%! TEX root = CT.tex
% vim: tw=50
% 30/10/2024 09AM

  For the second triangular identity, we have
  \begin{picmath}
    \begin{tikzcd}
      G \ar[r, "\alpha_G"]
      \ar[rd, "\identity{G}", swap]
      & GFG \ar[r, "(GFG\beta)^{-1}"]
      \ar[d, "\alpha_G^{-1}"]
      & GFGFG \ar[d, "(GF\alpha_G)^{-1} =
      (\alpha_{GFG})^{-1}"] \\
      & G \ar[r, "(G\beta)^{-1}"]
      \ar[rd, "\identity{G}", swap]
      & GFG \ar[d, "G\beta"] \\
      &
      & G\beta
    \end{tikzcd}
  \end{picmath}
  Hence by \cref{thm:3.7} we have $(F \adjoint
  G)$. But $(\beta')^{-1}$ and $\alpha^{-1}$ also
  satisfy the triangular identities for and
  \gls{adjunc} $(G \adjoint F)$.
\end{proof}

\begin{fclemma}[]
  \label{lemma:3.9}
  % Lemma 3.9
  Assuming:
  - $(F : \mathcal{C}
  \to \mathcal{D} \adjoint G : \mathcal{D} \to
  \mathcal{C})$ an \gls{adjunc} with \gls{counit} $\eps$
  Thens:[(i)]
  - $G$ is \gls{faith} if and only if $\eps$
  is \gls{pws} \gls{epic}
  - $G$ is \gls{full} and \gls{faith} if and
  only if $\eps$ is an isomorphism
\end{fclemma}

\begin{proof}
  \phantom{}
  \begin{enumerate}[(1)]
    \item Given $g : B \to C$ in $\mathcal{D}$,
      $g\eps_B$ corresponds to $Gg$ under the
      \gls{adjunc}. So $\eps_B$ \gls{epic} if and
      only if $G$ acts injectively on morphisms
      with domain $B$ and specified codomain.
      Hence $\eps_B$ \gls{epic} for all $B$ if and
      only if $G$ is \gls{faith}.
    \item Similarly, $G$ \gls{full} and
      \gls{faith} if and only if for all $B$ and
      $C$ composition with $\eps_B$ is a
      bijection $\mathcal{D}(B, C) \to
      \mathcal{D}(FGB, C)$. This happens if and
      only if $\eps_b : FGB \to B$ is an
      isomorphism for all $B$. \qedhere
  \end{enumerate}
\end{proof}

\begin{fcdefn}[Reflection]
  \glsnoundefn{reflon}{reflection}{reflections}%
  \glsadjdefn{refliv}{reflective}{subcategory}%
  \glsnoundefn{reflsub}{reflective
  subcategory}{reflective subcategories}%
  % Definition 3.10
  By a \emph{reflection}, we mean an \gls{adjunc}
  such that the \gls{radj} is \gls{full} and
  \gls{faith} (equivalently: the \gls{counit} is
  an isomorphism).
  % satisfying the conditions of
  % \cref{lemma:3.9}(ii).
  We say $\mathcal{D}
  \subseteq \mathcal{C}$ is a \emph{reflective
  subcategory} if it's \gls{full} and the
  inclusion $\mathcal{D} \to \mathcal{C}$ has a
  \gls{ladj}.
\end{fcdefn}

\begin{example}
  \label{eg:3.11}
  % Example 3.11
  \phantom{}
  \begin{enumerate}[(a)]
    \item $\AbGp$ is \gls{refliv} in $\Gp$: the
      \gls{ladj} to the inclusion sends $G$ to $G
      / G'$ where $G'$ is the subgroup generated
      by commutators. Any homomorphism $G \to A$
      with $A$ abelian factors uniquely through
      the quotient map $G \to G / G'$.
    \item Recall that a group $G$ is
      \emph{torsion} if all elements have finite
      order, and \emph{torsion free} if its only
      element of finite order is $1$. In an
      abelian group $A$, the torsion leements form
      a subgroup $A_t$, and $A \mapsto A_t$ is
      \gls{radj} to the inclusion $\mathbf{tAbGp}
      \to \AbGp$, since any homomorphism $B \to A$
      whose $B$ is torsion takes values in $A_t$.
      Similarly, $A \mapsto A / A_t$ defines a
      \gls{ladj} to the inclusion $\mathbf{tfAbGp}
      \to \AbGp$.
    \item Let $\mathbf{KHaus} \subseteq \Top$ be
      the full sub\gls{cat} of compact Hausdorff
      spaces. $\mathbf{KHaus}$ is \gls{refliv} in
      $\Top$: the \gls{ladj} is the
      \emph{Stone-\v{C}ech compactification}
      $\beta$.
    \item Let $\mathbf{Seq} \subseteq \Top$ be the
      \gls{full} sub\gls{cat} of \emph{sequential
      spaces}, i.e. those in which all
      sequentially closed sets are closed. The
      inclusion $\mathbf{Seq} \to \Top$ has a
      \gls{radj} sending $X$ to $X_s$, the same
      set as $X$ with all sequentially closed sets
      declared to be closed. The identity mapping
      $X_s \to X$ is (continuous, and) the
      \gls{counit} of the \gls{adjunc}.
    \item The \gls{cat} $\mathbf{Preord}$ of
      preordered sets is \gls{refliv} in $\Cat$:
      the \gls{reflon} sends $\mathcal{C}$ to
      $\mathcal{C} / \simeq$ where $\simeq$ is the
      congruence identifying all paralell pairs in
      $\mathcal{C}$.
    \item Given a topological space $X$, the poset
      $\Omega(X)$ of open subsets of $X$ is
      co\gls{refliv} in $\mathcal{P}(X)$, since if
      $U$ is open and $A \subseteq X$ is
      arbitrary, we have $U \subseteq A$ if and
      only if $U \subseteq A^\circ$ (recall
      ${}^\circ$ denotes interior). Dually, the
      poset of closed subsets is \gls{refliv} in
      $\mathcal{P}(X)$.
  \end{enumerate}
\end{example}

\newpage
\section{Limits}

\begin{fcdefn}[Diagram]
  \glsnoundefn{diag}{diagram}{diagrams}%
  \glsnoundefn{vert}{vertex}{vertices}%
  \glsnoundefn{edge}{edge}{edges}%
  % Definition 4.1
  Let $J$ be a \gls{cat} (almost always small, and
  often finite). By a \emph{diagram of shape $J$}
  in a \gls{cat} $\mathcal{C}$, we mean a
  \gls{func} $D : J \to \mathcal{C}$. The objects
  $D(j)$, $j \in \ob J$ are called \emph{vertices}
  of $D$, and morphisms $D(\alpha)$, $\alpha \in
  \mor J$ are called \emph{edges} of $D$.
\end{fcdefn}

For example, if $J$ is the \gls{cat}
\begin{picmath}
  \begin{tikzcd}
    \bullet \ar[r] \ar[d] \ar[rd]
    & \bullet \ar[d] \\
    \bullet \ar[r]
    & \bullet
  \end{tikzcd}
\end{picmath}
a \gls{diag} of shape $J$ is a commutative square
in $\mathcal{C}$.

If $J$ is instead
\begin{picmath}
  \begin{tikzcd}
    \bullet \ar[r] \ar[d] \ar[rd, shift left=3pt]
    \ar[rd, shift right=3pt]
    & \bullet \ar[d] \\
    \bullet \ar[r]
    & \bullet
  \end{tikzcd}
\end{picmath}
then a \gls{diag} of shape $J$ is a
not-necessarily-commutative square.

\begin{fcdefn}[Cone, limit]
  \glssymboldefn{cone}%
  \glsnoundefn{cone}{cone}{cones}%
  \glsnoundefn{lim}{limit}{limits}%
  \glsnoundefn{colim}{colimit}{colimits}%
  % Definition 4.2
  Let $D : J \to \mathcal{C}$ be a \gls{diag}. A
  \emph{cone} over $D$ consists of an object $A$
  (its \emph{apex}) together with morphisms
  $\lambda_j : A \to D(j)$ for each $j \in \ob J$
  (the \emph{legs} of the cone) such that
  \begin{picmath}
    \begin{tikzcd}
      & A \ar[ld, "\lambda_j", swap] \ar[rd,
      "\lambda_{j'}"] \\
      D(j) \ar[rr, "D(\alpha)", swap]
      &
      & D(j')
    \end{tikzcd}
  \end{picmath}
  commutes for each $\alpha : j \to j'$ in $J$.

  A morphism of cones  $(A, (\lambda_j \mid j \in \ob
  J)) \to (B, (\mu_j \mid j \in \ob J))$ is a
  morphism $f : A \to B$ such that $\mu_j f =
  \lambda_j$ for all $j$. We have a \gls{cat}
  $\mathbf{Cone}(D)$ of cones over $D$; a
  \emph{limit} for $D$ is a terminal object of
  $\mathbf{Cone}(D)$.

  Dually, a \emph{colimit} for $D$ is an initial
  cone under $D$.
\end{fcdefn}

\begin{center}
  \includegraphics[width=0.6\linewidth]{images/1c0e65bc11344a0d.png}
\end{center}

\glssymboldefn{Deltadiag}%
If $\Delta : \mathcal{C} \to \funccat[J,
\mathcal{C}]$ is the \gls{func} sending $A$ to
the \emph{constant diagram} with all
\glspl{vert} $A$ then a cone over $D$ is a
\gls{natt} $\Delta A \to D$.

Also, $\Cone(D)$ is another name for $(\Delta
\darr D)$, defined as in \cref{thm:3.3}${}^\op$.

So by \cref{thm:3.3}, $C$ has \glspl{lim} for all
\glspl{diag} of shape $J$ if and only if $\Delta$
has a \gls{radj}.
