\documentclass[12pt]{amsart}
\title{Lecture 6}
\author{}

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\newtheorem*{ex}{Example}

\begin{document}
\maketitle

\PP{6.0}{Overview}
%----------------
Recall that last time we saw the examples of direct sums and products,
defined as solutions to a representability problem. \\

Given a category $\CalC$, an indexing set $I$ and a collection
of objects $\mathfrak{X}=\{X_i\}_{i\in I}$, their direct sum and
product are defined via the data of the following natural
isomorphisms:

\[
\begin{split}
\Hom_{\CalC}\left(\bigoplus_{i\in I}^{\CalC} X_i, -\right)
\Rightarrow \prod_{i\in I} \Hom_{\CalC}(X_i,-), \\
\Hom_{\CalC}\left(-,\prod_{i\in I}^{\CalC} X_i\right)
\Rightarrow \prod_{i\in I} \Hom_{\CalC}(-,X_i).
\end{split}
\]

Today we will discuss a variant of this theme, called 
{\em direct and inverse limits}. In mathematical literature,
these are also called {\em injective and projective limits}
or {\em colimits and limits}. In this course, we will stick
with direct and inverse limits.\\

\PP{6.1}{Direct and Inverse family of objects}
%--------------------------------------------
Let us assume that we are given a partially ordered set
$(I,\leq)$. Recall that this means: $I$ is a set, $\leq$
is a relation on $I$ (a relation is a subset of $I\times I$,
we write $i\leq j$ if $(i,j)$ is in the subset). This
relation is required to satisfy the following properties:\\

\begin{enumerate}
\item $i\leq i$ for every $i\in I$.
\item $i\leq j\leq k$ implies $i\leq k$, for every $i,j,k\in I$.
\item $i\leq j$ and $j\leq i$ implies $i=j$.
\end{enumerate}

In many textbooks, an additional axiom is imposed on this poset
(short for partially ordered set).\\

The poset $(I,\leq)$ is {\em right directed} (or just {\em directed})
if for every $i,j\in I$, there exists a $k\in I$ such that
$i\leq k$ and $j\leq k$.\\

I will not assume this property, by default, and will explicitly
mention that we are talking about a right directed poset, if needed.\\

\begin{defn}
A {\em direct system of objects} with respect to $(I,\leq)$,
valued in a category $\CalC$, consists of 
\begin{itemize}
\item a collection of objects $\{X_i\}_{i\in I}$ of $\CalC$, and
\item for every $i\leq j$, a morphism $\psi_{ji}:X_i\to X_j$,
\end{itemize}
such that (i) $\psi_{ii}=\Id_{X_i}$, and (ii) for every $i\leq j\leq k$ we have
$\psi_{kj}\circ \psi_{ji} = \psi_{ki}$.
\end{defn}

Similarly, an {\em inverse system of objects}
with respect to $(I,\leq)$, valued in a category $\CalC$, consists of 
\begin{itemize}
\item a collection of objects $\{X_i\}_{i\in I}$ of $\CalC$, and
\item for every $i\leq j$, a morphism $\varphi_{ij}:X_j\to X_i$,
\end{itemize}
such that (i) $\varphi_{ii}=\Id_{X_i}$ and
(ii) for every $i\leq j\leq k$ we have
$\varphi_{ij}\circ \varphi_{jk} = \varphi_{ik}$.\\

\PP{6.2}{Direct and inverse systems as functor}
%---------------------------------------------
A different way to think about direct and inverse systems of objects
in $\CalC$, is to define a category $\mathbf{I}$, associated
to the poset $(I,\leq)$ as follows. Its objects are elements of
$I$, and for every $i,j\in I$,
\[
\Hom_{\mathbf{I}}(i,j) = \left\{
\begin{array}{ll}
\text{Singleton } i\rightarrow j & \text{if } i\leq j,\\
\emptyset & \text{otherwise}.
\end{array}
\right.
\]

Composition of morphisms is uniquely given by the partial order,
and $\Id_i = i\rightarrow i$ is the only element in $\Hom_{\mathbf{I}}(i,i)$.\\

Now, we can easily see that
\begin{itemize}
\item Objects of the category of covariant functors $\Cov{\mathbf{I}}{\CalC}$
are {\em direct systems}.
\item Objects of the category of contravariant functors 
$\Contra{\mathbf{I}}{\CalC}$ are {\em inverse systems}.
\end{itemize}

The advantage of this viewpoint is that it makes the notion of
{\em morphisms between direct/inverse systems} obvious (as morphisms
in the respective category of functors).\\

\PP{6.3}{Direct limit}
%---------------------------------
Let $\mathfrak{X} = \{(X_i)_{i\in I}, (\psi_{ji}:X_i\to X_j)_{i\leq j}\}$ 
be a direct system of objects in a category
$\CalC$, with respect to a poset $(I,\leq)$.
Consider the following (covariant) functor:
\[
\h_{\mathfrak{X}} : \CalC \to \Sets,
\]
given on objects by:
\[
\h_{\mathfrak{X}}(Y) = \{(X_i\stackrel{f_i}{\longrightarrow} Y)_{i\in I} : 
\text{ for every } i\leq j,\ f_j \circ \psi_{ji} = f_i\}
\subset \prod_{i\in I} \Hom_{\CalC}(X_i,Y).
\]

\begin{defn}
If the above functor $\h_{\mathfrak{X}}$ is representable, the object
representing it is called {\em the direct limit of the direct
system} $\mathfrak{X}$. It is denoted by
$\Dlim{(I,\leq)} X_i$, or just $\ds\lim_{\longrightarrow} X_i$,
if the poset is clear from the context.
\end{defn}

To spell out the universal property of direct limits, we use the
same ideas as in \S 4.6. Namely, $X=\Dlim{(I,\leq)} X_i$ means
the following:
\begin{itemize}
\item We are given morphisms $\iota_j : X_j\to X$ for every $j\in I$
such that $\iota_k\circ \psi_{kj} = \iota_j$ for every $j\leq k$.
\item For every $Y\in \CalC$ and morphisms $g_j:X_j\to Y$
satisfying $g_k\circ \psi_{kj}=g_j$ for every $j\leq k$,
there is a unique morphism $g:X\to Y$ such that
$g\circ \iota_j = g_j$ for every $j\in I$.
\end{itemize}
\ \\

\PP{6.4}{Example of direct limits in $\Ab$}
%------------------------------------------
In practice, direct limit is obtained as a ``quotient" of direct
sum. Let us consider the example of the category of abelian
groups $\Ab$. The following lemma holds with the same proof
in the category $\Rmod{R}$.\\

\begin{lem}
Direct limits exist in the category of abelian groups.
\end{lem}
\begin{pf}
Let $(I,\leq)$ be an arbitrary poset and let $\mathfrak{X}$
be a direct system of abelian groups: $\{G_i\}_{i\in I}$
together with morphisms $\psi_{ji}:G_i\to G_j$, for every
$i\leq j$. As before, we assume that (i) $\psi_{ii}
=\Id_{G_i}$ and (ii) for every $i\leq j\leq k$, we have
$\psi_{kj}\circ\psi_{ji} = \psi_{ki}$. We begin by
constructing an abelian group $G$, together with canonical
morphisms $\iota_j : G_j\to G$. We will then show that
it satisfies the universal property of the direct limit.\\

Consider $\ds \wt{G}=\bigoplus_{i\in I} G_i$. Recall that
$\wt{G}$ comes together with canonical morphisms
$\wt{\iota}_j : G_j\to \wt{G}$, given by $\ds
\left(\wt{\iota}_j(x)\right)_k = \delta_{jk}x$.\\

Let $N$ be the subgroup of $\wt{G}$ generated by the following
set of elements:
\[
N = \left\langle \wt{\iota}_j(x) - \wt{\iota}_k(\psi_{kj}(x)) : \forall\ 
j\leq k\right\rangle.
\]
Define: $G = \wt{G}/N$, let $\pi:\wt{G}\to G$ be the natural
projection map. For each $j\in I$, we have canonical morphisms
$\iota_j : G_j \to G$ given as the composition
$\iota_j = \pi \circ \wt{\iota}_j$.\\

\noindent First thing we have to check is: for every $j\leq k$:
$\iota_k\circ \psi_{kj} = \iota_j$. This is almost by definition
of $N$, since for every $x\in G_j$:
\[
\wt{\iota}_k(\psi_{kj}(x)) - \wt{\iota}_j(x)\in N \Rightarrow
\pi(\wt{\iota}_k(\psi_{kj}(x)) - \wt{\iota}_j(x)) = 0.
\]
Hence $\iota_k(\psi_{kj}(x)) = \iota_j(x)$.\\

Now assume that we are given an abelian group $H$, together with
morphisms $f_j : G_j \to H$ satisfying the commutativity relations:
$f_k\circ \psi_{kj} = f_j$ for every $j\leq k$. Forgetting these
relations for a moment, by the universal property of direct sum,
we get a unique morphism $\wt{f} : \wt{G} \to H$ such that,
for every $j\in I$, $\wt{f}\circ \wt{\iota}_j = f_j$. Now, all
we have to show is that this morphism $\wt{f}$ factors through
$G=\wt{G}/N$:
\[
\xy
(0,20)*{\wt{G}}="a"; (30,20)*{H}="b";
(0,0)*{G}="c";
{\ar@{->>}_{\pi} "a"; "c"};
{\ar^{\wt{f}} "a"; "b"};
{\ar@{-->} "c"; "b"};
\endxy
\]
That is, $N\subset \Ker(\wt{f})$. To see this, take a typical
generator $\wt{\iota}_j(x) - \wt{\iota}_k(\psi_{kj}(x))$ of $N$
(here $j\leq k$). We have:
\[
\wt{f}\left(\wt{\iota}_j(x) - \wt{\iota}_k(\psi_{kj}(x))\right)
= f_j(x) - f_k(\psi_{kj}(x)) = 0,
\]
since $\wt{f}\circ \wt{\iota}_j = f_j$ and $f_j = f_k\circ\psi_{kj}$.
Also, for every $j\in I$: $f\circ \iota_j = f\circ \pi \circ \wt{\iota}_j
= \wt{f}\circ\wt{\iota}_j = f_j$, as we wanted.\\

The uniqueness of $f$ is a direct consequence of that of $\wt{f}$,
and is left as a very easy exercise.
\end{pf}
\ \\

\PP{6.5}{Example: germs of holomorphic functions}
%---------------------------------------------------------
In the theory of functions of a complex variable, one encounters
the following definition.\\ 

\begin{defn}
Let $z\in\C$. {\em Germs of holomorphic functions near} $z$, denoted by
$\mathcal{O}_z$, is the set of equivalence classes of pairs
$(f,U)$, where $U$ is an open set containing $z$, and $f:U\to\C$
is a holomorphic function, under the following equivalence
relation:
\[
(f,U)\sim (g,V) \text{ if there exists an open set } W\subset U\cap V
\text{ such that } f|_W \equiv g|_W\ .
\]
\end{defn}

$\mathcal{O}_z$ is in fact more than just a set, it is a commutative
algebra over $\C$. It is even easy to see that it is {\em local}
{\footnote{A commutative ring $A$ is said to be local, if it has
a unique maximal ideal, or, equivalently, the set of all non--invertible
elements of $A$ form an ideal.}},
with the unique maximal ideal given by:
\[
\mathfrak{m}_z = \{[(f,U)] : f(z)=0\}.
\]

This example is generalized to what is known as {\em stalks of a sheaf
near a point} (see the next section). We can view 
the ring of germs of holomorphic functions near $z\in\C$,
as a direct limit of an appropriate
direct system of commutative algebras over $\C$.\\

Consider a poset $I^z_{\C}$
whose elements are open sets in $\C$ which contain $z$, and the partial order
is given by reverse inclusion:
\[
U\leq V \text{ means } V\subset U.
\]
For $U\in I^z_{\C}$, let $\mathcal{O}(U)$ be the ring of all holomorphic functions
$f:U\to \C$. For every $U\leq V$, the corresponding ring homomorphism
$\rho^U_V : \mathcal{O}(U)\to \mathcal{O}(V)$ is merely restriction
of functions from the domain $U$ to $V$.\\

It is easy to see that $\{(\mathcal{O}(U))_{U\in I^z_{\C}},(\rho^U_V)_{U\leq U}\}$
is a direct system of commutative algebras over $\C$.
I leave it for you to convince yourself that:
\[
\mathcal{O}_z = \Dlim{(I^z_{\C},\leq)} \mathcal{O}(U)
\text{ often written as } \Dlim{z\in U} \mathcal{O}(U).
\]
\ \\

\PP{6.6}{Example: presheaves}
{\footnote{First defined by French mathematician Jean Leray (1906-1998)
during his time as a prisnor of war (1940-1945) in Edelbach, Austria.}}
%----------------------------------
The example of the previous section is generalized to arbitrary
topological spaces, as follows. Let $X$ be a topological space.
A {\em presheaf of commutative rings} $\mathcal{F}$ on $X$ is the data of:
\begin{itemize}
\item a commutative ring $\mathcal{F}(U)$ for every open set $U\subset X$, and
\item a unital ring homomorphism, called restriction map,
$\rho^U_V : \mathcal{F}(U)\to\mathcal{F}(V)$, for every $V\subset U$.
\end{itemize}
This data is subject to the following axioms:
(i) $\rho^U_U = \Id_{\mathcal{F}(U)}$, and (ii) for every
$W\subset V\subset U$, we have $\rho^V_W\circ \rho^U_V = \rho^U_W$.\\

Thus, it is clear that a presheaf is nothing but a direct system,
valued in an appropriate category (in the definition above, $\mathbf{CommRings}$)
with respect to the following poset $I_X$.
The elements of $I_X$ are open sets in $X$, and the partial
order is the reverse inclusion, as in the previous paragraph.\\

Analogous to the ring of germs of holomorphic functions, we
can consider a point $x\in X$, and restrict our poset $I_X^x$ to
consist of only those open sets which contain $x$. The direct limit
of the direct system $\{(\mathcal{F}(U))_{U\in I_X^x},
(\rho^U_V)_{U\leq V}\}$ is called {\em stalks of the presheaf}
$\mathcal{F}$ at $x\in X$:

\[
\mathcal{F}_x := \Dlim{x\in U} \mathcal{F}(U).
\]
\ \\

\PP{6.7}{Inverse limit}
%---------------------------------
Now let $\mathfrak{X} = \{(X_i)_{i\in I}, (\varphi_{ij}:X_j\to X_i)_{i\leq j}\}$ 
be an inverse system of objects in a category
$\CalC$, with respect to a poset $(I,\leq)$.
Consider the following (contravariant) functor:
\[
\h^{\mathfrak{X}} : \CalC \to \Sets,
\]
given on objects by:
\[
\h^{\mathfrak{X}}(Y) = \{(Y\stackrel{f_i}{\longrightarrow} X_i)_{i\in I} : 
\text{ for every } i\leq j,\ \varphi_{ij}\circ f_j = f_i\}
\subset \prod_{i\in I} \Hom_{\CalC}(Y,X_i).
\]

\begin{defn}
If the above functor $\h^{\mathfrak{X}}$ is representable, the object
representing it is called {\em the inverse limit of the inverse
system} $\mathfrak{X}$. It is denoted by
$\Ilim{(I,\leq)} X_i$, or just $\ds\lim_{\longleftarrow} X_i$,
if the poset is clear from the context.
\end{defn}

Again, $X=\Ilim{(I,\leq)} X_i$ means
the following:
\begin{itemize}
\item We are given morphisms $p_j : X\to X_j$ for every $j\in I$
such that $\varphi_{jk}\circ p_k = p_j$ for every $j\leq k$.
\item For every $Y\in \CalC$ and morphisms $f_j:Y\to X_j$
satisfying $\varphi_{jk}\circ f_k=f_j$ for every $j\leq k$,
there is a unique morphism $f:Y\to X$ such that
$p_j\circ f = f_j$ for every $j\in I$.
\end{itemize}

\end{document}
