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\section{Adjunctions}

\begin{fcdefn}[Adjunction, D. Kan 1958]
  \glssymboldefn{adjoint}%
  \glsnoundefn{adj}{adjoint}{adjoints}%
  \glsnoundefn{ladj}{left adjoint}{left adjoints}%
  \glsnoundefn{radj}{right adjoint}{right adjoints}%
  \glsnoundefn{adjunc}{adjunction}{adjunctions}%
  % Definition 3.1
  Let $\mathcal{C}$ and $\mathcal{D}$ be
  \glspl{cat}. An \emph{adjunction} between
  $\mathcal{C}$ and $\mathcal{D}$ consists of
  \glspl{func} $F : \mathcal{C} \to \mathcal{D}$
  and $G : \mathcal{D} \to \mathcal{C}$ together
  with, for each $A \in \ob \mathcal{C}$ and $B
  \in \ob \mathcal{D}$, a bijection between
  morphisms $FA \to B$ in $\mathcal{D}$ and
  morphisms $A \to GB$ in $\mathcal{C}$, which is
  natural in $A$ and $B$. (If $\mathcal{C}$ and
  $\mathcal{D}$ are \gls{locsm}, this means that
  $\mathcal{D}(F\bullet, \bullet)$ and
  $\mathcal{C}(\bullet, G\bullet)$ are
  \gls{natyi} \glspl{func} $\mathcal{C}^\op \times
  \mathcal{D} \to \Set$.)

  We say $F$ is \emph{left adjoint} to $G$, or $G$
  is \emph{right adjoint} to $F$, and we write $(F
  \dashv G)$.
\end{fcdefn}

\begin{example}
  \label{eg:3.2}
  % Example 3.2
  \phantom{}
  \begin{enumerate}[(a)]
    \item The free \gls{func} $F : \Set \to \Gp$
      is \gls{ladj} to the forgetful \gls{func}
      $\Gp \stackrel{U}{\to} \Set$. By definition,
      homomorphisms $FA \to G$ correspond to
      functions $A \to UG$; \gls{naty} in $A$ was
      built into the definition of $F$ in
      \cref{eg:1.5}(b) and \gls{naty} in $G$ is
      immediate.
    \item The forgetful \gls{func} $U : \Top \to
      \Set$ has a \gls{ladj} $D$, which equips a
      set $A$ with its discrete topology since any
      function $A \to UX$ is continuous as a map
      $DA \to X$. $U$ also has a \gls{radj} $I$
      given by the `ìndiscrete' topology.
    \item The \gls{func} $\ob : \Cat \to \Set$ has
      a \gls{ladj} $D$ given by discrete
      \glspl{cat}, and a \gls{radj} $I$: $IA$ is
      the \gls{cat} with objects $A$ and morphisms
      $a \to b$ for each $(a, b)$. $D$ also has a
      \gls{ladj} $\pi_0$: $\pi_0 \mathcal{C}$ is
      the set of \emph{connected components} of
      $\mathcal{C}$, i.e. the quotient of $\ob
      \mathcal{C}$ by the smallest equivalence
      relation which identifies $\dom f$ with
      $\cod f$ for all $f \in \mor \mathcal{C}$.
    \item Given a set $A$, we can regard
      $(\bullet) \times A$ as a \gls{func} $\Set
      \to \Set$. It has a \gls{radj}, namely
      $\Set(A, \bullet)$. Given $f : B \times A
      \to C$ we can regard it as a function
      $\lambda f : B \to \Set(A, C)$ by $\lambda
      f(b)(a) = f(b, a)$.

      \glsadjdefn{cc}{cartesian
      closed}{\gls{cat}}%
      We call a \gls{cat} $\mathcal{C}$
      \emph{cartesian closed} if it has binary
      \glspl{prod} as defined in \cref{eg:2.6}(f)
      and each $(\bullet) \times A$ has a
      \gls{radj} $(\bullet)^A$. For example,
      $\Cat$ is cartesian clsosed, with
      $\mathcal{D}^{\mathcal{C}}$ taken to be the
      $\funccat[\mathcal{C}, \mathcal{D}]$.
    \item Let $M = \{1, e\}$ be the $2$-element
      monoid with $e^2 = e$ (and identity $1$). We
      have a \gls{func} $F : \Set \to \funccat[M,
      \Set]$ sending $A$ to $(A, \identity{A})$
      and a \gls{func} $G : \funccat[M, \Set] \to
      \Set$ sending $(A, e)$ to $\{a \in A \mid ea
      = a\}$.

      We have $(F \adjoint G \adjoint F)$: $(F
      \adjoint G)$ since any $f : M \to (B, e)$
      takes values in $G(B, e)$ and any $g : (B,
      e) \to FA$ is determined by its restriction
      to $G(B, e)$ since $g(b) = g(e, b)$.
      However, note that this is not an
      \gls{equivc} of \glspl{cat}.
    \item \glsnoundefn{iobj}{initial object}{initial
      objects}%
      \glsnoundefn{tobj}{terminal object}{terminal
      objects}%
      \glsadjdefn{initial}{initial}{object}%
      \glsadjdefn{terminal}{terminal}{object}%
      Let $\mathbf{1}$ be the \gls{cat} with
      one object and one morphism (which must the identity on
      the only object). A \gls{ladj} for the
      unique \gls{func} $\mathcal{C} \to
      \mathbf{1}$ picks out an \emph{initial
      object} of $\mathcal{C}$, i.e. an object
      such that there is a unique $I \to A$ for
      each $A \in \ob \mathcal{C}$. Dually, a
      \gls{radj} for $\mathcal{C} \to \mathbf{1}$
      `is' a \emph{terminal object} of
      $\mathcal{C}$ (a terminal object is an
      initial object in $\mathcal{C}^\op$).

      Again, the example of $\Gp$ shows that these
      two can coincide.
    \item Suppose given $A \stackrel{f}{\to} B$ in
      $\Set$. We have order-preserving mappings
      $Pf : PA \to PB$ and $P^* f : PB \to PA$,
      and $(Pf \adjoint P^* f$ since $A' \subseteq
      f^{-1} B' \iff f(A') \subseteq B'$.
    \item Suppose given a relation $R \subseteq A
      \times B$. We define $(\bullet)^r : PA \to
      PB$ and $(\bullet)^l : PB \to PA$ by
      \begin{align*}
        (S)^r
        &= \{b \in B \mid (\forall a \in S) ((a, b)
        \in R)\} \\
        (T)^l
        &= \{a \in A \mid (\forall b \in T) ((a, b)
        \in R)\}
      \end{align*}
      \glsadjdefn{aor}{adjoint on the
      right}{\gls{func}}%
      These are \gls{cont} \glspl{func} and $S
      \subseteq T^l \iff S \times T \subseteq R
      \iff T \subseteq S^r$. We say $(\bullet)^r$
      and $(\bullet)^l$ are \emph{adjoint on the
      right}.
    \item $P^* : \Set^\op \to \Set$ is
      self-\gls{aor}, since functions $A \to PB$
      and functions $B \to PA$ both correspond to
      relations $R \subseteq A \times B$.
    \item $(\bullet)^* : \Vect_k^* \to \Vect_k$ is
      self-\gls{aor}, since linear maps $V \to
      W^*$ and $W \to V^*$ both correspond to
      bilinear maps $V \times W \to k$.
  \end{enumerate}
\end{example}

\begin{fcthm}[]
  \label{thm:3.3}
  \glssymboldefn{darr}%
  % Theorem 3.3
  Assuming:
  - $G : \mathcal{D} \to \mathcal{C}$ is a
  \gls{func}
  - for $A \in \ob \mathcal{C}$, let $(A
  \downarrow G)$ be the \gls{cat} whose objects
  are pairs $(B, f)$ where $B \in \ob
  \mathcal{D}$ and $f : A \to GB$, and whose
  morphisms $(B, f) \to (B', f')$ are morphisms $g
  : B \to B'$ making
  \begin{picmath}
    \begin{tikzcd}
      A \ar[r, "f"] \ar[rd, "f'", swap]
      & GB \ar[d, "Gg"] \\
      & GB'
    \end{tikzcd}
  \end{picmath}
  commute.
  Then:
  specifying a \gls{ladj} for $G$ is equivalent to
  specifying an \gls{iobj} of $(A \darr G)$ for each
  $A$.
\end{fcthm}