{2} Let $F:\CalC\to\CalD$ be a functor. Let $f$ be a morphism in
$\CalC$. Assume that $f$ has a left (resp. right) inverse. Prove that
the same is true for $F(f)$. Is the similar assertion true, where
instead we have $f$ injective (resp. surjective)?\\

\problem{3} Give an example for each of the following:
\begin{enumerate}
\item A functor which is faithful and dense but not full.
\item A functor which is full and dense, but not faithful.
\item A functor which is faithful and full, but not dense.\\
\end{enumerate}

\problem{4} Consider the following category, denoted in \S 0.7
by $\mathcal{F}(\mathbf{Ab})$.
Its objects are {\em filetered abelian groups}: $G_{\bullet} = 
G_0\supseteq G_1\supseteq \ldots$ where $G_j$ is an abelian group
and $G_{j+1}$ is a subgroup of $G_j$ (for every $j\geq 0$). Morphisms
are defined by:
\[
\Hom(G_{\bullet},H_{\bullet}) = \{ f : G_0 \to H_0
\text{ group homomorphism such that } f(G_j)\subset H_j,\ 
\forall\  j\geq 0\}
\]
Give an example of a morphism in $\mathcal{F}(\mathbf{Ab})$ which is a 
bijection but not an isomorphism.\\

\problem{5} Let $M$ be a monoid and let $\ul{M}$ denote the category which
has only one object, say $p$, and $\End_{\ul{M}}(p) = M$. Prove that
functors between two such categories $\ul{M}$ and $\ul{N}$ are
same as homomorphisms of monoids $F : M\to N$. Verify that natural
transformations between two functor $F,G$ correspond to elements
$b\in N$ such that $b.F(x) = G(x).b$ for every $x\in M$.\\

\problem{6} Let $G$ be a group and let $G\mbox{-}\mathbf{Sets}$ be the
category whose objects are sets together with a $G$--action, and morphisms
are set maps which commute with the $G$--action (see \S 2.7).
Prove that natural isomorphisms of the forgetful functor $F:
G\mbox{-}\mathbf{Sets} \to \mathbf{Sets}$ are given by elements of the group $G$.
That is, $\Aut(F) = G$.\\

\problem{7} Let $\CalC$ be a category. Recall from \S 0.10 that for an object $X\in\CalC$,
we have two (covariant and contravariant respectively) functors
$h_X, h^X : \CalC\to\mathbf{Sets}$. Given a morphism $f:X\to Y$
in $\CalC$, prove that $h_f : h_Y\Rightarrow h_X$
and $h^f : h^X\Rightarrow h^Y$, defined in \S 1.5, are natural 
transformations.\\

\problem{8} Let $G$ be a group and $H<G$ a subgroup. Verify
that $\operatorname{Ind}_H^G : H\mbox{-}\mathbf{Sets} \to
G\mbox{-}\mathbf{Sets}$ defined in \S 2.7 is a functor.
Prove that $(\operatorname{Ind}_H^G,\operatorname{Res}_H^G)$
is an adjoint pair (see \S 2.7 for the definition of these
functors).\\

\problem{9} Recall the definition of the category $\mathbf{Mat}_K$
from \S 1.7, where $K$ is a fixed field. Recall that we 
have a functor $F : \mathbf{Mat}_K\to \mathbf{Vect}_K^{\fd}$. 
\begin{enumerate}
\item Prove that $F$ is an equivalence of categories.
\item Prove that constructing a functor $G:\mathbf{Vect}_K^{\fd}
\to \mathbf{Mat}_K$ together with natural isomorphisms
\[
\phi:\Id_{\mathbf{Mat}_K} \Rightarrow G\circ F \qquad\text{and}\qquad
\psi:\Id_{\mathbf{Vect}_K^{\fd}} \Rightarrow F\circ G,
\]
is same as making a choice of a basis $B_G(V)$ for every
$V\in\mathbf{Vect}_K^{\fd}$.
\item Let $G_1,G_2$ be two such functors, obtained by choosing $B_1(V),
B_2(V)$ respectively, two bases of $V$, for every finite--dimensional
vector space $V$. Prove that the change of basis matrix
provides a natural isomorphism between $G_1$ and $G_2$.\\
\end{enumerate}

\problem{10} Let $R$ be a unital ring, and let $R\mbox{-}\operatorname{mod}$
be the category of left $R$--modules. Consider the forgetful
functor $F:R\mbox{-}\operatorname{mod}\to \mathbf{Ab}$.
What are the natural transformations $F\Rightarrow F$?\\

\problem{11} Let $\mathbf{Gps}$ be the category of groups
and $\mathbf{Ab}$ the category of abelian groups. Consider
the functor of natural inclusion $I : \mathbf{Ab}\to\mathbf{Gps}$.
Let $A:\mathbf{Gps}\to\mathbf{Ab}$ be given, on objects by:
\[
A : G \mapsto G/[G,G],
\]
where $[G,G]$ is the subgroup of $G$ generated by commutators
$\{aba^{-1}b^{-1} : a,b\in G\}$ (recall that it is automatically
normal). Prove that $(A,I)$ is an adjoint pair.\\

\problem{12} Recall the definition of a pair of adjoint functors
from \S 2.4. Let $F_1:\CalC\to\CalD$ and $F_2:\CalD\to\CalC$
be functors such that $(F_1,F_2)$ is an adjoint pair.
\begin{enumerate}
\item Prove that there is a natural transformation 
$\eta : \Id_{\CalC} \Rightarrow F_2\circ F_1$
(it is called {\em the unit of adjunction}.)
\item Prove that there is a natural transformation
$\varepsilon : F_1\circ F_2 \Rightarrow \Id_{\CalD}$
(it is called {\em the counit of adjunction}.)
\end{enumerate}


\end{document}
