%ch1.tex


\chapter{Basics of Extension and Lifting Problems}
\begin{center}
{\small\em To boldly go where no map has gone before}
\end{center}

\section{Existence problems}
We begin with some metamathematics.
All problems about the existence of maps can be cast into one of the
following two forms, which are in a sense mutually dual.

\noindent
{\bf The Extension Problem}\index{extension problem} \    %%% NB index entry tag
Given an inclusion $ A \stackrel{i}{\hookrightarrow} X $, and a map
$ A \stackrel{f}{\rightarrow} Y $,
does there exist a map $f^{\dagger}:X\to Y$ such that
$f^{\dagger}$ agrees with $f$ on $A$?

Here the appropriate source category for maps should be clear from the
context and, moreover, commutativity through a
candidate $f^{\dagger}$ is precisely
the restriction requirement; that is,
\begin{equation}\label{fdag}
f^{\dagger}   :  f^{\dagger}\circ i = f^{\dagger}|_A = f\,. 
\end{equation}
If such an $f^{\dagger}$\footnote{${}^{\dagger}$ suggests striving
for perfection, crusading} 
exists as specified in (\ref{fdag}), then it is called an 
{\bf extension}\index{extension!of a map|bi} of $f$ and is said to 
{\bf extend}\index{extend|bi} $f$. In any diagrams, the presence of
a dotted arrow or an arrow carrying a ? indicates a pious hope, in no way
begging the question of its existence. Note that we shall usually
omit $\circ$ from composite maps.

\noindent
{\bf The Lifting Problem}\index{lifting problem} \
Given a pair of maps $E \stackrel{p}{\rightarrow}B$ and $X \stackrel{f}
{\rightarrow} B $,
does there exist a map 
\begin{equation}\label{fcirc}
f^{\circ} : X \to E, {\rm with} \ pf^{\circ} = f ?
\end{equation}

That {\em all\/} existence problems about maps are essentially of one
type or
the other from these two is seen as follows. Evidently, all existence problems
are representable by triangular diagrams\index{triangular diagrams} and it
is easily seen that there are only these six possibilities:
\begin{center}\begin{picture}(300,70)  %augch2 75
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\put(200,55){\vector(0,-1){30}}
\put(250,55){\vector(0,-1){30}}
\put(28,33){\small ?}
\put(78,33){\small ?}
\put(128,33){\small ?}
\put(178,33){\small ?}
\put(228,33){\small ?}
\put(278,33){\small ?}
\put(10,3){\bf 1}
\put(60,3){\bf 2}
\put(110,3){\bf 3}
\put(160,3){\bf 4}
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\put(260,3){\bf 6}
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\end{picture}\end{center}



\begin{figure}
\begin{picture}(300,220)(0,0)
\put(-20,-20){\resizebox{20 cm}{!}{\includegraphics{3dpdf}}}
\put(260,-10){\resizebox{15 cm}{!}{\includegraphics{contpdf}}}
\put(220,80){$\beta$}
\put(400,-10){$N$}
\put(260,170){$\beta$}
\put(90,15){$N$}
\end{picture}
\caption{{\em The log-gamma family of densities with central mean
$<N> \, = \frac{1}{2}$ as a surface and as a contour plot. }}
\label{pdf}
\end{figure}

\newpage
