|
Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (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}
% Local Variables:
% coding: raw-text-unix
% ee-anchor-format: "«%s»"
% End: