\documentclass{amsart}
\title{Algebra 2. Problem Set 3}
\author{}

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

\newcommand{\csets}{\mathbf{Sets}}
\newcommand{\AB}{\mathbf{Ab}}
\newcommand{\preord}[1]{\operatorname{Cat}(#1,\leq)}

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

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\newcommand{\invlim}[2]{\displaystyle \lim_{\begin{subarray}{c} \longleftarrow \\ (#1,\leq)\end{subarray}} #2\ }

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

\begin{document}
\maketitle

\problem{1} Let $(I,\leq)$ be a preordered set such that $i\leq j$ for every $i,j\in I$.
Let $\C$ be a category and let $\gX = \lp \{X_i\}_{i\in I}; \{\varphi_{ji}\}_{i\leq j}\rp$
be a direct system on $(I,\leq)$ with values in $\C$. Prove that, for every $i\in I$,
$\dirlim{I}{\gX}$ is isomorphic to $X_i$. Similarly let $
\gY = \lp \{Y_i\}_{i\in I}; \{\psi_{ij}\}_{i\leq j}\rp$ be an inverse
system over $(I,\leq)$ with values in $\C$. Prove that $\invlim{I}{\gY}$
is isomorphic to $Y_i$, for every $i\in I$. ({\em Hence, for questions regarding
direct/inverse limits, we can assume that the preordered set is in fact partially
ordered}).\\

\problem{2} Let $(I,\leq)$ be a partially ordered set with a unique maximal element,
say $i_0\in I$. Again, let $\gX$ be a direct system over $(I,\leq)$ with
values in $\C$. Prove that $\dirlim{I}{\gX}$ is isomorphic to $X_{i_0}$.
Similarly, let $\gY$ be an inverse system. Prove that $\invlim{I}{\gY}$
is isomorphic to $Y_{i_0}$.\\

\problem{3} Let $J$ be a set and let $\C$ be a category. Assume that for
every set of objects $\{X_j\}_{j\in J}$ of $\C$, the direct sum 
$\bigoplus_{j\in J} X_j$, and the direct product $\prod_{j\in J} X_j$
exist. Let $\C^J$ be the product category and 
we view $\bigoplus$ and $\prod$ as functors:
\[
\xy
(0,0)*{\C^J}; (40,0)*{\C};
{\ar (2,1)*{}; (38,1)*{}};
{\ar (2,-1)*{}; (38,-1)*{}};
(20,4)*{\bigoplus}; (20,-4)*{\prod};
\endxy
\]
Let $F : \C\to \C^J$
be the canonical inclusion functor. That is, $F(X)_j = X$ for every $j\in J$
and $X\in \C$; and for every morphism $f:X\to Y$ in $\C$, $F(f)_j = f$
for every $j\in J$. Prove that the following two pairs are adjoint functors:
\[
\lp\bigoplus \ ,\  F\rp \qquad \text{and} \qquad 
\lp F \ , \  \prod\rp
\]

\problem{4} Let $I = \{1,2,3,\ldots\}$ together with the usual order. Consider
the following inverse system on $I$ valued in $\AB$, denoted by 
$\mathfrak{Z} = \lp \{Z_n\}_{n\in I}; \{\varphi_{nm}\}_{n\leq m}\rp$: 
\begin{itemize}
\item $Z_n = \Z$ for every $n\in I$.
\item For every $n\leq m$, the group homomorphism $\varphi_{nm} : Z_m \to Z_n$ is given 
by $\varphi_{nm}(x) = 3^{m-n}x$.
\end{itemize}
Prove that $\invlim{I}{\mathfrak{Z}} = (0)$.\\

\problem{5} Given a preordered set $(I,\leq)$, consider the category, 
$\preord{I}$, whose objects are elements of $I$ and for every $i,j\in I$,
$\Hom_{\preord{I}}(i,j)$ is a singleton, say $\{a(i,j)\}$, if $i\leq j$;
and is empty if $i\not\leq j$. Verify that a direct (resp. inverse)
system on $(I,\leq)$, valued in a category $\C$, is an object of the category of 
covariant (resp. contravariant) functors from $\preord{I}$ to $\C$.
{\em Hence, we have a category of direct (resp. inverse) systems on $(I,\leq)$
valued in $\C$, denoted as follows:}\\
\begin{align*}
\C^{(I,\leq)} &:= \covariant{\preord{I}}{\C} \text{ category of direct systems}\\
\C_{(I,\leq)} &:= \contra{\preord{I}}{\C} \text{ category of inverse systems}
\end{align*}

\problem{6} Let $\C$ be a category and $(I,\leq)$ be a preordered set.
For each $X\in \C$, we can define a direct system $\gX$ where each $X_i = X$
and each $\psi_{ji} = \Id_{X}$. Prove that the direct limit (if exists) 
can be interpreted as a functor $\C^{(I,\leq)} \to \C$, and it is
adjoint (left or right?) to the functor thus obtained $F : \C
\to \C^{(I,\leq)}$. Obtain a similar assertion for the inverse limit.\\

\problem{7} Consider the inverse system of Problem 4 above, denoted there
by $\mathfrak{Z}$. Let $\mathfrak{Z}'$ be another inverse system over
the same ordered set $I=\{1,2,3,\ldots\}$ valued in $\AB$, where
each $Z'_n = \Z/2\Z$ and all the morphisms are identities.
Prove that the natural surjection $\Z\to \Z/2\Z$ is a morphism in
the category of inverse systems $\AB_{(I,\leq)}$. ({\em This is one
of the standard examples of the fact that inverse limit of surjections
need not be a surjection.}).\\

\problem{8} Prove that the inverse limit of injective morphisms is
injective. Prove that the direct limit of surjective morphisms is surjective.\\

\problem{9} Let $\gX$ be a direct system over a preordered set $(I,\leq)$
valued in $\csets$. Let $X := \sqcup_{i\in I} X_i / \sim$ where the equivalence
relation is:
\[
x\in X_i \sim \psi_{ji}(x)\in X_j \text{ for every } i\leq j
\]
Prove that $X$ is isomorphic to $\dirlim{I}{\gX}$.\\

\problem{10} With the set up of Problem 9 above, assume that $I$ is {\em right
directed} and each $X_i$ has a structure of a group and each $\psi_{ji}$
is a group homomorphism. Prove that $X$ has
a natural structure of a group which makes 
it isomorphic to the direct limit of $\gX$ in the category of groups.
\end{document}
