% (find-angg "LATEX/2009abcats.tex")
% (find-dn4ex "edrx08.sty")
% (find-angg ".emacs.templates" "s2008a")
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2009abcats.tex && latex    2009abcats.tex"))
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2009abcats.tex && pdflatex 2009abcats.tex"))
% (eev "cd ~/LATEX/ && Scp 2009abcats.{dvi,pdf} edrx@angg.twu.net:slow_html/LATEX/")
% (find-dvipage "~/LATEX/2009abcats.dvi")
% (find-pspage  "~/LATEX/2009abcats.pdf")
% (find-zsh0 "cd ~/LATEX/ && dvips -P pk  -o 2009abcats.ps 2009abcats.dvi")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o 2009abcats.ps 2009abcats.dvi")
% (find-zsh0 "cd ~/LATEX/ && dvips -P pk -D 300 -o 2009abcats.ps 2009abcats.dvi")
% (find-pspage  "~/LATEX/2009abcats.ps")
% (find-zsh0 "cd ~/LATEX/ && dvips -D 300 -o tmp.ps tmp.dvi")
% (find-pspage  "~/LATEX/tmp.ps")
% (ee-cp "~/LATEX/2009abcats.pdf" (ee-twupfile "LATEX/2009abcats.pdf") 'over)
% (ee-cp "~/LATEX/2009abcats.pdf" (ee-twusfile "LATEX/2009abcats.pdf") 'over)

% «.0-and-1-as-kers-and-coks»	(to "0-and-1-as-kers-and-coks")
% «.ker-and-cok-as-inv»		(to "ker-and-cok-as-inv")
% «.intersecs-of-subs»		(to "intersecs-of-subs")
% «.equalizers»			(to "equalizers")
% «.images»			(to "images")
% «.cok-epi-eq-0»		(to "cok-epi-eq-0")
% «.unique-factorization»	(to "unique-factorization")

\documentclass[oneside]{book}
\usepackage[latin1]{inputenc}
\usepackage{edrx08}       % (find-dn4ex "edrx08.sty")
%L process "edrx08.sty"  -- (find-dn4ex "edrx08.sty")
\input edrxheadfoot.tex   % (find-dn4ex "edrxheadfoot.tex")
\begin{document}

\input 2009abcats.dnt

%
% (eedn4-51-bounded)

\def\Kers{\operatorname{Kers}}
\def\Coks{\operatorname{Coks}}
\def\ker{\operatorname{ker}}
\def\cok{\operatorname{cok}}

\def\Ker{\operatorname{Ker}}
\def\Cok{\operatorname{Cok}}
\def\Im{\operatorname{Im}}
\def\Coim{\operatorname{Coim}}

\def\sm#1{\begin{smallmatrix}#1\end{smallmatrix}}
\def\kerrule#1#2#3{\sm{#1 \\ #2 & #3}}
\def\cokrule#1#2#3{\sm{#1 & #2 \\ & #3}}


% (find-freydabcatspage (+ 26 -7) "Contents")

Notes on chapter 2 of Freyd's ``Abelian Categories'' book (1964).

\bsk

Index of the slides:
\msk
% To update the list of slides uncomment this line:
\makelos{tmp.los}
% then rerun LaTeX on this file, and insert the contents of "tmp.los"
% below, by hand (i.e., with "insert-file"):
% (find-fline "tmp.los")
% (insert-file "tmp.los")
\tocline {0 and 1 as kernels and cokernels} {2}
\tocline {ker and cok are inverse functions} {3}
\tocline {Intersection of subobjects} {4}
\tocline {Difference kernels (a.k.a. equalizers)} {5}
\tocline {Images} {6}
\tocline {A map is epi iff its cok is 0} {7}

\newpage
% --------------------
% «0-and-1-as-kers-and-coks»  (to ".0-and-1-as-kers-and-coks")
% (s "0 and 1 as kernels and cokernels" "0-and-1-as-kers-and-coks")
\myslide {0 and 1 as kernels and cokernels} {0-and-1-as-kers-and-coks}

Lemma: for $m$ a monic and $e$ an epi,

\ssk

$\begin{array}{lcl}
 0 Ý \Kers m,   &&   0 Ý \Coks e, \\
 0 Ý \Kers 1,   &&   0 Ý \Coks 1, \\
 1 Ý \Kers 0,   &&   1 Ý \Coks 0. \\
 \end{array}
$

\msk

Proof: check:

%D diagram monic-and-epi-lemmas
%D 2Dx     100    +25    +25   +10    +25    +25
%D 2D  100 A0                  B0 ->> B1 ->> B2
%D 2D      |  \                          \    |
%D 2D      |   \                          \   |
%D 2D      v    v                          v  v
%D 2D  +25 A1 >-> A2 >-> A3                  B3
%D 2D
%D 2D  +15 C0                  D0 ->> D1 ->> D2
%D 2D      |  \                          \    |
%D 2D      |   \                          \   |
%D 2D      v    v                          v  v
%D 2D  +25 C1 >-> C2 >-> C3                  D3
%D 2D
%D 2D  +15 E0                  F0 --> F1 ->> F2
%D 2D      |  \                          \    |
%D 2D      |   \                          \   |
%D 2D      v    v                          v  v
%D 2D  +25 E1 >-> E2 --> E3                  F3
%D 2D
%D (( A0 .tex= X   A1 .tex= 0   A2 .tex= A   A3 .tex= B
%D    @ 0 @ 1  -> @ 0 @ 2  ->
%D    @ 1 @ 2 >-> .plabel= b 0
%D    @ 2 @ 3 >-> .plabel= b m
%D ))
%D (( C0 .tex= X   C1 .tex= 0   C2 .tex= A   C3 .tex= A
%D    @ 0 @ 1  -> @ 0 @ 2  ->
%D    @ 1 @ 2 >-> .plabel= b 0
%D    @ 2 @ 3 >-> .plabel= b 1
%D ))
%D (( E0 .tex= X   E1 .tex= A   E2 .tex= A   E3 .tex= 0
%D    @ 0 @ 1  -> @ 0 @ 2  ->
%D    @ 1 @ 2 >-> .plabel= b 1
%D    @ 2 @ 3  -> .plabel= b 0
%D ))
%D (( B0 .tex= A   B1 .tex= B   B2 .tex= 0   B3 .tex= Y
%D    @ 0 @ 1 ->> .plabel= a e
%D    @ 1 @ 2 ->> .plabel= a 0
%D    @ 1 @ 3  -> @ 2 @ 3  ->
%D ))
%D (( D0 .tex= B   D1 .tex= B   D2 .tex= 0   D3 .tex= Y
%D    @ 0 @ 1 ->> .plabel= a 1
%D    @ 1 @ 2 ->> .plabel= a 0
%D    @ 1 @ 3  -> @ 2 @ 3  ->
%D ))
%D (( F0 .tex= 0   F1 .tex= B   F2 .tex= B   F3 .tex= Y
%D    @ 0 @ 1  -> .plabel= a 0
%D    @ 1 @ 2 ->> .plabel= a 1
%D    @ 1 @ 3  -> @ 2 @ 3  ->
%D ))
%D enddiagram
%D
$\diag{monic-and-epi-lemmas}$



\newpage
% --------------------
% «ker-and-cok-as-inv»  (to ".ker-and-cok-as-inv")
% (s "ker and cok are inverse functions" "ker-and-cok-as-inv")
\myslide {ker and cok are inverse functions} {ker-and-cok-as-inv}

Theorem (Freyd's 2.11): if $a$ is a monic then $a \cong \ker \cok a$.

Corollary: $\ker$ and $\cok$ are inverse functions.

\ssk

Proof: choose $b$ such that $a \in \Kers b$. Then:

\msk

%D diagram 211
%D 2Dx     100   +30   +30
%D 2D  100 A'          C
%D 2D        v       ^ |
%D 2D         \     ^  |
%D 2D          v   /   |
%D 2D  +30       A     |
%D 2D          ^   \   |
%D 2D         /     v  |
%D 2D        ^       v v
%D 2D  +30 K           B
%D 2D
%D (( A' A >-> .plabel= a a
%D    A  C ->> .plabel= m \sm{c\,Ý\\\Coks"a}
%D    K  A >-> .plabel= m \sm{k\,Ý\\\Kers"c}
%D    A  B  -> .plabel= a b
%D    C  B  -> .plabel= r c\bsl"b
%D    A' K >-> sl_ .plabel= l a/k
%D    A' K <-< sl^ .plabel= r k/a
%D ))
%D enddiagram
%D
$\diag{211}$

\msk

The logical layer is:
%:
%:             cÝ\Coks"a         aÝ\Kers"b
%:             ----------        ----------
%:  kÝ\Kers"c    ac=0               ab=0      cÝ\Coks"a
%:  -----------------\kerrule"akc   --------------------\cokrule"acb
%:        Î(a/k)                       Î(c\bsl"b)
%:
%:        ^211a                          ^211b
%:
%:              kÝ\Kers"c                     
%:              ----------                    
%:                 kc=0                       
%:              ---------------------------   
%:  aÝ\Kers"b   kb=kc(c\bsl"b)=0(c\bsl"b)=0
%:  ----------------------------------------\kerrule"kab
%:        Î(k/a)
%:
%:        ^211c
%:
$$\ded{211a} \qquad \ded{211b}$$
$$\ded{211c}$$

% $ac=0 \funto Î(a/k)$
% $ab=0 \funto Î(c \bsl b)$
% $kc = 0 \funto kb = kc(c \bsl b) = 0 (c \bsl b) = 0 \funto Î(k/a)$



\bsk

Theorem (Freyd's 2.12): if $a$ is monic and epi then $a$ is an iso.

Proof: choose $b$ such that $a \in \Kers b$. Then:

\msk

%D diagram 212
%D 2Dx     100   +30   +30
%D 2D  100 A'          C
%D 2D        v       ^ |
%D 2D         \     ^  |
%D 2D          v   /   |
%D 2D  +30       A     |
%D 2D          ^   \   |
%D 2D         /     v  |
%D 2D        ^       v v
%D 2D  +30 K           B
%D 2D
%D (( C  .tex= 0
%D    K  .tex= A
%D    A' A >-> sl^ .plabel= a a
%D    A' A ->> sl_
%D    A  C ->> .plabel= m \sm{0\,Ý\\\Coks"a}
%D    K  A >-> .plabel= m \sm{1\,Ý\\\Kers"0}
%D    A  B ->> .plabel= a b
%D    C  B  -> .plabel= r 0
%D    A' K >-> sl_ .plabel= l a/1
%D    A' K <-< sl^ .plabel= r 1/a
%D ))
%D enddiagram
%D
$\diag{212}$



\newpage
% --------------------
% «intersecs-of-subs»  (to ".intersecs-of-subs")
% (s "Intersection of subobjects" "intersecs-of-subs")
\myslide {Intersection of subobjects} {intersecs-of-subs}

% (find-freydabcatspage (+ 26 37) "Theorem 2.13")
% (find-freydabcatspage (+ 26 38) "We shall prove a stronger property.")

Theorem (Freyd's 2.13): every pair of subobjects has a pullback.

Corollary: every pair of subobjects has an intersection.

Proof:

%D diagram 213
%D 2Dx     100   +50      +40     +40
%D 2D  100 X     
%D 2D        \
%D 2D         \
%D 2D          v
%D 2D  +25      A12 >--> A_2
%D 2D            v        v
%D 2D            |        |
%D 2D            v        v
%D 2D  +40      A_1 >---> A ----> C
%D 2D
%D (( A12 .tex= A_{12}
%D    A_1  A  >-> .plabel= b a_1
%D    A_2  A  >-> .plabel= l a_2
%D    A    C  ->> .plabel= b \sm{c\,Ý\\\Coks"a_1}
%D    A_2  C   -> .plabel= a a_2c
%D    A12 A_2 >-> .plabel= b \sm{p_2\,Ý\\\Kers"a_2c}
%D    A12 A_1 >-> .plabel= m \sm{p_1\,:=\\(p_2a_2)/a_1}
%D    X   A_1  -> .plabel= l x_1
%D    X   A_2  -> .plabel= a x_2
%D    X   A12  -> .PLABEL= _(0.72) \sm{x\,:=\\x_2/p_2}
%D ))
%D enddiagram
%D
$$\diag{213}$$

The logical layer is:

%:  p_2Ý\Kers"a_2c   cÝ\Coks"a_1
%:  ---------------  ------------
%:     p_2a_2c=0     a_1Ý\Kers"c
%:     -------------------------\kerrule{p_2a_2}{a_1}c
%:         Î(p_2a_2/a_1)
%:
%:         ^213a
%:
$$\ded{213a}$$

%:
%:                    cÝ\Coks"a_1
%:                    -----------
%:   x_2a_2=x_1a_1     a_1c=0
%:  ---------------   ---------
%:  x_2a_2c=x_1a_1c   x_1a_1c=0
%:  ---------------------------
%:        x_2a_2c=0              p_2Ý\Kers"a_2c
%:        --------------------------------------\kerrule{x_2}{p_2}{a_2c}
%:        Î(x_2/p_2)
%:
%:        ^213b
%:
%:
$$\ded{213b}$$

%:
%:   p_1a_1=p_2a_2      xp_2=x_2     
%:  ---------------  --------------  
%:  xp_1a_1=xp_2a_2  xp_2a_2=x_2a_2  
%:  -----------------------
%:      xp_1a_1=x_2a_2       x_2a_2=x_1a_1
%:      ----------------------------------
%:              xp_1a_1=x_1a_1              a_1\text{"monic}
%:              --------------------------------------------
%:                              xp_1=x_1
%:
%:                              ^213c
%:
$$\ded{213c}$$

To see that $x$ is unique use that $p_1$ and $p_2$ are monic.

The subobject $A_{12} \monicto A$ is the intersection of $A_1 \monicto A$ and $A_2 \monicto A$.

Notation: $A_1 Ì A_2 := A_{12}$.



\newpage
% --------------------
% «equalizers»  (to ".equalizers")
% (s "Difference kernels (a.k.a. equalizers)" "equalizers")
\myslide {Difference kernels (a.k.a. equalizers)} {equalizers}

(Freyd's 2.14)

% (find-es "xypic" "two-and-three")



We can construct the equalizer
$\Ker(x-y) \diagxyto/ >->/^p A \two/->`->/^x_y B$

using a product and a pullback of (split) monics:

%D diagram 2.14
%D 2Dx       100          +40     +30
%D 2D  100 \Ker(x-y) >-> A{}
%D 2D        v            v
%D 2D        |            |
%D 2D        v            v
%D 2D  +30  {}A >------> A×B ---> B
%D 2D                     |
%D 2D                     |
%D 2D                     v
%D 2D  +30                A
%D (( \Ker(x-y) A{}
%D        {}A   A×B  B
%D               A
%D    @ 0 @ 1 >-> .plabel= a p_2
%D    @ 0 @ 2 >-> .plabel= l p_1   @ 1 @ 3 >-> .plabel= r \ang{1,y}
%D    @ 2 @ 3 >-> .plabel= a \ang{1,x}  @ 3 @ 4 -> .plabel= a \pi_2
%D    @ 3 @ 5  -> .plabel= r \pi_1
%D    @ 0 relplace 8 7 \pbsymbol{7}
%D ))
%D enddiagram
%D
$$\diag{2.14}$$

Note that

$\ang{1,x}\pi_1 = 1$ \qquad $\ang{1,x}\pi_2 = x$

$\ang{1,y}\pi_1 = 1$ \qquad $\ang{1,y}\pi_2 = y$ 

so:

%:
%:      p_1\ang{1,x}=p_2\ang{1,y}             p_1\ang{1,x}=p_2\ang{1,y}         
%:  -----------------------------------   -----------------------------------   
%:  p_1\ang{1,x}\pi_1=p_2\ang{1,y}\pi_1   p_1\ang{1,x}\pi_2=p_2\ang{1,y}\pi_2   
%:  -----------------------------------   -----------------------------------   
%:           p_1=p_2                               p_1x=p_2y                      
%:
%:           ^2.14a                                ^2.14b
%:
$$\ded{2.14a} \qquad \ded{2.14b}$$

(How do I show that $\Ker(x-y) \to A$ is monic?)

\msk

Now we can construct arbitrary pullbacks (Freyd's 2.15):

%D diagram 2.15
%D 2Dx     100     +50    +20 +20
%D 2D  100                A
%D 2D                 
%D 2D  +20 K >---> A×B        C  
%D 2D
%D 2D  +20                B
%D 2D
%D (( K .tex= \begin{matrix}A×_{C}B:=\\\Ker(\pi_1f-\pi_2g)\end{matrix}
%D    K A×B >->
%D    A×B A -> .plabel= a \pi_1 A C -> .plabel= r f
%D    A×B B -> .plabel= b \pi_2 B C -> .plabel= r g
%D ))
%D enddiagram
%D
$$\diag{2.15}$$

% (find-freydabcatspage (+ 26 21) "Difference kernels and cokernels")


\newpage
% --------------------
% «images»  (to ".images")
% (s "Images" "images")
\myslide {Images} {images}

Lemma: in the diagram below $ac = 0$ iff $a$ factors through $s$.

%D diagram 2.16a
%D 2Dx     100   +30   +30
%D 2D  100 A
%D 2D
%D 2D  +30 S >-> B ->> C
%D 2D
%D (( A S  .> .plabel= l a/s
%D    A B  -> .plabel= r a
%D    S B >-> .plabel= b \sm{s\;Ý\\\Kers"c}
%D    B C ->> .plabel= b \sm{c\;Ý\\\Coks"s}
%D ))
%D enddiagram
%D
$$\diag{2.16a}$$

Theorem (Freyd's 2.16):

define the {\sl image of $A \ton{a} B$}, $\Im(a) \monicto B$,

as the kernel of the cokernel of $a$;

then $\Im(a)$ is the ``smallest subobject of $B$ through which $a$ factors'',

i.e., every factorization of $a$ through a subobject,

$A \to S \monicto B$, can be further factored as

$A \to \Im(a) \monicto S \monicto B$.

Here is the construction:

%D diagram 2.16b
%D 2Dx     100  +40     +40    +40
%D 2D  100 A ---------> B
%D 2D
%D 2D  +25    \Im(a)         \Cok(a)
%D 2D
%D 2D  +35      S            \Cok(m')
%D 2D
%D (( A B -> .plabel= a a
%D  # B \Cok(a) ->> .plabel= b \sm{c\;Ý\\\Coks"a}
%D    A \Im(a)             -> .plabel= a e=a/m
%D      \Im(a) B          >-> .plabel= a m
%D             B \Cok(a)  ->> .plabel= a c
%D    A   S                -> .plabel= b e'
%D        S    B          >-> .plabel= b m'
%D             B \Cok(m') ->> .plabel= b c'
%D  # A S -> S B .plabel= b s >-> B \Cok(s) ->>
%D    \Im(a) S >-> .plabel= m m/m'
%D    \Cok(a) \Cok(m') ->> .plabel= r c\bsl"c'
%D ))
%D enddiagram
%D
$$\diag{2.16b}$$

and here is its logical layer:

%:
%:  c'Ý\Coks"m'                         mÝ\Kers"c              
%:  -----------                         -----------            
%:   m'c'=0                              mc=0                  
%:  --------                            --------               
%:  e'm'c'=0                            mc(c\bsl"c')=0         
%:  --------                            --------               
%:   ac'=0     cÝ\Coks"a                 mc'=0     m'Ý\Kers"c' 
%:   ------------------\cokrule"ac{c'}   ---------------------\kerrule"m{m'}{c'}  
%:       Î(c\bsl"c')                         Î(m/m')           
%:                                                             
%:         ^2.16c                             ^2.16d           
%:
$$\ded{2.16c} \qquad \ded{2.16d}$$




\newpage
% --------------------
% «cok-epi-eq-0»  (to ".cok-epi-eq-0")
% (s "A map is epi iff its cok is 0" "cok-epi-eq-0")
\myslide {A map is epi iff its cok is 0} {cok-epi-eq-0}

We know from our first basic lemmas that if $a$ is epi then $0 \in \Coks(a)$:


%D diagram 2.17a
%D 2Dx     100   +20   +20
%D 2D  100 A ->> B ->> 0
%D 2D
%D 2D  +20             Y
%D 2D
%D (( A B ->> .plabel= a a  B 0 ->> .plabel= a 0
%D    B Y -> 0 Y ->
%D ))
%D enddiagram
%D
$$\diag{2.17a}$$

Now we can prove a converse for this ---

namely, that if $0 \in \Coks a$ then $a$ is epi (Freyd's 2.17).

This needs a big construction, with a big logical layer:

%:>->\monicto 
%:->>\epito 

%D diagram 2.17b
%D 2Dx     100  +30     +30    +30
%D 2D  100                     C
%D 2D
%D 2D  +40 A ---------> B    \Cok(a)
%D 2D
%D 2D  +15    \Im(a)
%D 2D
%D 2D  +25   \Ker(x-y)
%D 2D
%D (( A C ->
%D    B C -> sl^ .plabel= a x
%D    B C -> sl_ .plabel= b y
%D    A B -> .plabel= a a
%D    B \Cok(a) ->>
%D    A    \Im(a) ->    \Im(a) B >->
%D    A \Ker(x-y) -> \Ker(x-y) B >->
%D    \Im(a) \Ker(x-y) >->
%D ))
%D enddiagram
%:
%:   (B->>\Cok(a))=0
%:   ---------------
%:   (\Im(a)>->B)=1
%:  -----------------
%:  (\Ker(x-y)>->B)=1
%:  -----------------
%:        x=y
%:  -----------------
%:  (A->B)\text{"epi}
%:
%:  ^2.17c
%:
$$\cdiag{2.17b} \qquad \cded{2.17c}$$


\newpage
% --------------------
% «unique-factorization»  (to ".unique-factorization")
% (s "Unique factorization" "unique-factorization")
\myslide {Unique factorization} {unique-factorization}

Theorem (Freyd's 2.19):

$\Im(a) \cong \Coim(a)$,

and any epi-monic factorization of $A \ton{a} B$ is

isomorphic to $\Im(a)$.


%D diagram 2.19a
%D 2Dx     100   +30     +30
%D 2D  100     \Coim(a)
%D 2D
%D 2D  +20 A             B
%D 2D
%D 2D  +20      \Im(a)
%D 2D
%D (( A \Coim(a) ->>  \Coim(a) B >->
%D    A \Im(a)   ->>  \Im(a)   B >->
%D    \Coim(a) \Im(a) ->> sl_
%D    \Coim(a) \Im(a) >-> sl^
%D ))
%D enddiagram
%D
$$\diag{2.19a}$$

%D diagram 2.19b
%D 2Dx     100   +30     +30
%D 2D  100     \Coim(a)
%D 2D
%D 2D  +20 A     X       B
%D 2D
%D 2D  +20      \Im(a)
%D 2D
%D (( A \Coim(a) ->>  \Coim(a) B >->
%D    A \Im(a)   ->>  \Im(a)   B >->
%D    A X        ->>  X        B >->
%D    \Coim(a) X ->>
%D    X   \Im(a) >->
%D ))
%D enddiagram
%D
$$\diag{2.19b}$$



% (find-freydabcatspage (+ 26 44) "Unique factorization")







%

\end{document}

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