%! TEX root = CT.tex
% vim: tw=50
% 01/11/2024 09AM

\begin{example}
  \label{eg:4.3}
  % Example 4.3
  \phantom{}
  \begin{enumerate}[(a)]
    \item Suppose $J = \emptyset$. If $D :
      \emptyset \to \mathcal{C}$, then $\Cone(D)
      \cong \mathcal{C}$, so a \gls{lim} for $D$
      is a \gls{tobj}.
    \item \glsnoundefn{span}{span}{spans}%
      If $J = \bullet \bullet$, a \gls{diag}
      of shape $J$ is a pair $A, B$, and a
      \gls{cone} over it is a \emph{span}
      \begin{picmath}
        \begin{tikzcd}
          & C \ar[ld] \ar[rd] \\
          A
          &
          & B
        \end{tikzcd}
      \end{picmath}
      A \gls{lim} for it is a \gls{catprod}
      \begin{picmath}
        \begin{tikzcd}
          & A \times B \ar[ld, swap, "\pi_1"]
          \ar[rd, "\pi_2"] \\
          A
          &
          & B
        \end{tikzcd}
      \end{picmath}
      Dually, a \gls{colim} for it is a
      \gls{coprod}
      \begin{picmath}
        \begin{tikzcd}
          A \ar[rd]
          &
          & B \ar[ld] \\
          & A + B
        \end{tikzcd}
      \end{picmath}
    \item If $J$ is a (\gls{small}) discrete
      \gls{cat}, a (co)\gls{lim} for $(A_j \mid j
      \in J)$ is a (co)\gls{prod} $\prod_{j \in J}
      A_j$ ($\sum_{j \in J} A_j$).
    \item If $J$ is $\bullet \twofuncs{}{}
      \bullet$, then a \gls{diag} of shape $J$ is
      a parallel pair $A \twofuncs fg B$. A
      \gls{cone} over it consists of
      \begin{picmath}
        \begin{tikzcd}
          & C \ar[ld, swap, "h"] \ar[rd, "k"] \\
          A
          &
          & B
        \end{tikzcd}
      \end{picmath}
      satisfying $fh = k = gh$, or equivalently of
      $C \stackrel{h}{\to} A$ satisfying $fh =
      gh$. So a \gls{lim} for $A \twofuncs fg B$
      is an \gls{equaler} for $(f, g)$, as defined
      in \cref{eg:2.6}(g).
    \item \glsnoundefn{cospan}{cospan}{cospans}%
      \glsnoundefn{pullb}{pullback}{pullbacks}%
      \glsnoundefn{pushb}{pushout}{pushouts}%
      If $J$ is
      \begin{picmath}
        \begin{tikzcd}
          & \bullet \ar[d] \\
          \bullet \ar[r]
          & \bullet
        \end{tikzcd}
      \end{picmath}
      then a \gls{diag} of shape $J$ is a
      \emph{cospan}
      \begin{picmath}
        \begin{tikzcd}
          & A \ar[d, "f"] \\
          B \ar[r, "g", swap]
          & C
        \end{tikzcd}
      \end{picmath}
      A cone over it has $3$ legs, but if we omit
      the (redundant) middle one, it's a span
      \begin{picmath}
        \begin{tikzcd}
          D \ar[r, "h"] \ar[d, "k", swap]
          & A \\
          B
        \end{tikzcd}
      \end{picmath}
      completing the cospan to a commutative
      square.

      A \gls{lim} for
      \begin{picmath}
        \begin{tikzcd}
          & A \ar[d, "f"] \\
          B \ar["g", r, swap]
          & C
        \end{tikzcd}
      \end{picmath}
      is called a \emph{pullback} for $(f, g)$. If
      $\mathcal{C}$ has binary \glspl{prod} and
      \glspl{equaler}, we can construct pullbacks
      by forming the \gls{equaler} $A \times B
      \twofuncs{f\pi_1}{g\pi_2} C$. Dually,
      \glspl{colim} of shape $J^\op$ are called
      \emph{pushouts}.
    \item If $M = \{1, e\}$ is the $2$-element
      with $e^2 = e$, a \gls{diag} of shape $M$ is
      an object $A$ equipped with an idempotent $A
      \stackrel{e}{\to} A$. A \gls{lim}
      (respectively \gls{colim}) for $(A, e)$ is
      the \gls{monic} (respectively \gls{epic})
      part of a splitting of $e$.

      Note that the \gls{func} $\Set
      \stackrel{F}{\to} \funccat[M, \Set]$ in
      \cref{eg:3.2}(e) is $\Diag$, so this
      explains the coincidence of left and
      \glspl{radj}.
    \item \glsnoundefn{dirseq}{direct
      sequence}{direct sequences}%
      \glsnoundefn{dirlim}{direct limit}{direct
      limits}%
      \glsnoundefn{invseq}{inverse
      sequence}{inverse sequences}%
      \glsnoundefn{invlim}{inverse limit}{inverse
      limits}%
      Suppose $J = \Nbb$ is the ordered set of
      natural numbers. A \gls{diag} of shape
      $\Nbb$ is a \emph{direct sequence}
      \[
        A_0 \to A_1 \to A_2 \to A_3 \to \cdots
      ,\]
      and a \gls{colim} for it is called a
      \emph{direct limit} $A_\infty$.

      Dually, we have \emph{inverse sequences}
      \[
        \cdots \to A_2 \to A_1 \to A_0
      ,\]
      and their \glspl{lim} are called
      \emph{inverse limits}.

      For example in topology, an infinite
      dimensional CW-complex $X$ is the direct
      limit of its $n$-skeletons $X_n$. In
      algebra, the ring of $p$-adic integers is
      the \gls{lim} of the inverse sequence
      \[
        \cdots \to \Zbb / p^3 \Zbb \to \Zbb / p^2 \Zbb
        \to \Zbb / p\Zbb \to \{0\}
      \]
      in $\Rng$.
  \end{enumerate}
\end{example}

\begin{fcprop}[]
  \label{prop:4.4}
  % Proposition 4.4
  Assuming:
  - $\mathcal{C}$ a \gls{cat}
  Thens:[(i)]
  - If $\mathcal{C}$ has \glspl{equaler} and
  all \gls{small} \glspl{prod} (including
  empty \gls{prod}), then
  $\mathcal{C}$ has all \gls{small}
  \glspl{lim}.
  - If $\mathcal{C}$ has \glspl{equaler} and
  all finite \glspl{prod} (including empty
  \gls{prod}), then $\mathcal{C}$
  has all finite \glspl{lim}.
  - If $\mathcal{C}$ has \glspl{pullb} and a
  \gls{tobj}, then $\mathcal{C}$ has all
  finite \glspl{lim}.
\end{fcprop}

\begin{proof}
  \phantom{}
  \begin{enumerate}[(i) \& ()]
    \item[(i) \& (ii)] Let $D : J \to \mathcal{C}$
      be a \gls{diag}. Form the \glspl{prod} 
      \begin{align*}
        P
        &= \prod_{j \in \ob J} D(j) \\
        Q
        &= \prod_{\alpha \in \mor J} D(\cod \alpha)
      \end{align*}
      We have morphisms $P \twofuncs fg Q$ defined
      by $\pi_\alpha f = \pi_{\cod \alpha}$ and
      $\pi_\alpha g = D(\alpha) \pi_{\dom \alpha}$
      for all $\alpha$. Let $e \stackrel{e}{\to}
      P$ be an \gls{equaler} for $(f, g)$. The
      morphisms $\lambda_j = \pi_j e : E \to P \to
      D(j)$ form a \gls{cone} over $D$, since for
      any $\alpha : j \to j'$ we have
      \[
        D(\alpha)\lambda_j = D(\alpha) \pi_j e
        = \pi_\alpha ge = \pi_\alpha fe = \pi_{j'}
        e = \lambda{j'}
      .\]
      It is a \gls{lim}: given any \gls{cone} $(A,
      (\mu_j \mid j \in \ob J))$ over $D$, the
      $\mu_j$ form a \gls{cone} over the discrete
      \gls{diag} with vertices $D(j)$, so they
      induce a unique $\mu : A \to P$. Then $f \mu
      = g \mu$ since the $\mu_j$s form a
      \gls{cone} over $D$, so $\mu$ factors
      uniquely as $e\nu$, and $\nu$ is the unique
      factorisation of $(\mu_j \mid j \in \ob J)$
      through $(\lambda_j \mid j \in \ob J)$.
    \item[(iii)] If $1$ is a \gls{tobj} of
      $\mathcal{C}$, then we can construct $A
      \times B$ as the \gls{pullb} of
      \begin{picmath}
        \begin{tikzcd}
          & A \ar[d] \\
          B \ar[r]
          & 1
        \end{tikzcd}
      \end{picmath}
      Then we can construct $\prod_{i = 1}^n A_i$
      as $A_1 \times (A_2 \times ( \cdots \times
      A_n) \cdots ))$.

      To form an \gls{equaler} of $A \twofuncs fg
      B$, consider the \gls{pullb} of
      \begin{picmath}
        \begin{tikzcd}
          & A \ar[d, "(\identity{A}{,} f)"] \\
          A \ar[r, swap, "(\identity{A}{,} g)"]
          & A \times B
        \end{tikzcd}
      \end{picmath}
      Any \gls{cone}
      \begin{picmath}
        \begin{tikzcd}
          C \ar[r, "h"] \ar[d, swap, "k"]
          & A \\
          A
        \end{tikzcd}
      \end{picmath}
      over this has $h = k = \pi_1(\identity{A},
      g)k = \pi_1(\identity{A}, f)h$. So a
      \gls{lim} \gls{cone} has the universal
      property of an \gls{equaler} for $(f, g)$.
      \qedhere
  \end{enumerate}
\end{proof}

\begin{fcdefn}[Limit preserving / reflecting /
  creating]
  \glsadjdefn{preses}{preserves}{\gls{func}}%
  \glsadjdefn{prese}{preserve}{\gls{func}}%
  \glsadjdefn{reflects}{reflects}{\gls{func}}%
  \glsadjdefn{reflect}{reflect}{\gls{func}}%
  \glsadjdefn{creates}{creates}{\gls{func}}%
  \label{defn:4.5}
  % Definition 4.5
  Let $F : \mathcal{C} \to \mathcal{D}$ be a
  \gls{func}.
  \begin{cenum}[(a)]
    \item We say $F$ \emph{preserves} \glspl{lim} of
      shape $J$ if, given $D : J \to \mathcal{C}$
      and a limit \gls{cone} $(L, (\lambda_j \mid
      j \in \ob J))$ for it, $(FL, (F\lambda_j
      \mid j \in \ob J))$ is a \gls{lim} for $FD :
      J \to \mathcal{D}$.
    \item We say $F$ \emph{reflects} \glspl{lim}
      of shape $J$ if given $D : J \to
      \mathcal{C}$, any \gls{cone} over $D$ which
      maps to a \gls{lim} \gls{cone} in
      $\mathcal{D}$ is a \gls{lim} in
      $\mathcal{C}$.
    \item We say $F$ \emph{creates} \glspl{lim} of
      shape $J$ if, given $D : J \to \mathcal{C}$
      and a \gls{lim} \gls{cone} $(L, (\lambda_j
      \mid j \in \ob J))$ over $FD$, there exists
      a \gls{cone} over $D$ whose image under $F$
      is $\cong (L, (\lambda_j))$, and any such
      \gls{cone} is a \gls{lim} in $\mathcal{C}$.
  \end{cenum}
\end{fcdefn}
