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\begin{center}{\bf M901, Assignment 3: Due Friday, September 23, 2011}\end{center}

{\it Instructions}: Do any three problems.

First we establish some background.

Let ${\mathcal S}$ be the category of sets.

Let ${\mathcal V}$ be the category of finite dimensional real vector spaces.
Let $\operatorname{Id_{\mathcal V}}$ be the identity functor from
${\mathcal V}$ to ${\mathcal V}$. 
Let ${\bf R}$ be the real numbers, regarded as a real vector space.
Let $M_{\bf R}$ be the covariant functor
from ${\mathcal V}$ to ${\mathcal V}$ where for any object $W$ of ${\mathcal V}$
we set $M_{\bf R}(W) = \operatorname{Mor}_{\mathcal V}({\bf R},W)$ and for any morphism
$W\stackrel{f}{\to}W'$ we define $M_{\bf R}(f)$ to be the morphism 
$M_{\bf R}(W)\to M_{\bf R}(W')$
defined for any $\lambda\in M_{\bf R}(W)$ as $M_{\bf R}(f)(\lambda)=f\circ \lambda$.
We also define a morphism 
$\Phi_W\in \operatorname{Mor}_{\mathcal V}(M_{\bf R}(W),\operatorname{Id_{\mathcal V}}(W))$ 
by $\Phi_W(\lambda)=\lambda(1)$. 
(Usually we regard representation functors to be functors to the category of sets,
but the linear transformations from 
one vector space to another is itself a vector space, and composition of linear transformations
is linear, so in this case $M_{\bf R}$ defines a covariant functor
from ${\mathcal V}$ to itself. Our usual definition for the representation functor 
would be this functor composed
with the forgetful functor to the category of sets. Note that $\Phi_W$ is also a linear
transformation.)

Problem 1 is what is needed to verify that $\Phi$ is a 
natural transformation from $M_{\bf R}$ to $\operatorname{Id_{\mathcal V}}$.
Problem 2 is what is needed to show that $\Phi$ is in fact a natural equivalence.

Problems 3 and 5 give examples of contravariant functors, but in Problem 3 the functor is representable
whereas in Problem 5 it is not. I changed the context to finite abelian groups to simplify
showing non-representability in Problem 5 (as compared with working with
finite dimensional vector spaces). I added Problem 4 since the ideas there can be useful for Problem 5.
(I'm saving for the next homework assignment the problem of showing the functor in Problem 5
is a categorical product in the functor category ${\mathcal A}^{\mathcal S}$ of contravariant
functors from ${\mathcal A}$ to ${\mathcal S}$.)
Problem 6 is to make sure everyone is familiar with the proof of a basic fact about dual spaces.

In preparation for Problems 3 and 5,
let ${\mathcal A}$ be the category of finite abelian groups.
For any finite abelian group $C$, let $M^C$ be the contravariant functor
from ${\mathcal A}$ to ${\mathcal S}$ where for any object $B$ of ${\mathcal A}$
we set $M^C(B) = \operatorname{Mor}_{\mathcal A}(B,C)$ and for any morphism
$B\stackrel{f}{\to}B'$ we define $M^C(f)$ to be the morphism 
$M^C(B')\to M^C(B)$
defined for any $\lambda\in M^C(B')$ as $M^C(f)(\lambda)=\lambda \circ f$. 
Now let $C_1$ and $C_2$ be objects of ${\mathcal A}$.
Define a contravariant functor $F$ from ${\mathcal A}$ to 
${\mathcal S}$ on any object $B$ of ${\mathcal A}$
as $F(B)=M^{C_1}(B)\times M^{C_2}(B)$, and for any morphism $B\stackrel{f}{\to}B'$
define $F(B')\stackrel{F(f)}{\to}F(B)$ by
$(F(f))((\lambda_1,\lambda_2))=(\lambda_1\circ f,\lambda_2\circ f)$.
For each finite abelian group $D$, define a map $\Psi_D:F(D)\to M^{C_1\times C_2}$
by $\Psi_D((a_1,a_2))=u_{(a_1,a_2)}$ where $u_{(a_1,a_2)}:D\to C_1\times C_2$ is the
homomorphism defined by $u_{(a_1,a_2)}:d\mapsto (a_1(d),a_2(d))$. (You may assume without proof 
that the usual direct product
$C_1\times C_2$
is a categorical product of $C_1$ and $C_2$
in the category of finite abelian groups, and thus that $u_{(a_1,a_2)}$ is the unique homomorphism
whose existence is guaranteed by the universal property of the categorical product
$C_1\times C_2$, given homomorphisms $a:D\to C_1$ and $b:D\to C_2$. I.e., 
$u_{(a_1,a_2)}:D\to C_1\times C_2$ is the unique homomorphism
such that $\pi_i\circ u_{(a_1,a_2)}=a_i$ for $i=1,2$, where $\pi_i:C_1\times C_2\to C_i$
are the projections.)

\begin{itemize}

\item[(1)] For any morphism $W\stackrel{f}{\to}W'$ in ${\mathcal V}$, show that the following diagram commutes:
$$
\begin{matrix}
M_{\bf R}(W)  &\xrightarrow{M_{\bf R}(f)} & M_{\bf R}(W')  \\
\\
\downarrow\rlap{$\Phi_W$}  & {} & \llap{$\Phi_{W'}$}\downarrow  \\
\\
\operatorname{Id_{\mathcal V}}(W)  &\xrightarrow{\operatorname{Id_{\mathcal V}}(f)} & \operatorname{Id_{\mathcal V}}(W').
\end{matrix}
$$
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\item[(2)] For any finite dimensional real vector space $W$, show that $\Phi_W$ is bijective.
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\item[(3)] Show that $\Psi$ defines a natural transformation from $F$ to $M^{C_1\times C_2}$
which is an isomorphism $\Psi_D$ for each finite abelian group $D$
(which implies that $\Psi$ is a natural equivalence and hence that
$F$ is representable and in particular that 
$C_1\times C_2$ represents $F$).
\vskip\baselineskip

\item[(4)] Let $C_1, C_2, \ldots$ be non-trivial finite abelian groups (i.e., each one has order bigger than 1).
Let $C=C_1\times C_2\times \cdots$ be the direct product with the usual abelian group structure
of component-wise addition. (You may assume here and in Problem 5
that $C$ together with the usual 
projection homomorphisms is a categorical product in the category of all (not necessarily 
finite) abelian groups.)
For each finite abelian group $B$, let $H(B)$ be the set of
group homomorphisms $B\to C$. 
\vskip\baselineskip

\begin{itemize}
\item[(a)] Show that either one can chose a cyclic group $P$ of prime order such that $|H(P)|$ is infinite or one can choose 
a sequence of cyclic groups $P_1, P_2, \ldots$ of prime orders such that $|P_1|<|P_2|<\cdots$
and such that $\sup_j\{|H(P_j)|\}$ is infinite.
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\item[(b)] Show that the groups $C_i$ and $B$ can be chosen such that $|H(B)|=1$ (i.e., such that $H(B)$
has a single element).
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\end{itemize}
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\item[(5)] Let $C_1, C_2, \ldots$ be non-trivial finite abelian groups (i.e., each one has order bigger than 1).
We now define a functor $G$ from ${\mathcal A}$ to ${\mathcal S}$ generalizing $F$; in particular,
let $G(B)=\Pi_{i}\ M^{C_i}(B)$ be the set-theoretic categorical product of the sets $M^{C_i}(B)$ and for any 
homomorphism $B\stackrel{f}{\to}B'$, define $G(f)$ to be the morphism 
$\Pi_{i}\ M^{C_i}(B')\to \Pi_{i}\ M^{C_i}(B)$
defined as $(G(f))(\lambda_1,\lambda_2,\ldots )=(\lambda_1\circ f,\lambda_2\circ f,\ldots )$. 
Show that $G$ is not representable.
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\item[(6)] For any finite dimensional vector space $W$ over a field $K$, show that
$W$ and its dual $W^*$ have the same dimension. (Recall that the dual space
is the set of linear transformations $W\to K$. It is a $K$-vector space
with respect to addition and scalar multiplication defined as follows.
Given any $\lambda_1,\lambda_2\in W^*$ and $a\in K$,
we define addition $\lambda_1+\lambda_2$ by setting 
$(\lambda_1+\lambda_2)(w)=\lambda_1(w)+\lambda_2(w)$ for all $w\in W$ and we define
scalar multiplication of $\lambda_1$ by $a$ by $(a\lambda_1)(w)=a(\lambda_1(w))$
for all $w\in W$.)
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\end{itemize}

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