\documentclass{article}
\input{packages}
\input{def}

\lecture{Adjunctions}

\begin{document}
\maketitle

\begin{definition}[Adjunction for a 2-Category $\cat{C}$]
A tuple $\langle \ob{C}, \ob{D}, \mo{f}, \mo{g}, \eta, \varepsilon, \prf{f}, \prf{g} \rangle$ whose components have the following types:
\begin{framed}
\begin{description}
\item[$\ob{C}$] is an object of $\cat{C}$
\item[$\ob{D}$] is an object of $\cat{C}$
\item[$\mo{f}$] is a morphism of $\cat{C}$ from $\ob{C}$ to $\ob{D}$
\item[$\mo{g}$] is a morphism of $\cat{C}$ from $\ob{D}$ to $\ob{C}$
\item[$\eta$] is a 2-cell of $\cat{C}$ from $\ob{C}$ to $\mo{f} \cocomp \mo{g}$
\item[$\varepsilon$] is a 2-cell of $\cat{C}$ from $\mo{g} \cocomp \mo{f}$ to $\ob{D}$
\item[$\prf{f}$] is a proof that
\begin{tikzpicture}[baseline=(f2.base)]
\node(f1) at (0,0) {$\mo{f}$};
\node(unit)[circle,draw] at (-.5,1) {$\eta$};
\node(counit)[circle,draw] at (.5,2) {$\varepsilon$};
\node(f2) at (0,3) {$\mo{f}$};
\draw[->] (f1) -- ++(0,.5) .. controls +(0,.5) and ($(counit.south east) + (.5,-.5)$) .. (counit.south east);
\draw[->] (unit) -- node[above]{$\mo{g}$} (counit);
\draw[<-] (f2) -- ++(0,-.5) .. controls +(0,-.5) and ($(unit.north west) + (-.5,.5)$) .. (unit.north west);
\node at (-1,2.5) {$\ob{C}$};
\node at (1,.5) {$\ob{D}$};
\end{tikzpicture}
equals
\begin{tikzpicture}[baseline=(f2.base)]
\node(f1) at (0,0) {$\mo{f}$};
\node(f2) at (0,3) {$\mo{f}$};
\draw[->] (f1) -- (f2);
\node at (-.5,1.5) {$\ob{C}$};
\node at (.5,1.5) {$\ob{D}$};
\end{tikzpicture}
. In other words,
\begin{tikzpicture}[baseline=(c2.base)]
\node(c1) at (0,-1) {$\ob{C}$};
\node(c2) at (-1,1) {$\ob{C}$};
\node(d1) at (1,-1) {$\ob{D}$};
\node(d2) at (0,1) {$\ob{D}$};
\draw (c1) to[bend left=30] node(c){} (c2);
\draw (d1) to[bend right=30] node(d){} (d2);
\draw[->] (c1) -- node[below]{$\mo{f}$} (d1);
\draw[->] (c2) -- node[above]{$\mo{f}$} (d2);
\draw[->] (d2) .. controls (0,.5) and (-.5,.5) .. (0,0) node[below]{$\mo{g}$} .. controls (.5,-.5) and (0,-.5) .. (c1);
\draw[double,double equal sign distance,-implies,shorten >=3mm,shorten <=2mm] (c) -- node[above left]{$\eta$} (d2);
\draw[double,double equal sign distance,implies-,shorten >=3mm,shorten <=2mm] (d) -- node[below right]{$\varepsilon$} (c1);
\end{tikzpicture}
equals
\begin{tikzpicture}[baseline=(c2.base)]
\node(c1) at (0,-1) {$\ob{C}$};
\node(c2) at (-1,1) {$\ob{C}$};
\node(d1) at (1,-1) {$\ob{D}$};
\node(d2) at (0,1) {$\ob{D}$};
\draw (c1) to[bend left=30] node(c){} (c2);
\draw (d1) to[bend right=30] node(d){} (d2);
\draw[->] (c1) -- node[below]{$\mo{f}$} node(f1){} (d1);
\draw[->] (c2) -- node[above]{$\mo{f}$} node(f2){} (d2);
\draw[double,double equal sign distance,shorten >=2mm,shorten <=2mm] (f1) -- (f2);
\end{tikzpicture}
.
\item[$\prf{g}$] is a proof that
\begin{tikzpicture}[baseline=(g2.base)]
\node(g1) at (0,0) {$\mo{g}$};
\node(unit)[circle,draw] at (.5,1) {$\eta$};
\node(counit)[circle,draw] at (-.5,2) {$\varepsilon$};
\node(g2) at (0,3) {$\mo{g}$};
\draw[->] (g1) -- ++(0,.5) .. controls +(0,.5) and ($(counit.south west) + (-.5,-.5)$) .. (counit.south west);
\draw[->] (unit) -- node[above]{$\mo{f}$} (counit);
\draw[<-] (g2) -- ++(0,-.5) .. controls +(0,-.5) and ($(unit.north east) + (.5,.5)$) .. (unit.north east);
\node at (-1,2.5) {$\ob{D}$};
\node at (1,.5) {$\ob{C}$};
\end{tikzpicture}
equals
\begin{tikzpicture}[baseline=(g2.base)]
\node(g1) at (0,0) {$\mo{g}$};
\node(g2) at (0,3) {$\mo{g}$};
\draw[->] (g1) -- (g2);
\node at (-.5,1.5) {$\ob{D}$};
\node at (.5,1.5) {$\ob{C}$};
\end{tikzpicture}
. In other words,
\begin{tikzpicture}[baseline=(c2.base)]
\node(c1) at (0,-1) {$\ob{C}$};
\node(c2) at (1,1) {$\ob{C}$};
\node(d1) at (-1,-1) {$\ob{D}$};
\node(d2) at (0,1) {$\ob{D}$};
\draw (c1) to[bend right=30] node(c){} (c2);
\draw (d1) to[bend left=30] node(d){} (d2);
\draw[->] (d1) -- node[below]{$\mo{g}$} (c1);
\draw[->] (d2) -- node[above]{$\mo{g}$} (c2);
\draw[<-] (d2) .. controls (0,.5) and (.5,.5) .. (0,0) node[above]{$\mo{f}$} .. controls (-.5,-.5) and (0,-.5) .. (c1);
\draw[double,double equal sign distance,-implies,shorten >=3mm,shorten <=2mm] (c) -- node[above right]{$\eta$} (d2);
\draw[double,double equal sign distance,implies-,shorten >=3mm,shorten <=2mm] (d) -- node[below left]{$\varepsilon$} (c1);
\end{tikzpicture}
equals
\begin{tikzpicture}[baseline=(c2.base)]
\node(c1) at (0,-1) {$\ob{C}$};
\node(c2) at (1,1) {$\ob{C}$};
\node(d1) at (-1,-1) {$\ob{D}$};
\node(d2) at (0,1) {$\ob{D}$};
\draw (c1) to[bend right=30] node(c){} (c2);
\draw (d1) to[bend left=30] node(d){} (d2);
\draw[->] (d1) -- node[below]{$\mo{g}$} node(g1){} (c1);
\draw[->] (d2) -- node[above]{$\mo{g}$} node(g2){} (c2);
\draw[double,double equal sign distance,shorten >=2mm,shorten <=2mm] (g1) -- (g2);
\end{tikzpicture}
.
\end{description}
\end{framed}
\end{definition}

\begin{definition}[Left/Right Adjoint]
$\mo{f}$ above is called the left adjoint, and $\mo{g}$ above is called the right adjoint.
A morphism of a 2-category is a left/right adjoint if it is the left/right adjoint of some adjunction.
\end{definition}

\begin{exercise}
Prove that there is a bijection between adjunctions in $\cat{Cat}$ and adjunctions via transpositions.
\end{exercise}

\begin{example}
Consider $\cat{Prost}$.
Suppose we had a pair of preordered sets $\langle C, \leq \rangle$ and $\langle D, \leq \rangle$, and we want to make an adjunction out of some relation-preserving functions $f : C \to D$ and $g : D \to C$.
Then $\eta$ exists if and only if $\forall c : C.\; c \leq g(f(c))$, and $\beta$ exists if and only if $\forall d : D.\; f(g(d)) \leq g$.
If $\eta$ and $\beta$ exist, then $\prf{f}$ and $\prf{g}$ are trivial since $\cat{Prost}$ is a \emph{locally thin} 2-category.
Such a situation is called a monotone Galois connection.
\end{example}

\end{document}