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% 06/11/2024 09AM

Can we represent an \gls{iobj} as a \gls{lim}?

\begin{fclemma}[]
  \label{lemma:4.11}
  % Lemma 4.11
  Assuming:
  - $\mathcal{C}$ a \gls{cat}
  Then:
  specifying an \gls{iobj}
  of $\mathcal{C}$ is equivalent to specifying a
  \gls{lim} for $\identity{\mathcal{C}} : \mathcal{C} \to
  \mathcal{C}$.
\end{fclemma}

\begin{proof}
  First suppose $I$ is \glsref[iobj]{initial}. The
  unique morphisms $I \to A$, $A \in \ob
  \mathcal{C}$, form a \gls{cone} over
  $\identity{\mathcal{C}}$, and it's a \gls{lim}
  \gls{cone} since if $(A, (f_B : A \to B \mid B
  \in \ob \mathcal{C}))$ is any \gls{cone} over
  $\identity{\mathcal{C}}$, then $f_I$ is its
  unique factorisation through the one with apex
  $I$.

  \glsadjdefn{wiobj}{weakly}{initial}%
  Conversely, suppose given a \gls{lim} $(I, (f_A
  : I \to A \mid A \in \ob \mathcal{C}))$ for
  $\identity{\mathcal{C}}$. Then $I$ is weakly
  \glsref[iobj]{initial} (i.e. it admits morphisms
  to every object of $\mathcal{C}$); and if $g : I
  \to A$ then $g f_I = f_A$. In particular, $f_A
  f_I = f_A$ for all $A$, so $f_I$ is a
  factorisation of the \gls{lim} \gls{cone}
  through itself, so $f_I = \identity{I}$ and $I$
  is \glsref[iobj]{initial}.
\end{proof}

The `primitive' Adjoint Functor Theorem follows
from \cref{lemma:4.10}, \cref{lemma:4.11} and
\cref{thm:3.3}. But it only applies to preorders
(see Example Sheet).

\begin{fcthm}[General Adjoint Functor Theorem]
  \label{gaft}
  \glsnoundefn{ssc}{solution-set
  condition}{solution-set conditions}\glsnoundefn{sset}{solution-set}{solution-sets}%
  % Theorem 4.12
  Assuming:
  - $\mathcal{D}$ is \gls{comp} and \gls{locsm}
  Then:
  $G : \mathcal{D} \to \mathcal{C}$ has a
  \gls{ladj} if and only if $G$ preserves
  \gls{small} \glspl{lim} and satisfies the
  \emph{solution-set condition}: for every $A \in
  \ob \mathcal{C}$, there's a set $\{(B_i, f_i)
  \mid i \in I\}$ of objects of $(A \darr G)$
  which is collectively \gls{wiobj}
  \glsref[iobj]{initial}.
\end{fcthm}

\begin{proof}
  \phantom{}
  \begin{enumerate}[$\Rightarrow$]
    \item[$\Rightarrow$] $G$ preserves \glspl{lim}
      by \cref{lemma:4.9}, and $\{(FA, \eta_A)\}$
      is a singleton \gls{sset} for each $A$.
    \item[$\Leftarrow$] By \cref{lemma:4.10}, the
      \glspl{cat} $(A \darr G)$ are \gls{comp},
      and they're \gls{locsm} since $\mathcal{D}$
      is.

      So we need to show: if $\mathcal{A}$ is
      \gls{comp} and \gls{locsm}, and has a
      \gls{wiobj} \glsref[iobj]{initial} set
      $\{A_i \mid i \in I\}$, then it has an
      \gls{iobj}. First form $P = \prod_{i \in I}
      A_i$; then $P$ is \gls{wiobj}
      \glsref[iobj]{initial}. Now form the
      \gls{lim} of the \gls{diag} with
      \glspl{vert} $P$ and $P'$, with the
      morphisms $P \to P'$ being all endomorphisms
      of $P$.

      Writing $I \stackrel{i}{\to} P$ for
      this, $I$ is still \gls{wiobj}
      \glsref[iobj]{initial}. Suppose given $I
      \twofuncs fg B$; let $E \stackrel{e}{\to} I$
      be their \gls{equaler}. There exists some $h
      : P \to E$. Now $ieh : P \to P$, but we also
      have $\identity{P} : P \to P$, so $i =
      \identity{P} i = iehi$. But $i$ is
      \gls{monic}, so we get $ehi = \identity{I}$,
      so $e$ is \gls{splitme} \gls{epic}, and hence $f =
      g$.
      \qedhere
  \end{enumerate}
\end{proof}

\begin{example}
  % Example 4.13
  \phantom{}
  \begin{enumerate}[(a)]
    \item Consider the forgetful \gls{func} $U :
      \Gp \to \Set$. $\Gp$ has and $U$
      \gls{preses} all \gls{small} \glspl{lim} by
      \cref{eg:4.7}(a), and $\Gp$ is \gls{locsm}.
      Given $A$, any $A \stackrel{f}{\to} UG$
      factors through $A \to UG'$ where $G'$ is
      the subgroup generated by $\{f(a) \mid a \in
      A\}$. Also $\card G' \le \max \{\aleph_0,
      \card A\}$. Let $B$ be a set of this
      cardinality: considering all subsets $B'
      \subseteq B$, all group structures on $B'$
      and all functions $A \to B'$, we get a
      \gls{sset} at $A$.
    \item Let $\mathbf{CLat}$ be the \gls{cat} of
      complete lattices (posets with all joins and
      all meets). $U : \mathbf{CLat} \to \Set$
      creates \glspl{lim} just like $U : \Gp \to
      \Set$.

      In 1965, A. Hales showed that there exist
      arbitrarily large complete lattices
      generated by 3 element subsets, so the
      \gls{ssc} fails for $A = \{a, b, c\}$.

      Now also that $\mathbf{CLat}$ doesn't have a
      \gls{coprod} for 3 copies of $\{0, a, 1\}$.
  \end{enumerate}
\end{example}

\begin{fcdefn}[Subobject]
  \glsnoundefn{subobj}{subobject}{subobjects}%
  \glssymboldefn{Sub}%
  \glsadjdefn{wp}{well-powered}{\gls{cat}}%
  \glsadjdefn{wcp}{well-copowered}{\gls{cat}}%
  % Definition 4.14
  By a subobject of $A \in \ob \mathcal{C}$, we
  mean a \gls{mono} $A' \monic A$. We order
  subobjects by $(A' \monic A) \le (A'' \monic A)$
  if there exists
  \begin{picmath}
    \begin{tikzcd}
      A' \ar[rr] \ar[rd, tail]
      &
      & A'' \ar[ld, tail] \\
      & A
    \end{tikzcd}
  \end{picmath}
  We write $\Subinternal_{\mathcal{C}}(A)$ for this
  preorder.

  We say $\mathcal{C}$ is well-powered if every
  $\Sub_{\mathcal{C}}(A)$ is equivalent to a
  \gls{small} poset.
\end{fcdefn}

For example, $\Set$ is \gls{wp} since the
inclusions $A' \subseteq A$ form a representative
set of \glspl{subobj} of $A$. It is \gls{wcp}
since isomorphism classes of \glspl{epi} $A \epic
B$ correspond to equivalence relations on $A$.

\begin{fclemma}[]
  \label{lemma:4.15}
  % Lemma 4.15
  Assuming:
  - a \gls{pullb} diagram
  \begin{picmath}
    \begin{tikzcd}
      P \ar[r, "h"] \ar[d, "k"]
      & A \ar[d, tail, "f"] \\
      B \ar[r, "g"]
      & C
    \end{tikzcd}
  \end{picmath}
  where $f$ is \gls{monic}
  Then:
  $k$ is \gls{monic}.
\end{fclemma}

\begin{proof}
  Suppose given $D \twofuncs lm P$ with $kl = km$.
  Then $fhl = gkl = gkm = fhm$, but $f$ is
  \gls{monic} so $hl = hm$. So $l$ and $m$ are
  both factorisations of
  \begin{picmath}
    \begin{tikzcd}
      D \ar[r, "hl"] \ar[d, "kl"]
      & A \\
      B
    \end{tikzcd}
  \end{picmath}
  through the \gls{pullb}, and hence $l = m$.
\end{proof}

\begin{fcthm}[Special Adjoint Functor Theorem]
  \label{saft}
  % Theorem 4.16
  Assuming:
  - $\mathcal{C}$ and $\mathcal{D}$ are \gls{locsm}
  - $\mathcal{D}$ is \gls{comp} and \gls{wp}
  - $\mathcal{D}$ has a co\gls{seping} set of
  objects
  Then:
  \begin{iffc}
    \lhs
    $G : \mathcal{D} \to \mathcal{C}$ has a
    \gls{ladj}
    \rhs
    it \gls{preses} all
    \gls{small} \glspl{lim}.
  \end{iffc}
\end{fcthm}