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\begin{document}
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\section*{VII Singular Homology}
\textbf{ VII.1 Categories and Functors}
\begin{defn} A category $\mathcal{C}$ consists of \\
(1) A class of objects $X$\\
(2) $\forall $ ordered pair $X , Y$ of objects, a set $hom(X,Y)$
of morphisms \\(denoted by $f :X\rightarrow Y $) s.t. $\forall
f\in hom(X,Y) , g\in hom(Y,Z)$,\\ their composite $g\circ f \in
hom(X,Z)$ is defined and satisfies\\
$\hspace*{1em}$(associativity) $
f\in hom(X,Y) , g\in hom(Y,Z) , h\in hom(X,Z) \\
\hspace*{6em}\Rightarrow h\circ(g\circ f) = (h\circ g)\circ f$\\
$\hspace*{1em}(\exists$ of id )$\forall X$ : object, $\exists 1_X
\in hom(X,X)$ called an identity morphism \\
$\hspace*{4em}$s.t. $1_X \circ f=f$ and $g\circ 1_X=g,\\
\hspace*{4em}\forall f\in hom(Y,X)$ and $\forall g\in hom(X,Y)$
for $\forall Y :$ object.
\end{defn}

1. id morphism is unique. ($\because 1_X = 1_X \circ 1'_X = 1'_X$)

\begin{defn} $g\circ f = 1_X \Rightarrow g$ is called a left
inverse of $f.\\ \hspace*{9.5em} f$ is called a right inverse of
$g$.\end{defn}

2. If $f$ has a left inverse $g$ and right inverse $g'$, then
$g=g'.\\( \because g'=1_X \circ g'= (g\circ f)\circ g' =
g\circ(f\circ g' ) = g\circ 1_X = g)$\\
$f$ has an inverse. $\Rightarrow f$ is called an equivalence.

\begin{defn} A covariant(contravariant resp.) functor $F$ from a category $\mathcal{C}$ to a category $\mathcal{D}$ is a
function assigning to each object $X$ of $\mathcal{C}$, an object
$F(X)$ of $\mathcal{D}$ and assigning to each morphism $f:
X\rightarrow Y$ , a morphism\\ $F(f) :F(X) \rightarrow F(Y)$ s.t.
(1) $F(1_X ) = 1_{F(X)} , \forall X \\ \hspace*{5em}
(\leftarrow\hspace{0.5em}$resp.)$\hspace{2em}$ (2) $F(g\circ f) =
F(g)\circ F(f) (= F(f)\circ F(g)$ resp.)
\end{defn}

\textbf{ Note.} $f$ : equivalence. $\Rightarrow F(f) : $
equivalence.\\($\because F(g)\circ F(f) = F(g\circ f) = F(1_X ) = 1_{F(X)})$\\

\textbf{Example} The category of sets and functions
(=$\mathcal{S}$)\\ The category of
topological spaces and continuous functions (=$\mathcal{T}$)\\
The
category of groups and homomorphisms (=$\mathcal{G}$)\\
The
category of abelian groups and homomorphisms (=$\mathcal{A}$)\\
The category of $R$-modules and homomorphisms (=$\mathcal{M}$)\\
The category of based topological spaces $(X, x_0)$ and continuous
functions preserving base point
$(X,x_0 )\rightarrow (Y,y_0 )\hspace{1em}(=\mathcal{T }_0$)\\

\newpage
The category of pairs of topological spaces and pairs of
continuous functions\\ $(X,Y)\overset{(f,g)}{\rightarrow}(X' , Y')
\hspace{1em}(=\mathcal{T}\times\mathcal{T})$\\
The
category of simplicial complexes and simplicial maps \\
The
category of chain complexes and chain maps\\
Given $(X,x_0 )$, the category of covering spaces and morphisms\\

\textbf{Examples of Functors}\\

1. $F : \mathcal{T} \times \mathcal{T} \rightarrow \mathcal{T}\\
\hspace*{3em}(X,Y) \mapsto X\times Y \\ \hspace*{4em}\downarrow
(f,g) \hspace{1em}\downarrow (f\times g)(x\times y)=(f(x),g(y))\\
\hspace*{3em} (X' , Y') \mapsto X'\times Y'$\\

2. Forgetful functor :$\mathcal{T} \rightarrow \mathcal{S}$ and $\mathcal{G}\rightarrow\mathcal{S}\\
\hspace*{9em} X\mapsto "X"$(underlying set)\\
$\hspace*{9em}\downarrow f \hspace{1em} \downarrow "f"$(underlying
set function)\\ $\hspace*{9em} Y\mapsto "Y"$\\

3. $\mathcal{T}_0 \overset{F=\pi _1}{\longrightarrow}\mathcal{G}\\
(X,x_0 )\mapsto \pi_1(X,x_0 )\\
\hspace*{0.5em}\downarrow f\hspace{2em} \downarrow \pi_1(f) =f_*
\\ (Y,y_0 )\mapsto \pi_1(Y,y_0)$\\

4. Cat. simplicial cxs and simplicial maps
$\overset{F}{\rightarrow}$Cat. of chain cxs and chain maps.\\
$\hspace*{14em}K \hspace{3em}\mapsto\hspace{3em} \mathcal{C}(K) = \{C_p (K) ,\partial\}\\
\hspace*{13.5em}\downarrow f\hspace{7em}\downarrow f_\sharp\\
\hspace*{14em}L\hspace{3em}\mapsto \hspace{3em}\mathcal{C}(L) = \{
C_p (L),\partial\}$\\

5. Cat. simplicial cxs and simplicial maps
$\overset{H_p}{\rightarrow}$ Cat. of abel. gps and homs.\\
$\hspace*{14em}K \hspace{3em}\mapsto\hspace{3em} H_p(K) \\
\hspace*{16em}F
\searrow\hspace{2em}\nearrow\\
\hspace*{13.5em}\downarrow f\hspace{2.5em} \mathcal{C}(K)\hspace{2.5em}\downarrow f_*\\
\hspace*{14em}L\hspace{3em}\mapsto \hspace{3em}H_p(L)\\
\hspace*{16.5em}\searrow\hspace{0.7em}\downarrow\hspace{0.7em}\nearrow\\\hspace*{18em}\mathcal{C}(L)$\\
\newpage
6. Cat. of vector sps and linear trs $\overset{F}{\rightarrow}$
Cat. of vector sps and linear
trs\\
$\hspace*{11em}V \hspace{3em}\mapsto\hspace{3em} V^*\hspace{1em}\alp\circ f\\
\hspace*{10.5em}\downarrow f\hspace{7em}\uparrow f^*\hspace{1em}\uparrow\\
\hspace*{11em}W\hspace{3em}\mapsto \hspace{3em}W^* \hspace{1em}\alp$\\
This is a contravariant functor.\\

\textbf{Natural Transformation}
\begin{defn} $\mathcal{C}\overset{F}{\underset{G}{\rightrightarrows}}
\mathcal{D}$ , two functors from a category $\mathcal{C}$ to a
category $\mathcal{D}$.\\
A natural transformations $T$ from $F$ to $G$ is a function :$Ob(\mathcal{C})\rightarrow Mor(\mathcal{D})$ \\
s.t. $F(X)\overset{F(f)}{\mapsto} F(Y)\hspace{1em}$ commutes for $\forall X, Y \in Ob
(\mathcal{C})\hspace{4em} X \mapsto T_X \hspace{3em}$\\
$\hspace*{2.5em}\downarrow T_X \hspace{0.5em}\curvearrowright\hspace{0.5em} \downarrow T_Y\\
\hspace*{2em}G(X)\overset{G(f)}{\mapsto} G(Y)$\end{defn}

If $T_X$ is an equivalence, $\forall X\in Ob(\mathcal{C}),$ then
$T$ is called a natural equivalence between two functors.\\

example: Let $(X,Y)\overset{F}{\mapsto} X\times Y$ \\
$\hspace*{9em}
\overset{G}{\mapsto} Y\times X $\\
Then $T_{(X,Y)} : X\times Y \rightarrow Y\times X$ is a natural
equivalence i.e.,\\
$\hspace*{7em} (x,y)\mapsto (y,x)$\\
$F: X\times
Y \overset{f\times g}{\longrightarrow}X'\times Y'\\
\hspace*{2.5em}\downarrow T_{(X, Y)}\hspace{1em}\downarrow T_{(X', Y')}\hspace{2em}$ commutes $\forall X,Y.$\\
$G:Y\times X \overset{g\times f}{\longrightarrow} Y'\times X'$

\end{document}