%! TEX root = CT.tex
% vim: tw=50
% 04/11/2024 09AM

\glsadjdefn{comp}{complete}{\gls{cat}}%
\glsadjdefn{compness}{completeness}{\gls{cat}}%
We say a \gls{cat} $\mathcal{C}$ is
\emph{complete} if it has all \gls{small}
\glspl{lim}.

\begin{corollary}[]
  % Corollary 4.6
  In each of the statements of \cref{prop:4.4}, we
  may replace `$\mathcal{C}$ has' by either
  `$\mathcal{D}$ has and $G : \mathcal{D} \to
  \mathcal{C}$ \gls{preses}' or `$\mathcal{C}$ has
  and $\mathcal{D} \to \mathcal{C}$ \gls{creates}'.
\end{corollary}

\begin{proof}
  Exercise.
\end{proof}

\begin{example}
  \label{eg:4.7}
  % Example 4.7
  \phantom{}
  \begin{enumerate}[(a)]
    \item The \gls{func} $\Gp \to \Set$
      \gls{creates}
      all \gls{small} \glspl{lim}: given a family
      of groups $\{G_i \mid i \in I\}$, there's a
      unique structure on $\prod_{i \in I} G_i$
      making the projections into homomorphisms,
      and it's a \gls{prod} in $\Gp$. Similarly
      for \glspl{equaler}. But $\Gp \to
      \Set$ doesn't \gls{prese} or \gls{reflect}
      \glspl{coprod}.
    \item The forgetful \gls{func} $\Top \to \Set$
      \gls{preses} \gls{small} \glspl{lim} and
      \glspl{colim}, but doesn't \gls{reflect} them.
    \item The inclusion $\AbGp \to \Gp$
      \gls{reflects} \glspl{coprod}, but doesn't
      \gls{prese} them.

      A \gls{coprod} $A * B$ in $\Gp$ is
      nonabelian if both $A$ and $B$ are
      nontrivial. So the only \glspl{cone} in
      $\AbGp$ thot could map to \gls{coprod}
      \glspl{cone} in $\Gp$ are those where either
      $A$ or $B$ is trivial. But if $A = \{1\}$
      then $A \times B \cong B$ in either
      \gls{cat}.
    \item If $\mathcal{D}$ is a \gls{reflsub}
      of $\mathcal{C}$, the inclusion
      $\mathcal{D} \to \mathcal{C}$ \gls{creates}
      any \glspl{lim} which exist.

      Given $D : J \to \mathcal{D}$ and a
      \gls{lim} \gls{cone} $(L, (x_j \mid j \in
      \ob J))$ for it in $\mathcal{C}$, the
      morphisms $FL \stackrel{F x_j}{\to} FD(j)
      \stackrel{\eta_{D(j)}^{-1}}{\to} D(j)$
      (where $F$ is the \gls{ladj}, and $\eta$ is
      the \gls{unit}) form a \gls{cone} over $D$,
      so they induce a unique $u : FL \to L$. Now
      $u\eta_L : L \to L$ is $\identity{L}$ since
      it's a factorisation of the \gls{lim}
      through itself. So $\eta_L u \eta_L =
      \eta_L$, i.e. $\eta_L u$ is a factorisation
      of $\eta_L$ through itself, so $\eta_L u =
      \identity{FL}$. So the
      $\eta_{D(j)}^{-1}(F\lambda_j)$ form a
      \gls{lim} \gls{cone} in $\mathcal{C}$, and
      hence in $\mathcal{D}$.
    \item If $\mathcal{D}$ has \glspl{lim} of
      shape $J$, so does $\funccat[\mathcal{C},
      \mathcal{D}]$ for any $\mathcal{C}$, and the
      forgetful \gls{func} $\funccat[\mathcal{C},
      \mathcal{D}] \to \mathcal{D}^{\ob
      \mathcal{C}}$ \gls{creates} them (strictly).

      Given $D : J \to \funccat[\mathcal{C},
      \mathcal{D}]$, we can regard it as a
      \gls{func} $J \times \mathcal{C} \to
      \mathcal{D}$. For each $A \in \ob
      \mathcal{C}$, $D(\bullet, A)$ is a
      \gls{diag} of shape $J$ in $\mathcal{D}$, so
      has a \gls{lim} $(LA, (\lambda_{j, A} : LA
      \to D(j, A) \mid j \in \ob J))$. Given $f :
      A \to B$ in $\mathcal{C}$, the composites
      $LA \stackrel{\lambda_{j, A}}{\to} D(j, A)
      \stackrel{D(j, f)}{\to} D(j, B)$ form a
      \gls{cone} over $D(\bullet, B)$, so induce a
      unique $Lf : LA \to LB$.
      \glsref[func]{Functoriality} of $L$ follows
      fro uniqueness, and this is the unique way
      of making $L$ into a \gls{func} which lifts
      the $\lambda_{j, \bullet}$ to a \gls{cone} in
      $\funccat[\mathcal{C}, \mathcal{D}]$.

      The fact that it's a \gls{lim} \gls{cone} is
      straightforward.
  \end{enumerate}
\end{example}

\begin{remark}
  \label{remark:4.8}
  % Remark 4.8
  In any \gls{cat}, $A \stackrel{f}{\to} B$ is
  \gls{monic} if and only if
  \begin{picmath}
    \begin{tikzcd}
      A \ar[r, "\identity{A}"]
      \ar[d, "\identity{A}"]
      & A \ar[d, "f"] \\
      A \ar[r, "f"]
      & B
    \end{tikzcd}
  \end{picmath}
  is a \gls{pullb}. Hence, if $\mathcal{D}$ has
  \glspl{pullb}, then any \gls{mono} in
  $\funccat[\mathcal{C}, \mathcal{D}]$ is
  \gls{pws} \gls{monic}, since its \gls{pullb}
  along itself is contsructed \gls{pws}.
\end{remark}

\begin{fclemma}[]
  \label{lemma:4.9}
  % Lemma 4.9
  Assuming:
  - $G : \mathcal{D} \to \mathcal{C}$ has a
  \gls{ladj}
  Then:
  $G$ \gls{preses} all \glspl{lim} which exist in
  $\mathcal{D}$.
\end{fclemma}

\begin{proof}[Proof 1]
  Suppose $(F \adjoint G)$, and suppose
  $\mathcal{C}$ and $\mathcal{D}$ have \glspl{lim}
  of shape $J$. Then the diagram
  \begin{picmath}
    \begin{tikzcd}
      \mathcal{C} \ar[r, "F"]
      \ar[d, "\Diag"]
      & \mathcal{D} \ar[d, "\Diag"] \\
      \funccat[J, "\mathcal{C}"] \ar[r, "{[J, F]}"]
      & \funccat[J, "\mathcal{D}"]
    \end{tikzcd}
  \end{picmath}
  commutes, and all the \glspl{func} in it have
  \glspl{radj}, so
  \begin{picmath}
    \begin{tikzcd}
      \funccat[J, "\mathcal{D}"]
      \ar[r, "{[J, G]}"]
      \ar[d, "\lim_J"]
      & \funccat[J, "\mathcal{C}"] \\
      \mathcal{D} \ar[r, "G"]
      & \mathcal{C}
    \end{tikzcd}
  \end{picmath}
  commutes up to isomorphism by \cref{coro:3.6}.
\end{proof}

\begin{proof}[Proof 2]
  Suppose given $D : J \to \mathcal{D}$ and a
  \gls{lim} \gls{cone} $(L, (\lambda_j \mid j \in
  \ob J))$ over it. Give a \gls{cone} $(A, (\mu_j
  : A \to GD(j)))$ over $GD$, the transposes
  $\ol{\mu_j} : FA \to D(j)$ form a \gls{cone}
  over $D$ by \gls{naty} of the \gls{adjunc}, so
  induce a unique $\ol{\mu} : FA \to L$ such that
  $\lambda_j \ol{\mu} = \ol{\mu_j}$ for all $j$.

  Then $\mu : A \to GL$ is the unique morphism
  satisfying $(G \lambda_j) \mu = \mu_j$ for all
  $j$.
\end{proof}

\begin{fclemma}[]
  \label{lemma:4.10}
  % Lemma 4.10
  Assuming:
  - $J$ a \gls{diag} shape
  - $\mathcal{D}$ has all \glspl{lim} of shape $J$
  - $G : \mathcal{D} \to \mathcal{C}$ \gls{preses}
  all \glspl{lim} of shape $J$
  Then:
  for each $A \in \ob \mathcal{C}$, $(A \darr G)$
  has \glspl{lim} of shape $J$ and the forgetful
  \gls{func} $(A \darr G) \stackrel{U}{\to}
  \mathcal{D}$ creates them.
\end{fclemma}

\begin{proof}
  Suppose given $D : J \to (A \darr G)$; write
  $D(j) = (UD(j), f_j : A \to GUD(j))$ and let
  $(L, (\lambda_j \mid j \in \ob J))$ be a
  \gls{lim} for $UD$. Since the \glspl{edge} of $D$ are
  morphisms in $(A \darr G)$, the $f_j$ form a
  \gls{cone} over $GUD$, so there's a unique $f :
  A \to GL$ satisfying $(G\lambda_j)f = f_j$ for
  all $j$.

  So $(L, f)$ is the unique lifting of $L$ to an
  object of $(A \darr G)$ which makes the
  $\lambda_j$ into morphisms $(L, f) \to (UD(j),
  f_j)$ in $(A \darr G)$. The fact that these
  morphisms form a \gls{lim} \gls{cone} is
  straightforward.
\end{proof}