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    \item The \emph{Hurewicz homomorphism} links
      the homotopy and homology groups of a space
      $X$. Elements of $\pi_n(X, x)$ are homotopy
      classes of basepoint-preserving maps $S^n
      \stackrel{f}{\to} X$. If we think of $S^n$
      as $\partial \Delta^{n + 1}$, $f$ defines a
      singular $n$-cycle on $X$ and homotopic maps
      differ by an $n$-boundary, so we get a
      well-defined map $\pi_n(X, x)
      \stackrel{h_n}{\to} H_n(X)$. $h_n$ is a
      homomorphism, and it's a \gls{natt} $\pi_n
      \to H_n U$, where $U$ is the forgetful
      functor $\Top_* \to \Top$.
  \end{enumerate}
\end{example}

We have isomorphisms of \glspl{cat}: e.g. $F
: \Rel \to \Rel^\op$ defined by $FA = A$,
$FR = R^o = \{(b, a) \st (a, b) \in R\}$ is
its own inverse.

But we have a weaker notion of equivalence of
\glspl{cat}.

\glsadjdefn{natyi}{naturally
isomorphic}{\glspl{cat}}%
\begin{fclemma}[]
  \label{lemma:1.8}
  \glsnoundefn{nati}{natural isomorphism}{natural
  isomorphisms}%
  % Lemma 1.8
  Assuming:
  - $\alpha : F \to G$ is a \gls{natt} between
  \glspl{func} $\mathcal{C} \twofuncs{}{} \mathcal{D}$
  Then:
  $\alpha$ is an isomorphism in
  $\funccat[\mathcal{C}, \mathcal{D}]$ if and only if
  $\alpha_A$ is an isomorphism in $\mathcal{D}$
  for each $A$.
\end{fclemma}

\begin{proof}
  \phantom{}
  \begin{enumerate}[$\Rightarrow$]
    \item[$\Rightarrow$] Obvious since composition
      in $\funccat[\mathcal{C}, \mathcal{D}]$.
    \item[$\Leftarrow$] Suppose each $\alpha_A$
      has an inverse $\beta_A$. Given $A
      \stackrel{f}{\to} B$ in $\mathcal{C}$, in
      the diagram
      \begin{picmath}
        \begin{tikzcd}
          GA \ar[r, "Gf"] \ar[d, "\beta_A"]
          & GB \ar[d, swap, "\beta_B"]
          \\
          FA \ar[r, "Ff"]
          \ar[u, bend left=45, "\alpha_A"]
          & FB
          \ar[u, bend right=45, swap, "\alpha_B"]
        \end{tikzcd}
      \end{picmath}
      we have $\beta_B(Gf) = \beta_B(Gf) \alpha_A
      \beta_A = \beta_B \alpha_B(Ff)\beta_A =
      (Ff)\beta_A$. \qedhere
  \end{enumerate}
\end{proof}

\begin{fcdefn}[Equivalence of categories]
  \label{defn:1.9}
  % Definition 1.9
  \glsnoundefn{catp}{categorical
  property}{categorical properties}%
  \glsnoundefn{equivc}{equivalence}{equivalences}%
  \glsadjdefn{equivt}{equivalent}{\glspl{cat}}%
  \glssymboldefn{equiv}%
  Let $\mathcal{C}$ and $\mathcal{D}$ be
  \glspl{cat}. An \emph{equivalence} between
  $\mathcal{C}$ and $\mathcal{D}$ consists of
  \glspl{func} $F : \mathcal{C} \to \mathcal{D}$
  and $G : \mathcal{D} \to \mathcal{C}$ together
  with \glspl{nati} $\alpha : \identity{
  \mathcal{C}} \to GF$, $\beta : FG \to
  \identity{ \mathcal{D}}$. We write $\mathcal{C}
  \equiv \mathcal{D}$ if there exists an
  \gls{equivc} between $\mathcal{C}$ and
  $\mathcal{D}$.

  We say $P$ is a \emph{categorical property} if
  \[
    (\text{$\mathcal{C}$ has $P$ and $\mathcal{C}
    \equiv \mathcal{D}$}) \implies
    \text{$\mathcal{D}$ has $P$}
  .\]
\end{fcdefn}

\begin{example}
  % Example 1.10
  \phantom{}
  \begin{enumerate}[(a)]
    \item The \gls{cat} $\Part$ of sets and
      partial functions is \gls{equivt} to $\Set_*$
      (the \gls{cat} of pointed sets). We define
      $F : \Set_* \to \Part$ by $F(A, a) = A
      \setminus \{a\}$ and if $f : (A, a) \to (B,
      b)$, with $(Ff)(x) = f(x)$ if $f(x) \neq b$
      and undefined otherwise. Then define $G :
      \Part \to \Set_*$ by $G(A) = (A \cup \{A\},
      A)$ and if $f : A \partf B$, then
      \[
        Gf(x) = \begin{cases}
          f(x) & \text{if $x \in A$ and $f(x)$ is
          defined} \\
          B & \text{otherwise}
        \end{cases}
      .\]
      Then $FG = \identity{\Part}$; $GF \neq
      \identity{\Set_*}$, but there is an
      isomorphism $\identity{\Set_*} \to GF$.
      Note that $\Part \not\cong \Set_*$.
    \item We have an \gls{equivc}
      $\mathbf{fdVect}_k \equivc
      \mathbf{fdVect}_k^\op$: both \glspl{func} are
      $(\bullet)^*$, and both isomorphisms are
      $\alpha : \identity{\mathbf{fdVect}_k} \to
      (\bullet)^{**}$.
    \item We have an \gls{equivc}
      $\mathbf{fdVect}_k \equivc \Mat_k$: we
      define $F : \Mat_k \to \mathbf{fdVect}_k$ by
      $F(n) = k^n$, $F(n \stackrel{A}{\to} p)$ is
      the linear map $k^n \to k^p$ represented by
      $A$ (with respect to standard bases). TO
      define $G$, choose a basis for each $V$, and
      define $G(V) = \dim V$,
      \[
        G(V \stackrel{f}{\to} W) = \text{matrix
        representing $f$ with respect to chosen
      bases}
      .\]
      $GF = \identity{\Mat_k}$; the choice of
      bases yields isomorphisms $k^{\dim V} \to V$
      for each $V$, which form a \gls{natt} $FG
      \to \identity{\mathbf{fdVect}_k}$.
  \end{enumerate}
\end{example}

\begin{fcdefn}[Faithful / full / essentially
  surjective]
  % Definition 1.11
  \glsadjdefn{faith}{faithful}{\gls{func}}%
  \glsadjdefn{full}{full}{\gls{func}}%
  \glsadjdefn{esss}{essentially surjective}{\gls{func}}%
  \glsadjdefn{essi}{essentially injective}{\gls{func}}%
  Let $F : \mathcal{C} \to \mathcal{D}$ be a
  \gls{func}.
  \begin{cenum}[(a)]
    \item We say $F$ is \emph{faithful} if, given
      $f$ and $g$ in $\mor \mathcal{C}$, $(Ff =
      Fg$, $\dom f = \dom g$, $\cod f = \cod g)
      \implies f = g$.
    \item We say $F$ is \emph{full} if, for every
      $g : FA \to FB$ in $\mathcal{D}$, there
      exists $f : A \to B$ in $\mathcal{C}$ with
      $Ff = g$.
    \item We say $F$ is \emph{essentially
      surjective} if, for any $B \in \ob
      \mathcal{D}$, there exists $A \in \ob
      \mathcal{C}$ with $FA \cong B$.
  \end{cenum}
  Note that if $F$ is full and faithful, it's
  essentially injective: given $FA
  \customstackrel[\cong]{g}{\to} FB$ in $\mathcal{D}$,
  the unique $A \stackrel{f}{\to} B$ with $Ff = g$
  is an isomorphism.

  We say $\mathcal{D} \subseteq \mathcal{C}$ is a
  \emph{full subcategory} if the inclusion
  $\mathcal{D} \to \mathcal{C}$ is a full
  \gls{func}.
\end{fcdefn}

\begin{fclemma}[]
  % Lemma 1.12
  Assuming:
  - $F : \mathcal{C} \to \mathcal{D}$
  Then:
  $F$ is part of an \gls{equivc} $\mathcal{C}
  \equivc \mathcal{D}$ if and only if $F$ is
  \gls{full}, \gls{faith}, \gls{esss}.
\end{fclemma}

\begin{iffproof}
  \rightimpl
  Suppose give $G$,
  $\alpha$ and $\beta$ as in \cref{defn:1.9}.
  Then $\beta_B : FGB \to B$ witnesses the
  fact that $F$ is \gls{esss}. If $A
  \twofuncs{f}{g} B$ satisfy $Fg = Fg$, then
  $GFf = GFg$; but $f = \alpha_B^{-1}(GFf)
  \alpha_A$, so $f = g$. Suppose given $FA
  \stackrel{g}{\to} FB$; then $f =
  \alpha_B^{-1} (Gg) \alpha_A$ satisfies $GFf
  = Gf$ but $G$ is \gls{faith} for the same
  reason as $F$, so $Ff = g$.

  \leftimpl
  For each $B \in \ob
  \mathcal{D}$, chose $GB \in \ob \mathcal{C}$
  and an isomorphism $\beta_B : FGB \to B$.
  Given $B \stackrel{g}{\to} C$, define $Gg :
  GB \to GC$  to be the unique morphism such
  that $FGg = \beta_C^{-1} g \beta_B$.
  Functoriality follows from uniqueness, and
  \gls{naty} of $\beta$.
  We define $\alpha_A : A \to GFA$ to be the
  unique morphism such that $F\alpha_A =
  \beta_{FA}^{-1} : FA \to FGFA$. $\alpha_A$
  is an isomorphism, and \glspl{natsq} for
  $\alpha$ are mapped by $F$ to \glspl{natsq}
  for $\beta^{-1}$, so they commute. \qedhere
\end{iffproof}
