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% 04/12/2024 09AM

\begin{fclemma}[]
  \label{lemma:7.14}
  % Lemma 7.14
  Assuming:
  - $\mathcal{C}$ a \gls{small} \gls{regc} and
  \gls{ats} \gls{cat}
  Then:
  there exists an isomorphism
  \glsref[reflects]{reflecting} \gls{reg}
  \gls{func} $\mathcal{C} \to \hat{\mathcal{C}}$
  where $\mathcal{C}$ is \gls{cap}. Hence in
  particular, there is an
  isomorphism-\glsref[reflects]{reflecting}
  \gls{reg} \gls{func} $\mathcal{C} \to \Set$.
\end{fclemma}

\begin{proof}
  Consider the sequence
  \[
    \mathcal{C} = \mathcal{C}_0 \to \mathcal{C}_1
    \to \mathcal{C}_2 \to \cdots
  ,\]
  where each $\mathcal{C}_{n + 1}$ is obtained
  from $\mathcal{C}_n$ by the construction of
  \cref{lemma:7.13}.

  We define $\hat{\mathcal{C}}$ to be the
  pseudo-\gls{colim} of this sequence: objects are
  pairs $(n, A)$ where $A \in \ob \mathcal{C}_n$,
  and morphisms $(n, A) \to (m, B)$ are
  \gls{reped} by pairs $(p, f)$ where $p \ge
  \max \{m, n\}$ and $F : IA \to I'B$ in
  $\mathcal{C}_p$, modulo the identification of
  $(p, f)$ with $(p', f')$ if $p \le p'$ and $f' =
  If$.

  The proof that $\mathcal{C}$ is \gls{regc}, and
  that the embeddings $\mathcal{C}_n \to
  \hat{\mathcal{C}}$ are
  isomorphism-\glsref[reflects]{reflecting}
  \gls{reg} \glspl{func}, is as in
  \cref{lemma:7.13}.

  Given any non-invertible \gls{mono} $A' \monic
  A$ in $\hat{\mathcal{C}}$, it lives in
  $\mathcal{C}_n$ for some $n$, so there exists
  $1 \to A$ in $\mathcal{C}_{n + 1}$ not factoring
  through $A' \monic A$.

  But if $A \stackrel{f}{\to} B$ isn't \gls{monic}
  in $\hat{\mathcal{C}}$, the legs $R \twofuncs ab
  A$ of its kernel-pair aren't equal, so there
  exists $1 \stackrel{r}{\to} R$ not factoring
  through their equation, so $1 \twofuncs{ar}{br}
  A$ are distinct but have the same composite with
  $f$.

  So $\hat{\mathcal{C}}(1, \blank)$ \gls{reflects}
  \glspl{mono} and hence \gls{reflects}
  isomorphisms.
\end{proof}

\begin{fcthm}[]
  \label{thm:7.15}
  % Theorem 7.15
  Assuming:
  - $\mathcal{C}$ \gls{small} and \gls{regc}
  Then:
  there exists a set $I$ and an
  isomorphism-\glsref[reflects]{reflecting}
  \gls{reg} \gls{func} $\mathcal{C} \to \Set^I$.
\end{fcthm}

\begin{proof}
  Let $I$ be a representative set of
  \glspl{subobj} of $1$ in $\mathcal{C}$, and for
  each $U \in I$ consider the composite
  \[
    \mathcal{C} \stackrel{(!_U)^*}{\to}
    \mathcal{C} / U \to (\mathcal{C} / U)_\tv
    \to \widehat{(\mathcal{C} / U)_\tv} \to \Set
  ,\]
  where the third factor is the \gls{func} of
  \cref{lemma:7.14} and the fourth is \gls{reped}
  by $1$.

  Given any non-invertible morphism $A
  \stackrel{f}{\to} B$ in $\mathcal{C}$, if $U$ is
  the \gls{supp} of $B$ then $(!)_H^* f$ remains
  non-invertible in $\mathcal{C} / U$ and its
  codomain is \gls{ws} there, so it remains
  non-invertible in $(\mathcal{C} / U)_\tv$ and
  hence in $\Set$.

  So these \glspl{func} collectively
  \glsref[reflects]{reflect} isomorphisms.
\end{proof}

\begin{remark}
  \label{remark:7.16}
  % Remark 7.16
  \phantom{}
  \begin{enumerate}[(a)]
    \item Barr's original embedding theorem
      produces a \gls{full} and \gls{faith}
      \gls{reg} \gls{func} $\mathcal{C} \to
      \funccat[\mathcal{D}, \Set]$ for some
      \gls{small} \gls{cat} $\mathcal{D}$.
      Moreover if $\mathcal{C}$ is \gls{ats} we
      can take $\mathcal{D}$ to be a monoid.
    \item \cref{thm:7.15} yields a `meta theorem'
      saying that `anything we can prove in $\Set$
      is true in all \gls{reg} \glspl{cat}'.

      For example to prove \cref{prop:7.5}
      (\gls{cover} implies \gls{reg} \gls{epic}),
      given a \gls{cover} $A \stackrel{f}{\covers}
      B$ in a \gls{regc} \gls{cat} $\mathcal{C}$,
      and a $A \stackrel{g}{\to} C$ having equal
      composites with the kernel-pair $R
      \twofuncs{}{} A$ of $f$, we can cut down to
      a \gls{small} sub\gls{cat} $\mathcal{C}'$
      containing $f$ and $g$ and closed under
      finite \glspl{lim} and \glspl{image}, and
      then show that the first component of $I
      \stackrel{(h, k)}{\monic} A \times C$
      becomes an isomorphism in $\Set^I$.
    \item Abelian \glspl{cat} are \gls{regc}
      \glspl{cat} enriched over $\AbGp$ (i.e. for
      any two objects $A$ and $B$, $\mathcal{A}(A,
      B)$ has an abelian group structure and
      composition distributes over addition).

      Abelian \glspl{cat} are \gls{ts} since their
      \glspl{tobj} are \gls{initial}, so for any
      \gls{small} abelian $\mathcal{A}$ we get an
      isomorphism-\glsref[reflects]{reflecting}
      \gls{reg} \gls{func} $\mathcal{A} \to \Set$
      and hence an
      isomorphism-\glsref[reflects]{reflecting}
      \gls{func} $\mathcal{A} \cong
      \AbGp(\mathcal{A}) \to \AbGp(\Set) = \AbGp$.
  \end{enumerate}
\end{remark}
