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Warning: this is an htmlized version!
The original is here, and the conversion rules are here. |
% (find-LATEX "2020maclane-moerdijk.tex")
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% file:///home/edrx/LATEX/2020maclane-moerdijk.pdf
% file:///tmp/2020maclane-moerdijk.pdf
% file:///tmp/pen/2020maclane-moerdijk.pdf
% http://angg.twu.net/LATEX/2020maclane-moerdijk.pdf
% (find-LATEX "2019.mk")
% «.title» (to "title")
% «.Set-C-op» (to "Set-C-op")
% «.yoneda» (to "yoneda")
% «.Omega-in-presheaf» (to "Omega-in-presheaf")
% «.sieve-on-C» (to "sieve-on-C")
% «.cat-of-elements» (to "cat-of-elements")
% «.sieves-and-sheaves» (to "sieves-and-sheaves")
% «.O-Bottle» (to "O-Bottle")
% «.OO-House» (to "OO-House")
% «.sieves» (to "sieves")
% «.top-sheaves-in-my-notation» (to "top-sheaves-in-my-notation")
% «.sheaves-on-a-site» (to "sheaves-on-a-site")
% «.LT-subsumes-groth» (to "LT-subsumes-groth")
% «.localic-topoi» (to "localic-topoi")
% «.spaces-from-locales» (to "spaces-from-locales")
% «.5._localic_topoi» (to "5._localic_topoi")
\documentclass[oneside,12pt]{article}
\usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\usepackage{indentfirst}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage[x11names,svgnames]{xcolor} % (find-es "tex" "xcolor")
%\usepackage{colorweb} % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (find-dn6 "preamble6.lua" "preamble0")
\usepackage{proof} % For derivation trees ("%:" lines)
\input diagxy % For 2D diagrams ("%D" lines)
\xyoption{curve} % For the ".curve=" feature in 2D diagrams
%
\usepackage{edrx15} % (find-LATEX "edrx15.sty")
\input edrxaccents.tex % (find-LATEX "edrxaccents.tex")
\input edrxchars.tex % (find-LATEX "edrxchars.tex")
\input edrxheadfoot.tex % (find-LATEX "edrxheadfoot.tex")
\input edrxgac2.tex % (find-LATEX "edrxgac2.tex")
\input 2017planar-has-defs.tex % (find-LATEX "2017planar-has-defs.tex")
%
\usepackage[backend=biber,
style=alphabetic]{biblatex} % (find-es "tex" "biber")
%\addbibresource{catsem-u.bib} % (find-LATEX "catsem-u.bib")
%\addbibresource{2019notes-yoneda.bib} % (find-LATEX "2019notes-yoneda.bib")
\addbibresource{catsem-slides.bib} % (find-LATEX "catsem-slides.bib")
%
% (find-es "tex" "geometry")
\begin{document}
\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"} % (find-LATEX "dednat6load.lua")
\long\def\ColorRed #1{{\color{Red1}#1}}
\long\def\ColorViolet#1{{\color{MagentaVioletLight}#1}}
\long\def\ColorViolet#1{{\color{Violet!50!black}#1}}
\long\def\ColorGreen #1{{\color{SpringDarkHard}#1}}
\long\def\ColorGreen #1{{\color{SpringGreen4}#1}}
\long\def\ColorGreen #1{{\color{SpringGreenDark}#1}}
\long\def\ColorGray #1{{\color{GrayLight}#1}}
\long\def\ColorGray #1{{\color{black!30!white}#1}}
%\def\Sets {\mathbf{Sets}}
\def\Nat {\text{Nat}}
\def\phop {{}^{\phantom{\op}}}
\def\Cop {{\catC^\op}}
\def\SetsCop {\Sets^{\catC^\op}}
\def\hatC {{\widehat\catC}}
\def\OXop {\Opens(X)^\op}
\def\SetsOXop{\Sets^{\Opens(X)^\op}}
\def\univ {\mathsf{univ}}
\def\Spaces {\mathbf{Spaces}}
\def\Locales {\mathbf{Locales}}
\def\Frames {\mathbf{Frames}}
\def\Top {\mathbf{Top}}
\def\Bund {\mathbf{Bund}}
\def\Loc {\mathrm{Loc}}
\def\Ker {\operatorname{Ker}}
\def\Pt {\operatorname{pt}}
\def\Sh {\operatorname{Sh}}
\def\PSh {\operatorname{PSh}}
\def\acz {\setlength{\arraycolsep}{0pt}}
% _____ _ _ _
% |_ _(_) |_| | ___
% | | | | __| |/ _ \
% | | | | |_| | __/
% |_| |_|\__|_|\___|
%
% «title» (to ".title")
{\setlength{\parindent}{0em}
\footnotesize
Notes on MacLane and Moerdijk's
``Sheaves in Geometry and Logic - A First Introduction to Topos Theory'' (1994)
\url{https://www.springer.com/gp/book/9780387977102}
\ssk
These notes are at:
\url{http://angg.twu.net/LATEX/2020maclane-moerdijk.pdf}
\ssk
See:
\url{http://angg.twu.net/LATEX/2020favorite-conventions.pdf}
\url{http://angg.twu.net/math-b.html\#favorite-conventions}
I wrote these notes mostly to test if the conventions above
are good enough.
}
\section*{1. Categories of functors}
% ____ _ /\ ____ /\
% / ___| ___| |_ |/\| / ___| |/\| ___ _ __
% \___ \ / _ \ __| | | / _ \| '_ \
% ___) | __/ |_ | |___ | (_) | |_) |
% |____/ \___|\__| \____| \___/| .__/
% |_|
%
% «Set-C-op» (to ".Set-C-op")
% (find-books "__cats/__cats.el" "maclane-moerdijk")
% (find-maclanemoerdijkpage (+ 11 25) "(viii)")
(Page 25):
(viii) $\Set^{\catC^\op}$, where...
%D diagram p25-a
%D 2Dx 100 +25 +20 +25 +25 +15 +25
%D 2D 100 A0 A1 C0 C1 E0 F0 F1
%D 2D
%D 2D +20 A2 A3 C2 C3 E1 F2 F3
%D 2D
%D 2D +12 B0 D0
%D 2D +8 B1 B2 D1 D2 G0 G1
%D 2D
%D ren A0 A1 A2 A3 B0 B1 B2 ==> C PC D PD \catC \phop\catC^\op \Set
%D ren C0 C1 C2 C3 D0 D1 D2 ==> C P'C D P'D \catC \phop\catC^\op \Set
%D ren E0 E1 F0 F1 F2 F3 G0 G1 ==> C D PC P'C PD P'D P P'
%D
%D (( A0 A1 |->
%D A0 A2 <- .plabel= l f
%D A1 A3 -> .plabel= r Pf
%D A2 A3 |->
%D A0 A3 harrownodes nil 20 nil |->
%D B0 place
%D B1 B2 -> .plabel= a P
%D
%D C0 C1 |->
%D C0 C2 <- .plabel= l f
%D C1 C3 -> .plabel= r P'f
%D C2 C3 |->
%D C0 C3 harrownodes nil 20 nil |->
%D D0 place
%D D1 D2 -> .plabel= a P'
%D
%D E0 E1 <- .plabel= l f
%D F0 F1 -> .plabel= a θ_C
%D F0 F2 -> .plabel= l Pf
%D F1 F3 -> .plabel= r P'f
%D F2 F3 -> .plabel= a θ_D
%D G0 G1 -> .plabel= a θ
%D ))
%D enddiagram
%D
$$\pu
\diag{p25-a}
$$
\bsk
%D diagram p25-b
%D 2Dx 100 +25 +25 +40
%D 2D 100 A0 A1 C0
%D 2D
%D 2D +20 A2 A3 C2 C3
%D 2D
%D 2D +20 A4 A5 C4 C5
%D 2D
%D 2D +12 B0
%D 2D +8 B1 B2
%D 2D
%D ren A0 A1 A2 A3 A4 A5 ==> C PC D PD E PE
%D ren B0 B1 B2 ==> \catC \phop\catC^\op \Set
%D ren C0 C2 C4 ==> x x·f (x·f)·g
%D ren C3 C5 ==> =x|f =x·(f∘g)
%D
%D (( A0 A1 |->
%D A0 A2 <- .plabel= l f
%D A1 A3 -> .plabel= r Pf
%D A0 A3 harrownodes nil 20 nil |->
%D A2 A3 |->
%D A2 A4 <- .plabel= l g
%D A3 A5 -> .plabel= r Pg
%D A2 A5 harrownodes nil 20 nil |->
%D A4 A5 |->
%D B0 place
%D B1 B2 -> .plabel= a P
%D
%D C0 C2 |-> C2 C4 |->
%D C3 place C5 place
%D ))
%D enddiagram
%D
$$\pu
\diag{p25-b}
$$
\newpage
% __ __ _
% \ \ / /__ _ __ ___ __| | __ _
% \ V / _ \| '_ \ / _ \/ _` |/ _` |
% | | (_) | | | | __/ (_| | (_| |
% |_|\___/|_| |_|\___|\__,_|\__,_|
%
% «yoneda» (to ".yoneda")
% (mmop 3 "yoneda")
% (mmoa "yoneda")
% (find-books "__cats/__cats.el" "maclane-moerdijk")
% (find-maclanemoerdijkpage (+ 11 26) "Each object C of C gives rise")
(Page 26):
Each object $C$ of $\catC$ gives rise to a presheaf $𝐛y(C)$ on
$\catC$...
%D diagram p26-a
%D 2Dx 100 +30 +40 +35 +20 +20
%D 2D 100 A0 A1 C0 D0 E0 E1
%D 2D
%D 2D +20 A2 A3 C1 D1 E2
%D 2D
%D 2D +12 B0
%D 2D +8 B1 B2
%D 2D
%D ren A0 A1 A2 A3 ==> D 𝐛y(C)(D) D' 𝐛y(C)(D')
%D ren B0 B1 B2 ==> \catC \phop\catC^\op \Set
%D ren C0 C1 ==> =\Hom_\catC(D,C) =\Hom_\catC(D',C)
%D ren D0 D1 ==> u u∘α
%D ren E0 E1 E2 ==> D C D'
%D
%D (( A0 A1 |->
%D A0 A2 <- .plabel= l α
%D A1 A3 -> .plabel= r 𝐛y(C)(α)
%D A2 A3 |->
%D A0 A3 harrownodes nil 20 nil |->
%D B0 place
%D B1 B2 -> .plabel= a 𝐛y(C)
%D
%D C0 C1 -> .plabel= r (∘α)
%D D0 D1 |->
%D E0 E1 -> .plabel= a u
%D E0 E2 <- .plabel= l α
%D E1 E2 <- .plabel= r u∘α
%D ))
%D enddiagram
%D
$$\pu
\diag{p26-a}
$$
%D diagram p26-b
%D 2Dx 100 +25 +35 +50 +35 +20 +20
%D 2D 100 A0 A1 C0 D0 E0 F0 F1
%D 2D
%D 2D +20 A2 A3 C1 D1 E1 F2
%D 2D
%D 2D +12 B0
%D 2D +8 B1 B2
%D 2D
%D ren A0 A1 A2 A3 ==> C_1 𝐛y(C_1) C_2 𝐛y(C_2)
%D ren B0 B1 B2 ==> \catC \catC \Set^{\catC^\op}
%D ren C0 C1 ==> =\Hom_\catC(-,C_1) =\Hom_\catC(-,C_2)
%D ren D0 D1 ==> \Hom_\catC(D,C_1) \Hom_\catC(D,C_2)
%D ren E0 E1 ==> v f∘v
%D ren F0 F1 F2 ==> D C_1 C_2
%D
%D (( A0 A1 |->
%D A0 A2 -> .plabel= l f
%D A1 A3 -> .plabel= r 𝐛y(f)
%D A2 A3 |->
%D A0 A3 harrownodes nil 20 nil |->
%D B0 place
%D B1 B2 -> .plabel= a 𝐛y
%D C0 C1 -> .plabel= r (f∘)
%D D0 D1 -> .plabel= r (f∘)
%D E0 E1 |->
%D F0 F1 -> .plabel= a v
%D F1 F2 -> .plabel= r f
%D F0 F2 -> .plabel= l f∘v
%D ))
%D enddiagram
%D
$$\pu
\diag{p26-b}
$$
%
%D diagram p26-yoneda
%D 2Dx 100 +45
%D 2D 100 A1
%D 2D
%D 2D +20 A2 A3
%D 2D
%D 2D +20 A4 A5
%D 2D
%D 2D +20 B0 B1
%D 2D
%D 2D +15 C0 C1
%D 2D
%D 2D +20 C2
%D 2D
%D 2D +20
%D 2D
%D ren A1 A2 A3 A4 A5 ==> 1 C PC D PD
%D ren B0 B1 ==> \phop\catC^\op \Set
%D ren C0 C1 C2 ==> \Hom_\catC(-,C) \Set(1,P-) P
%D
%D (( A1 A3 -> .plabel= r \sm{\nameof{θ_α}=\\\nameof{α_C(1_C)}}
%D A2 A3 |->
%D A2 A4 <-
%D A3 A5 ->
%D A2 A5 harrownodes nil 20 nil |->
%D A4 A5 |->
%D B0 relplace 0 -8 \catC
%D B0 B1 -> .plabel= a P
%D C0 C1 ->
%D C1 C2 <->
%D C0 C2 -> .plabel= b α
%D C0 relplace -35 0 𝐛y(C)=
%D ))
%D enddiagram
%D
$$\pu
\diag{p26-yoneda}
$$
\newpage
(Page 37):
% «Omega-in-presheaf» (to ".Omega-in-presheaf")
% (mmop 3 "Omega-in-presheaf")
% (mmoa "Omega-in-presheaf")
% (find-books "__cats/__cats.el" "maclane-moerdijk")
% (find-maclanemoerdijkpage (+ 11 37) "For an arbitrary presheaf category hatC")
For an arbitrary presheaf category $\hatC = \SetsCop$, if there is a
subobject classifier $Ω$, it must, in particular, classify the
subobjects of each representable presheaf $𝐛yC = \Hom_\catC(-,C): \Cop
→ \Sets$. Therefore,
%
$$\begin{array}{rcl}
\Sub_\hatC(\Hom_\catC(-,C)) & ≅ & \Hom_\hatC(\Hom_\catC(-,C),Ω) \\
& ≅ & \Nat(\Hom_\catC(-,C),Ω) \\
\end{array}
$$
%
By the Yoneda Lemma [see \S1(6) above], the set on the right is (up to
isomorphism) $Ω(C)$. Thus the subobject classifier $Ω$, if it exists,
must be the functor $Ω: \Cop → \Sets$ with object function
%
$$\begin{array}{rcl}
Ω(C) & = & \Sub_\hatC(\Hom_\catC(-,C)) \\
& = & \setofst{S}{S \text{ is a subfunctor of }\Hom_\catC(-,C)}, \\
\end{array}
$$
%
and with a suitable mapping function.
%D diagram p37-1
%D 2Dx 100 +50 +50 +55
%D 2D 100 A1 D1
%D 2D
%D 2D +20 A2 A3 D2 D3
%D 2D
%D 2D +20 A4 A5 D4 D5
%D 2D
%D 2D +20 B0 B1 E0 E1
%D 2D
%D 2D +15 C0 C1 F0 F1
%D 2D
%D 2D +20 C2 F2
%D 2D
%D ren A1 A2 A3 A4 A5 ==> 1 C Ω(C) D Ω(D)
%D ren B0 B1 C0 C1 C2 ==> \phop\Cop \Sets \Hom_\catC(-,C) \Set(1,Ω(-)) Ω(-)=Ω
%D ren D1 D2 D3 D4 D5 ==> 1 Ω \Sub(Ω) X \Sub(X)
%D ren E0 E1 F0 F1 F2 ==> \phop\hatC^\op \Sets \Hom_\hatC(-,Ω) \Sets(1,\Sub(-)) \Sub(-)
%D
%D (( A1 A3 ->
%D A2 A3 |->
%D A2 A4 <-
%D A3 A5 ->
%D A2 A5 harrownodes nil 20 nil |->
%D A4 A5 |->
%D B0 B1 -> .plabel= a Ω
%D C0 C1 ->
%D C1 C2 <->
%D C0 C2 ->
%D B0 relplace 0 -8 \catC
%D C0 relplace -30 0 𝐛yC=
%D ))
%D (( D1 D3 -> .plabel= r \sm{\nameof{⊤}\\(\univ)}
%D D2 D3 |->
%D D2 D4 <-
%D D3 D5 ->
%D D2 D5 harrownodes nil 20 nil |->
%D D4 D5 |->
%D E0 E1 -> .plabel= a \Sub
%D F0 F1 <->
%D F1 F2 <->
%D F0 F2 <->
%D E0 relplace 0 -8 \hatC
%D ))
%D enddiagram
%D
$$\pu
\diag{p37-1}
$$
\bsk
$$\begin{array}{rcl}
Ω(C) & ≅ & \Sets(1,Ω(C)) \\
& ≅ & \Hom_\hatC(𝐛yC,Ω) \\
& ≅ & \Sub_\hatC(𝐛yC) \\
& = & \Sub_\hatC(\Hom_\catC(-,C)) \\
& = & \setofst{S}{S \text{ is a subfunctor of }\Hom_\catC(-,C)}, \\
\end{array}
$$
\newpage
% «sieve-on-C» (to ".sieve-on-C")
% (mmop 4 "sieve-on-C")
% (mmoa "sieve-on-C")
% (find-maclanemoerdijkpage (+ 11 38) "Sieve on C =")
(Page 38:)
A sieve on $C$ is...
\def\cod{\mathop{\text{cod}}}
%\def\thinpsm #1{\setlength{\arraycolsep}{0pt}\psm{#1}}
%\def\thinpsm #1{\setlength{\arraycolsep}{-10pt}\psm{#1}}
\def\thinpsm #1{\psm{#1}}
\def\thinpsmtwo #1#2#3{\thinpsm{  \\ #2 \!\!\! &↑ \\  }}
\def\thinpsmthree#1#2#3#4#5{\thinpsm{  \\ #2 \!\!\! &↑ \\  \\ #4 \!\!\! &↑ \\  }}
$$\begin{array}{rcl}
t_C &=& \setofst{g}{\cod g = C} \\
&=& \setofst{ \thinpsmtwo{C}{g}{D} }{\cod g = C} \\
S \text{ is a sieve on } C
&=& (S⊂t_C \text{ and } S \text{ is downward-closed} ) \\
&=& (S⊂t_C \text{ and }
∀\thinpsmthree{C}{g}{D}{h}{D'}.
\pmat{g∈S \\ ↓ \\ g∘h∈S}
) \\
Ω(C) &=& \setofst{S⊂t_C}{S \text{ is a sieve on } C} \\
&=& \setofst{S⊂t_C}{S \text{ is downward-closed}} \\
\end{array}
$$
%D diagram ??
%D 2Dx 100 +30 +30 +55
%D 2D 100 A0 |-> A1 C0 D0
%D 2D | | | |
%D 2D | |-> | | |
%D 2D | | | |
%D 2D +25 A2 |-> A3 C1 D1
%D 2D
%D 2D +13 B0'
%D 2D +7 B0 --> B1
%D 2D
%D ren A0 A1 A2 A3 ==> C Ω(C) B Ω(B)
%D ren B0' B0 B1 ==> \catC \phantom{{}^\op}\catC^\op \Set
%D ren C0 C1 ==> S g^*S=S·g
%D ren D0 D1 ==> \inOmC \inOmB
%D
%D (( A0 A1 |->
%D A0 A2 <- .plabel= l g
%D A1 A3 -> .plabel= r g^*
%D A2 A3 |->
%D A0 A3 harrownodes nil 20 nil |->
%D
%D B0' place
%D B0 B1 -> .plabel= a Ω
%D
%D C0 C1 |->
%D D0 D1 |->
%D ))
%D enddiagram
%D
$$\pu
\def\inOmC{\setofst{ \thinpsmtwo{C}{g}{D} }{ g∈S }}
\def\inOmB{\setofst{ \thinpsmtwo{B}{h}{D} }{ g∘h∈S }}
\diag{??}
$$
\newpage
% «cat-of-elements» (to ".cat-of-elements")
% (find-books "__cats/__cats.el" "maclane-moerdijk")
% (find-maclanemoerdijkpage (+ 11 41) "Given P, the index category J")
(Page 41):
Given $P$, the index category $J$ which serves to prove the
proposition is the so-called {\sl category of elements} of $P$,
denoted by $∫_\catC P$ or, more briefly, $∫P$. Its objects are all
pairs $(C,p)$ where $C$ is an object of $\catC$ and $p$ is an element
$p∈P(C)$. Its morphisms $(C',p)→(C,p)$ are those morphisms $u:C'→C$ of
$\catC$ for which $pu=p'$; in other words, $u$ must take the chosen
element $p$ in $P(C)$ ``back'' into $p'$ in $P(C')$:
%
$$(C',p)→(C,p) \qquad \text{by $u:C'→C$ with $pu=p'$.}$$
These morphisms ar composed by composing the underlying arrows $u$ of
$\catC$. This category has an evident projection functor
%
$$π_P: ∫_\catC P → \catC, \qquad (C,p) \mapsto C.
$$
%D diagram ??
%D 2Dx 100 +30 +65 +35 +20 +20 +20
%D 2D 100 A1
%D 2D
%D 2D +20 A2 A3 C0 C1
%D 2D
%D 2D +20 A4 A5 C2 C3
%D 2D
%D 2D +20 B0 B1 D0 D1
%D 2D
%D ren A1 A2 A3 A4 A5 ==> 1 C P(C) C' P(C')
%D ren B0 B1 ==> \phop\Cop \Sets
%D ren C0 C1 C2 C3 ==> (C,p) C (C',p') C'
%D ren D0 D1 ==> ∫_{\catC}P=∫P \catC
%D
%D (( A1 A3 -> .plabel= r \nameof{p}
%D A2 A3 |->
%D A2 A4 <- .plabel= l u
%D A3 A5 -> .plabel= r P(u)
%D A4 A5 |->
%D A2 A5 harrownodes nil 20 nil |->
%D A1 A5 -> .slide= 25pt .plabel= r \sm{\nameof{pu}:=\\\nameof{P(u)(p)}\;=\\\nameof{p'}}
%D B0 relplace 0 -8 \catC
%D B0 B1 -> .plabel= a P
%D ))
%D (( C0 C1 |->
%D C0 C2 <- .plabel= l u
%D C1 C3 <- .plabel= r u
%D C2 C3 |->
%D C0 C3 harrownodes nil 20 nil |->
%D D0 D1 -> .plabel= a π_P
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
\newpage
(Page 41):
Colimits over the category of elements can be used to construct a pair
of adjoint functors which will have many uses, as follows.
{\bf Theorem 2.} {\sl If $A: \catC→\calE$ is a functor from a small
category $\catC$ to a cocomplete category $\calE$, the functor $R$
from $\calE$ to presheaves given by
%
$$R(E): C \mapsto \Hom_\calE(A(C),E)$$
%
has a left adjoint $L:\SetsCop → \calE$ defined for each presheaf
$P$ in $\SetsCop$ as the colimit
%
$$L(P) = \Colim(∫ P \ton{π_P} \catC \ton{A} \calE).$$
}
\bsk
Here's how I found the type and a precise definition of $R_1$...
(It's too big! How do other people do this?)
%
\def\HomE{\Hom_\calE}
%
%D diagram ??
%D 2Dx 100 +25 +40 +35 +40 +20 +30
%D 2D 100 A0 - A1 C0 - C1 E0 F0 - F1
%D 2D | | | | | | |
%D 2D +20 A2 - A3 C2 - C3 E1 F2 - F3
%D 2D |
%D 2D +20 A4
%D 2D +7 D0' G0'
%D 2D +8 B0 - B1 D0 - D1 G0 - G1
%D 2D
%D ren A0 A1 A2 A3 A4 ==> C A(C) C' A(C') E
%D ren B0 B1 ==> \catC \calE
%D ren D0' D0 D1 ==> \catC \phop\Cop \Sets
%D ren C0 C1 C2 C3 ==> C \HomE(A(C),E) C' \HomE(A(C'),E)
%D ren E0 E1 ==> f{∘}Ag f
%D ren F0 F1 F2 F3 ==> C R(E)(C) C' R(E)(C')
%D ren G0' G0 G1 ==> \catC \phop\Cop \Sets
%D
%D (( A0 A1 |->
%D A0 A2 -> .plabel= l g
%D A1 A3 -> .plabel= r Ag
%D A0 A3 harrownodes nil 20 nil |->
%D A2 A3 |->
%D A3 A4 -> .plabel= r f
%D A1 A4 -> .slide= 20pt .plabel= r f∘Ag
%D B0 B1 -> .plabel= a A
%D
%D C0 C1 |->
%D C0 C2 -> .plabel= l g
%D C1 C3 <- .plabel= r (∘Ag)
%D C2 C3 |->
%D C0 C3 harrownodes nil 20 nil |->
%D D0' place
%D D0 D1 -> .plabel= a R(E)
%D
%D E0 E1 <-|
%D
%D F0 F1 |->
%D F0 F2 -> .plabel= l g
%D F1 F3 <- .plabel= r (∘Ag)
%D F2 F3 |->
%D F0 F3 harrownodes nil 20 nil |->
%D G0' place
%D G0 G1 -> .plabel= a R(E)
%D
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
%D diagram ??
%D 2Dx 100 +30 +45 +30 +30 +30 +25 +40 +25
%D 2D 100 A0 B0 - B1 D0 - D1 E0 F0 - F1
%D 2D | | | | | | | |
%D 2D +20 A1 B2 - B3 D2 - D3 E1 F2 - F3
%D 2D
%D 2D +15 G0 - G1 H0
%D 2D | | |
%D 2D +20 C0 - C1 G2 - G3 H1
%D 2D
%D ren A0 A1 ==> C C'
%D ren B0 B1 B2 B3 ==> \HomE(A(C),E) \HomE(A(C),E') \HomE(A(C'),E) \HomE(A(C'),E')
%D ren C0 C1 ==> E E'
%D ren D0 D1 D2 D3 ==> f{∘}AG h{∘}f{∘}Ag f h{∘}f
%D ren E0 E1 ==> C C'
%D ren F0 F1 F2 F3 ==> R(E)(C) R(E')(C) R(E)(C') R(E')(E')
%D ren G0 G1 G2 G3 ==> R(E) R(E') E E'
%D ren H0 H1 ==> \SetsCop \calE
%D
%D (( A0 A1 -> .plabel= a g
%D B0 B1 -> .plabel= a (h∘)
%D B0 B2 <- .plabel= l (∘Ag)
%D B1 B3 <- .plabel= r (∘Ag)
%D B2 B3 -> .plabel= a (h∘)
%D C0 C1 -> .plabel= a h
%D
%D D0 D1 |->
%D D0 D2 <-|
%D D1 D3 <-|
%D D2 D3 |->
%D
%D E0 E1 <- .plabel= l g
%D
%D F0 F1 -> .plabel= a (h∘)
%D F0 F2 <- .plabel= l (∘Ag)
%D F1 F3 <- .plabel= r (∘Ag)
%D F2 F3 -> .plabel= a (h∘)
%D
%D G0 G1 -> .plabel= a Rh
%D G0 G2 <-|
%D G1 G3 <-|
%D G2 G3 -> .plabel= b h
%D G0 G3 varrownodes nil 15 nil <-|
%D
%D H0 H1 <- .plabel= r R
%D ))
%D enddiagram
%D
$$\pu
\def\HomE{\Hom_\calE}
\def\HomE{\calE}
\diag{??}
$$
\newpage
% «sieves» (to ".sieves")
% (find-books "__cats/__cats.el" "maclane-moerdijk")
% (find-maclanemoerdijkpage (+ 11 24) "I. Categories of functors")
% (find-maclanemoerdijkpage (+ 11 37) "a sieve on C")
% http://angg.twu.net/MINICATS/sheaves_for_children__1_to_7.pdf
\cite{MacLaneMoerdijk}, (Page 37):
Given an object $C$ in the category $\catC$, a {\sl sieve} on $C$ is a
set $S$ of arrows with codomain $C$ such that $f∈S$ implies $f∘h∈S$.
A sieve $S$ can be seen as a subfunctor $S: \catC^\op → \Set$ of
$\Hom(-,C): \catC^\op → \Set$ --- or, more explicitly, as a natural
transformation $ι: S → \Hom(-,C)$ such that each $ι_A$ is an
inclusion.
%
%D diagram sieve-on-C
%D 2Dx 100 +20 +35 +20 +40 +35 +25
%D 2D 100 C
%D 2D
%D 2D +20 B A0 B0 B1 D0 D1
%D 2D
%D 2D +20 A A1 B2 B3 D2 D3
%D 2D
%D 2D +15 C0 C1
%D 2D
%D ren A0 B0 B1 ==> B S(B) \Hom(B,C)
%D ren A1 B2 B3 ==> A S(A) \Hom(A,C)
%D ren C0 C1 ==> S(-) \Hom(-,C)
%D ren D0 D1 ==> f f
%D ren D2 D3 ==> f∘h f∘h
%D
%D (( B C -> .plabel= l f∈S
%D A B -> .plabel= l h
%D A C -> .plabel= r f∘h∈S
%D C relplace -7 14 \dashrightarrow
%D
%D A0 A1 <- .plabel= l h
%D B0 B1 `-> .plabel= a ι_B
%D B0 B2 -> .plabel= l ∘h
%D B1 B3 -> .plabel= r ∘h
%D B2 B3 `-> .plabel= a ι_A
%D C0 C1 `-> .plabel= a ι
%D
%D D0 D1 |-> D0 D2 |-> D1 D3 |-> D2 D3 |->
%D ))
%D enddiagram
%D
$$\pu
\diag{sieve-on-C}
$$
\newpage
% (find-maclanemoerdijkpage (+ 11 70) "a sieve S on U")
(Page 70):
A sieve $S$ on an object $U$ of $\Opens(X)$ is a subfunctor of
$\Hom(-,U)$:
%
%D diagram sieve-on-a-topology
%D 2Dx 100 +20 +35 +20 +40 +35 +25
%D 2D 100 X
%D 2D
%D 2D +20 U
%D 2D
%D 2D +20 V A0 B0 B1 D0 D1
%D 2D
%D 2D +20 W A1 B2 B3 D2 D3
%D 2D
%D 2D +15 C0 C1
%D 2D
%D ren A0 B0 B1 ==> V S(V) \Hom(V,U)
%D ren A1 B2 B3 ==> W S(W) \Hom(W,U)
%D ren C0 C1 ==> S(-) \Hom(-,U)
%D ren D0 D1 ==> ι_{V,U} ι_{V,U}
%D ren D2 D3 ==> ι_{W,U} ι_{W,U}
%D
%D (( U X ->
%D V U -> .plabel= l ι_{V,U}∈S
%D W V -> .plabel= l ι_{W,V}
%D W U -> .plabel= r ι_{W,U}∈S
%D U relplace -7 14 \dashrightarrow
%D
%D A0 A1 <- .plabel= l ι_{W,V}
%D B0 B1 `-> .plabel= a ι_V
%D B0 B2 -> .plabel= l ∘ι_{W,V}
%D B1 B3 -> .plabel= r ∘ι_{W,V}
%D B2 B3 `-> .plabel= a ι_W
%D C0 C1 `-> .plabel= a ι
%D
%D D0 D1 |-> D0 D2 |-> D1 D3 |-> D2 D3 |->
%D ))
%D enddiagram
%D
$$\pu
\diag{sieve-on-a-topology}
$$
% «sieves-and-sheaves» (to ".sieves-and-sheaves")
% (find-maclanemoerdijkpage (+ 11 69) "2. Sieves and Sheaves")
% (find-maclanemoerdijkpage (+ 11 70) "a sieve S on U")
\subsection*{2. Sieves and Sheaves}
(From pages 69--70):
\begin{quotation}
On any space $X$, each open set $U$ determines a presheaf $\Hom(-,U)$ defined, for each open set $V$, by
%
$$\Hom(V,U) = \begin{cases}
1 & \text{if $V⊂U$,} \\
∅ & \text{otherwise.} \\
\end{cases}
$$
This presheaf is clearly a sheaf; it is the representable presheaf
$𝐛y(U) = \Hom(-,U)$ on the category $\Opens(X)$. Recall from section
I.4 that a {\sl sieve} $S$ on $U$ in this category is defined to be a
subfunctor of $\Hom(-,U)$. Replacing the sieve $S$ by the set (call it
$S$ again) of all those $V⊂U$ with $SV=1$, we may also describe a
sieve on $U$ as a subset $S⊂\Opens(U)$ of objects such that $V_0 ⊂ V ∈
S$ implies $V_0 ∈ S$. Each indexed family $\setofst{V_i⊂U}{i∈I}$ of
subsets of $U$ generates (= ``spans'') a sieve $S$ on $U$; namely, the
set $S$ consisting of all those open $V$ with $V ⊆ V_i$ for some $i$;
in particular, each $V_0 ⊂ U$ determines a {\sl principal sieve}
$(V_0)$ on $U$, consisting of all $V$ with $V⊆V_0$. It is not
difficult to see that a sieve $S$ on $U$ is principal iff the
subfunctor $S$ of $𝐛y(U)$ is a subsheaf (Exercise 1). A sieve $S$ on
$U$ is said to be a {\sl covering sieve} for $U$ when $U$ is the union
of all the open sets $V$ in $S$.
\end{quotation}
Let's see how to visualize this.
Definitions:
if $V∈\Opens(X)$ then $↓V = \setofst{W∈\Opens(X)}{W⊆V}$;
if $\calV⊆\Opens(X)$ then $↓\calV = \bigcup_{V∈\calV}(↓V)$.
%L house = ".1.|2.3|4.5"
%L mp = MixedPicture.new({def="dagHouse", meta="s", scale="5pt"}, z):zfunction(house):output()
%L local mpl = mpnew({zdef="bottlelr", scale="12pt", meta=""}, "12321L")
%L mpl:addlrs():output()
\pu
Let's use this topology from \cite[sections 12 and 13]{PH1}:
$X = H = \dagHouse•••••$ (the ``house'' DAG), and
%
$$\Opens(X) \;\; = \;\; \zha{bottlelr} \;\; .
$$
\newpage
Writing 0 for $∅$,
%R local y22, mid, lc, dnlc, y21 =
%R 1/ 0 \, 1/ 0 \, 1/ 0 \, 1/ 0 \, 1/ 0 \
%R | 1 | | 0 | | 0 | | 0 | | 0 |
%R | 1 1 | | 0 0 | | 0 0 | | 0 0 | | 1 0 |
%R |1 1 1| |1 1 1| |1 1 0| |1 1 0| |1 1 0|
%R | 1 1 | | 1 1 | | 0 0 | | 1 1 | | 1 1 |
%R \ 1 / \ 1 / \ 0 / \ 1 / \ 1 /
%R -- y22 :tozmp({zdef="y22", meta="s", size="6pt"}):addcells():output()
%R y22 :tozmp({zdef="y22", scale="8pt"}):addcells():output()
%R mid :tozmp({zdef="mid", scale="8pt"}):addcells():output()
%R lc :tozmp({zdef="lc", scale="8pt"}):addcells():output()
%R dnlc:tozmp({zdef="dnlc", scale="8pt"}):addcells():output()
%R y21 :tozmp({zdef="y21", scale="8pt"}):addcells():output()
%R y21 :tozmp({zdef="bott", scale="5pt"}):addbullets():output()
\pu
$𝐛y(22) = \Hom(-,22) = \zha{y22} \;\; ,$
This is also a sieve on 22: $S = \zha{dnlc} \;\; .$
Let $\calV = \setofst{V_i⊂U}{i∈I} = \zha{lc}$;
then $\calV$ spans ${↓}\calV = \zha{dnlc}$.
Note that this ${↓}\calV$ is not a principal sieve.
We have $\bigcup ({↓}\calV) = 21 ≠ 22$, so ${↓}\calV$ is not a covering sieve on $U$.
\bsk
A subset $\calV⊆\Opens(X)$ is a sieve on $X$ if and only if $\calV = {↓}\calV$.
Let's use the letters $\calA, \calB, \calC, \ldots$ to denote sieves on $X$.
For every sieve $\calA$ on $X$ we have:
$\calA$ is a covering sieve on $\bigcup\calA$,
and ${↓}\bigcup\calA$ is a principal sieve (generated by $\bigcup\calA$).
\bsk
The operation $\calA \mapsto \calA^* := {↓}\bigcup\calA$ takes sieves to principal sieves.
This operation obeys $\calA ⊆ \calA^* = \calA^{**}$.
Fact (true but not obvious): $\calA^* ∩ \calB^* = (\calA ∩ \calB)^*$.
\bsk
Now reread \cite[sections 12 and 13]{PH1}.
Remember that $\Opens(H) = \Opens(\dagHouse•••••) = \zha{bott}$.
In \cite[p.20]{DaveyPriestley} the operation `$\Opens$' is defined in
a different, but equivalent, way: if $X$ is an ordered set then
$\Opens(X)$ is the set of the down-sets of $X$, ordered by inclusion.
\newpage
% «O-Bottle» (to ".O-Bottle")
% «OO-House» (to ".OO-House")
% (find-LATEX "2020sheaves.tex" "OO-House")
With the definition in \cite{DaveyPriestley} it is easy to calculate
$\Opens(\Opens(H))$ as a set of down-sets, and then interpret it as a
topology. We have:
% (find-books "__alg/__alg.el" "davey-priestley")
% (find-daveypriestleypage (+ 10 24) "1.28 The ordered set O(P ) of down-sets")
% (find-daveypriestleytext (+ 10 24) "1.28 The ordered set O(P ) of down-sets")
% (find-LATEX "2017planar-has-defs.tex" "defzha-and-deftcg")
%
% Let B be the bottle ZHA.
% Label its nodes like this:
%
% L
% M
% L R
% L M R
% L R
% M
%
% I name the elements of O(B) by counting how many 1s
% they have on Ls, Ms, and Rs. The topology O(B) is:
%
% 433
% \
% 333
% |
% 323
% / \
% 322 223
% / \ / \
% 321 222 123
% \ / | \ /
% 221 212 122
% | X X |
% 211 121 112
% / \ | / \
% 210 111 012
% \ / \ /
% 110 011
% \ /
% 000
%
%
% (find-LATEX "2017planar-has-defs.tex" "defzha-and-deftcg")
\def\Def#1#2{\expandafter\def\csname myDef#1\endcsname{#2}}
\def\Get #1{\csname myDef#1\endcsname}
\def\Run #1{\csname myDef#1\endcsname}
\def\Defupperargs #1#2{\Def{21}{#1}\Def{12}{#2}}
\def\Defmiddleargs#1#2#3{\Def{20}{#1}\Def{11}{#2}\Def{02}{#3}}
\def\Deflowerargs #1#2{\Def{10}{#1}\Def{01}{#2}}
\def\Defallargs#1.#2.#3.#4.#5.#6{
\Def{32}{#1}
\Def{22}{#2}
\Defupperargs#3
\Defmiddleargs#4
\Deflowerargs#5
\Def{00}{#6}
}
\def\Setbottle#1#2{\Def{#1}{\Defallargs#2}}
\Setbottle{433}{1.1.11.111.11.1}
\Setbottle{333}{0.1.11.111.11.1}
\Setbottle{323}{0.0.11.111.11.1}
\Setbottle{322}{0.0.10.111.11.1}
\Setbottle{223}{0.0.01.111.11.1}
\Setbottle{321}{0.0.10.110.11.1}
\Setbottle{222}{0.0.00.111.11.1}
\Setbottle{123}{0.0.01.011.11.1}
\Setbottle{221}{0.0.00.110.11.1}
\Setbottle{212}{0.0.00.101.11.1}
\Setbottle{122}{0.0.00.011.11.1}
\Setbottle{211}{0.0.00.100.11.1}
\Setbottle{121}{0.0.00.010.11.1}
\Setbottle{112}{0.0.00.001.11.1}
\Setbottle{210}{0.0.00.100.10.1}
\Setbottle{111}{0.0.00.000.11.1}
\Setbottle{012}{0.0.00.001.01.1}
\Setbottle{110}{0.0.00.000.10.1}
\Setbottle{011}{0.0.00.000.01.1}
\Setbottle{010}{0.0.00.000.00.1}
\Setbottle{000}{0.0.00.000.00.0}
%% A simple test using matrices:
%
% \def\pbottle{\psm{
% \Get{32} \;\;\; \\
% \Get{22} \\
% \Get{21} \Get{12} \\
% \Get{20} \Get{11} \Get{02} \\
% \Get{10} \Get{01} \\
% \Get{00} \\
% }}
%
% \def\B#1#2#3{\Run{#1#2#3}\pbottle}
%
% $\def\S{\phantom{mmm}}
% \mat{ \B433 \phantom{mm} \\
% \B333 \\
% \B323 \\
% \B322 \S \B223 \\
% \B321 \S \B222 \S \B123 \\
% \B221 \B212 \B122 \\
% \B211 \B121 \B112 \\
% \B210 \S \B111 \S \B012 \\
% \B110 \S \B011 \\
% \B000 \\
% }
% $
\def\G #1#2{\Get{#1#2}}
\def\B#1#2#3{\Run{#1#2#3}\zha{bottlesieve}}
%R local bottlesieve =
%R 4/ !G32 \
%R | !G22 |
%R | !G21 !G12 |
%R |!G20 !G11 !G02|
%R | !G10 !G01 |
%R \ !G00 /
%R bottlesieve:tozmp({zdef="bottlesieve", meta="s", scale="4.5pt"}):addcells():output()
%R local bottlesieves =
%R 5/ !B433 \
%R | !B333 |
%R | !B323 |
%R | !B322 !B223 |
%R |!B321 !B222 !B123|
%R | !B221!B212!B122 |
%R | !B211!B121!B112 |
%R |!B210 !B111 !B012|
%R | !B110 !B011 |
%R \ !B000 |
%R
%R -- A bug fix:
%R -- (find-dn6 "zhas.lua" "MixedPicture")
%R -- (find-dn6 "zhas.lua" "MixedPicture" "addarrowsexcept =")
%R -- (find-dn6 "zhas.lua" "MixedPicture-cuts")
%R -- (find-dn6file "zhas.lua" "addxys =")
%R -- (find-dn6 "picture.lua" "V" " xy =")
%R
%R V.__index.xy = function (v) return pformat("(%s,%s)", v[1], v[2]) end
%R
%R MixedPicture.__index.
%R arrows = function (mp, w) return (mp.ar or mp.zha):arrows(w) end
%R
%R bottlesieves:tozmp({zdef="bottlesieves", meta="s", scale="42pt"})
%R :addcells()
%R :addarrowsexcept("w", "(0,4)0")
%R :output()
\pu
$$\Opens(\Opens(\dagHouse•••••))
\;\;=\;\;
\Opens(\zha{bott})
\;\;=\;\;
\zha{bottlesieves}
$$
The elements of $\Opens(\Opens(H))$ are exactly the sieves on $H$.
The operation $\calA \mto \calA^*$ that takes sieves on $H$ to
principal sieves is a J-operator on $\Opens(\Opens(H))$ (see
\cite{PH2}).
\newpage
% «top-sheaves-in-my-notation» (to ".top-sheaves-in-my-notation")
\section*{Topological sheaves in my notation}
\def\calVi{\calV_i}
\def\calVj{\calV_j}
\def\hij{h_i|_j}
\def\hji{h_j|_i}
%D diagram top-sheaves-my-notation-1
%D 2Dx 100 +20 +20 +20 +20 +20 +20 +20 +20 +20 +20 +20 +20
%D 2D 100 A0 C0
%D 2D \ \
%D 2D +20 A1 C1 E1 G1 I1
%D 2D / | / | / \ / \ / \
%D 2D +20 A2 | C2 | E2 E3 G2 G3 I2 I3
%D 2D \ | \ | \ / \ / \ |
%D 2D +20 A3 C3 E4 G4 I4 I5
%D 2D +15 B0' F0'
%D 2D +10 B0 ---> D0 F0 ------> H0
%D 2D
%D ren A0 A1 A2 A3 ==> X U V W
%D ren C0 C1 C2 C3 ==> F(X) F(U) F(V) F(W)
%D ren B0' B0 D0 ==> \Opens(X)\;\; \Opens(X)^\op \Set
%D ren F0' F0 H0 ==> \Opens(X)\;\; \Opens(X)^\op \Set
%D ren E1 E2 E3 E4 ==> \bigcup\calV \calVi \calVj \calVi∩\calVj
%D ren G1 G2 G3 G4 ==> F(\bigcup\calV) F(\calVi) F(\calVj) F(\calVi∩\calVj)
%D ren I1 I2 I3 I4 I5 ==> g h_i h_j \hij \hji
%D
%D (( A0 A1 <-
%D A1 A2 <-
%D A2 A3 <-
%D A1 A3 <-
%D C0 C1 ->
%D C1 C2 ->
%D C2 C3 ->
%D C1 C3 ->
%D B0' xy+= -10 0
%D B0 xy+= -10 0
%D D0 xy+= -10 0
%D B0' place
%D B0 D0 ->
%D
%D E1 E2 <-
%D E1 E3 <-
%D E2 E4 <-
%D E3 E4 <-
%D G1 G2 ->
%D G1 G3 ->
%D G2 G4 ->
%D G3 G4 ->
%D F0' place
%D F0 H0 ->
%D
%D I4 xy+= -10 0 I5 xy+= -10 0
%D I4 xy+= 4 0 I5 xy+= -4 0
%D I1 I2 |->
%D I1 I3 |->
%D I2 I4 |->
%D I3 I5 |->
%D ))
%D enddiagram
%D
$$\pu
\diag{top-sheaves-my-notation-1}
$$
$$
\begin{array}[t]{l}
I : \text{a set} \\
\calV : I → \Opens(X) \\
\bigcup\calV := \bigcup_{i∈I} \calV_i \\
g : F(\bigcup\calV) \\
h : (i:I) \to F(\calV_i) \\
i,j : I \\
g|_\calV : F_e(\calV) \\
g|_\calV := λi.F(ι:\calV_i→\bigcup\calV)(g) \\
\end{array}
%
\!\!\!\!\!\!\!\!
%
\begin{array}[t]{l}
\hij := F(ι:\calV_i∩\calV_j→\calV_i)(h_i) \\
\hji := F(ι:\calV_j∩\calV_j→\calV_j)(h_j) \\
F_0(\calV) := F(\bigcup\calV) \\
F_1(\calV) := (i:I) \to F(\calV_i) \\
F_e(\calV) := \setofst {h:(i:I)→F(\calV_i)} {∀(i,j:I).\hij=\hji} \\
F_2(\calV) := (i,j:I) \to F(\calV_i∩\calV_j) \\
\end{array}
$$
%D diagram top-sheaves-my-notation-2
%D 2Dx 100 +30 +30 +30 +30 +30
%D 2D 100 A0 B0
%D 2D | \ | \
%D 2D +22 | \ B1 B2
%D 2D +8 Ae - A1 - A2 B3 - B4
%D 2D +8 B5 - B6
%D 2D +8 B7 - B8
%D 2D
%D ren A0 Ae A1 A2 ==> F_0(\calV) F_e(\calV) F_1(\calV) F_2(\calV)
%D ren B0 B1 B2 B3 B4 ==> g g|_\calV g|_\calV h h
%D ren B5 B6 ==> h λi,j.\hij
%D ren B7 B8 ==> h λi,j.\hji
%D
%D (( A0 Ae ->
%D A0 A1 ->
%D Ae A1 ->
%D A1 A2 ->
%D
%D B0 B1 |->
%D # B0 B2 |->
%D B3 B4 |->
%D B5 B6 |->
%D B7 B8 |->
%D ))
%D enddiagram
%D
$$\pu
\diag{top-sheaves-my-notation-2}
$$
\newpage
% (find-maclanemoerdijkpage (+ 11 83) "5. Sheaves and cross-sections")
% (find-maclanemoerdijkpage (+ 11 87) "Corollary 4")
{\bf II.5. Sheaves and cross-sections}
(Page 87):
%D diagram ??
%D 2Dx 100 +40 +20
%D 2D 100 A0 - A1 C0
%D 2D | | |
%D 2D +20 A2 - A3 C1
%D 2D
%D 2D +15 B0 - B1
%D 2D
%D ren A0 A1 ==> ΓΛ_P P
%D ren A2 A3 ==> F F
%D ren B0 B1 ==> \Sh(X) \SetsOXop
%D ren C0 C1 ==> P ΓΛ_P
%D
%D (( A0 A1 <-|
%D A0 A2 -> .plabel= l σ
%D A1 A3 -> .plabel= r θ
%D A0 A3 harrownodes nil 20 nil <->
%D A2 A3 |->
%D
%D B0 B1 <- sl^ .plabel= a ΓΛ
%D B0 B1 >-> sl_ .plabel= b \text{inc}
%D # newnode: B1' at: @B1+v(45,0) .TeX= =\widehat{\Opens(X)}=\PSh(X)? place
%D newnode: B1' at: @B1+v(30,0) .TeX= =\widehat{\Opens(X)} place
%D
%D C0 C1 -> .plabel= r η
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
% (find-maclanemoerdijkpage (+ 11 88) "6. Sheaves as Étale Spaces")
(Page 88):
6. Sheaves as Étale Spaces
%D diagram ??
%D 2Dx 100 +25 +45 +20
%D 2D 100 L0 A0 - A1 R0
%D 2D | | | |
%D 2D +20 L1 A2 - A3 R1
%D 2D
%D 2D +15 B0' - B0 - B1
%D 2D
%D ren A0 A1 ==> ΛP P
%D ren A2 A3 ==> Y ΓY
%D ren B0 B1 ==> \Bund(X) \SetsOXop
%D ren L0 L1 ==> ΛΓY Y
%D ren R0 R1 ==> P ΓΛP
%D
%D (( A0 A1 <-|
%D A0 A2 -> # .plabel= l σ
%D A1 A3 -> # .plabel= r θ
%D A0 A3 harrownodes nil 20 nil <->
%D A2 A3 |->
%D
%D newnode: B0' at: @B0+v(-35,0) .TeX= \Top/X B0' B0 =
%D B0 B1 <- sl^ .plabel= a Λ
%D B0 B1 -> sl_ .plabel= b Γ
%D # newnode: B1' at: @B1+v(45,0) .TeX= =\widehat{\Opens(X)}=\PSh(X)? place
%D # newnode: B1' at: @B1+v(30,0) .TeX= =\widehat{\Opens(X)} place
%D
%D R0 R1 -> .plabel= r η_P
%D L0 L1 -> .plabel= l ε_Y
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
\newpage
% «sheaves-on-a-site» (to ".sheaves-on-a-site")
% (mmop 13 "sheaves-on-a-site")
% (mmoa "sheaves-on-a-site")
\subsubsection*{III.4. Sheaves on a Site}
(Page 121)
% (find-maclanemoerdijkpage (+ 11 121) "4. Sheaves on a Site")
% (find-maclanemoerdijkpage (+ 11 123) "Proposition 1")
Definition in page 122:
%
%D diagram ??
%D 2Dx 100 +20 +20 +20 +30 +30
%D 2D 100 A0 A1 B0 - B1 C0 D0
%D 2D | | / | |
%D 2D +20 A2 B2 C1 D1
%D 2D
%D ren A0 A1 A2 ==> C J(C) ∀S
%D ren B0 B1 B2 ==> 𝐛yC P S
%D ren C0 C1 ==> \Hom(𝐛yC,P) \Hom(S,P)
%D ren D0 D1 ==> g g∘i
%D
%D (( A0 place
%D A1 A2 <- .plabel= r ∈
%D B0 B1 -> .plabel= a ∃!g
%D B0 B2 <- .plabel= l i
%D B2 B1 -> .plabel= r ∀f
%D C0 C1 -> .plabel= r \sm{(∘i)\\\text{iso}}
%D D0 D1 |->
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
Archetypal case:
%
%D diagram ??
%D 2Dx 100 +20 +20 +20 +30 +30
%D 2D 100 A0 A1 B0 - B1 C0 D0
%D 2D | | / | |
%D 2D +20 A2 B2 C1 D1
%D 2D
%D ren A0 A1 A2 ==> U J(U) ∀\calU
%D ren B0 B1 B2 ==> {↓}U F \calU
%D ren C0 C1 ==> \Hom({↓U},F) \Hom(\calU,F)
%D ren D0 D1 ==> g g∘i
%D
%D (( A0 place
%D A1 A2 <- .plabel= r ∈
%D B0 B1 -> .plabel= a ∃!g
%D B0 B2 <- .plabel= l i
%D B2 B1 -> .plabel= r ∀f
%D C0 C1 -> .plabel= r \sm{(∘i)\\\text{iso}}
%D D0 D1 |->
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
Only the `$j$'s:
%
%D diagram ??
%D 2Dx 100 100 +20 +20 +30 +30
%D 2D 100 A0 A1 B0 - B1 C0 D0
%D 2D | | / | |
%D 2D +20 A2 B2 C1 D1
%D 2D
%D ren A0 A1 A2 ==> . P^* ∀P
%D ren B0 B1 B2 ==> P^* F P
%D ren C0 C1 ==> \Hom(P^*,F) \Hom(P,F)
%D ren D0 D1 ==> g g∘d
%D
%D (( A0 place
%D A1 A2 <- .plabel= l \sm{d\\\text{dense}}
%D B0 B1 -> .plabel= a ∃!g
%D B0 B2 <- .plabel= l d
%D B2 B1 -> .plabel= r ∀f
%D C0 C1 -> .plabel= r \sm{(∘d)\\\text{iso}}
%D D0 D1 |->
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
All dense truth values:
%
%D diagram ??
%D 2Dx 100 100 +20 +20 +30 +30
%D 2D 100 A0 A1 B0 - B1 C0 D0
%D 2D | | / | |
%D 2D +20 A2 B2 C1 D1
%D 2D
%D ren A0 A1 A2 ==> . ∀Q ∀P
%D ren B0 B1 B2 ==> Q F P
%D ren C0 C1 ==> \Hom(Q,F) \Hom(P,F)
%D ren D0 D1 ==> g g∘d
%D
%D (( A0 place
%D A1 A2 <- .plabel= l \sm{d\\\text{dense}}
%D B0 B1 -> .plabel= a ∃!g
%D B0 B2 <- .plabel= l d
%D B2 B1 -> .plabel= r ∀f
%D C0 C1 -> .plabel= r \sm{(∘d)\\\text{iso}}
%D D0 D1 |->
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
All dense maps (see Bell p.174):
%
% (find-books "__cats/__cats.el" "bell-lst")
% (find-belltpage (+ 14 174) "(mu-)sheaf")
%
%D diagram ??
%D 2Dx 100 100 +20 +20 +30 +30
%D 2D 100 A0 A1 B0 - B1 C0 D0
%D 2D | | / | |
%D 2D +20 A2 B2 C1 D1
%D 2D
%D ren A0 A1 A2 ==> . ∀B ∀A
%D ren B0 B1 B2 ==> B F A
%D ren C0 C1 ==> \Hom(B,F) \Hom(A,F)
%D ren D0 D1 ==> g g∘d
%D
%D (( A0 place
%D A1 A2 <- .plabel= l \sm{d\\\text{dense}}
%D B0 B1 -> .plabel= a ∃!g
%D B0 B2 <- .plabel= l d
%D B2 B1 -> .plabel= r ∀f
%D C0 C1 -> .plabel= r \sm{(∘d)\\\text{iso}}
%D D0 D1 |->
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
\newpage
\subsubsection*{III.5. The associated sheaf functor}
(Page 128):
% (find-maclanemoerdijkpage (+ 11 128) "5. The associated sheaf functor" "P^+")
%D diagram ??
%D 2Dx 100 +40 +20
%D 2D 100 A0 - A1 C0
%D 2D | | |
%D 2D +20 A2 - A3 C1
%D 2D
%D 2D +15 B0 - B1
%D 2D
%D ren A0 A1 ==> 𝐛a{P} P
%D ren A2 A3 ==> F F
%D ren B0 B1 ==> \Sh(\catC,J) \SetsCop
%D ren C0 C1 ==> P i𝐛aP
%D
%D (( A0 A1 <-|
%D A0 A2 -> .plabel= l ?
%D A1 A3 -> .plabel= r ?
%D A0 A3 harrownodes nil 20 nil <->
%D A2 A3 |->
%D
%D B0 B1 <- sl^ .plabel= a 𝐛a
%D B0 B1 >-> sl_ .plabel= b i
%D newnode: A0' at: @A0+v(-25,0) .TeX= (P^+)^+= place
%D newnode: B1' at: @B1+v(30,0) .TeX= =\widehat{\Opens(X)} place
%D
%D C0 C1 -> .plabel= r η
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
\newpage
% «LT-subsumes-groth» (to ".LT-subsumes-groth")
% (mmop 12 "LT-subsumes-groth")
% (mmoa "LT-subsumes-groth")
% (find-maclanemoerdijkpage (+ 11 233) "V.4 Lawvere-Tierney Subsumes Grothendieck")
\section*{V.4 Lawvere-Tierney Subsumes Grothendieck}
% _ _ _ _ _
% | | ___ ___ __ _| (_) ___ | |_ ___ _ __ ___ (_)
% | | / _ \ / __/ _` | | |/ __| | __/ _ \| '_ \ / _ \| |
% | |__| (_) | (_| (_| | | | (__ | || (_) | |_) | (_) | |
% |_____\___/ \___\__,_|_|_|\___| \__\___/| .__/ \___/|_|
% |_|
%
% «localic-topoi» (to ".localic-topoi")
% (find-books "__cats/__cats.el" "maclane-moerdijk")
% (find-maclanemoerdijkpage (+ 11 470) "IX. Localic Topoi")
% (find-maclanemoerdijkpage (+ 11 480) "4. Embeddings and Surjections of Locales")
% (find-maclanemoerdijkpage (+ 11 483) "nucleus")
% (find-maclanemoerdijkpage (+ 11 487) "5. Localic Topoi")
\section*{IX. Localic Topoi}
% (find-maclanemoerdijkpage (+ 11 471) "Lemma 1")
(Page 471)
Lemma IX.1.1:
%D diagram ??
%D 2Dx 100 +25
%D 2D 100 A0 - A1
%D 2D | |
%D 2D +20 A2 - A3
%D 2D
%D 2D +15 B0 - B1
%D 2D
%D ren A0 A1 ==> Φ(V) V
%D ren A2 A3 ==> U Ψ(U)
%D ren B0 B1 ==> A B
%D
%D (( A0 A1 <-|
%D A0 A2 ->
%D A1 A3 ->
%D A0 A3 harrownodes nil 20 nil <->
%D A2 A3 |->
%D B0 B1 <- sl^ .plabel= a Φ
%D B0 B1 -> sl_ .plabel= b Ψ
%D
%D newnode: A3' at: @A3+v(50,0) .TeX= =\bigvee\setofst{V∈B}{Φ(V)≤U} place
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
%D diagram ??
%D 2Dx 100 +30
%D 2D 100 A0 - A1
%D 2D | |
%D 2D +20 A2 - A3
%D 2D
%D 2D +15 B0 - B1
%D 2D
%D 2D +15 C0 - C1
%D 2D
%D ren A0 A1 ==> f^{-1}(V) V
%D ren A2 A3 ==> U f_*(U)
%D ren B0 B1 ==> \Opens(S) \Opens(T)
%D ren C0 C1 ==> S T
%D
%D (( A0 A1 <-|
%D A0 A2 ->
%D A1 A3 ->
%D A0 A3 harrownodes nil 20 nil <->
%D A2 A3 |->
%D B0 B1 <- sl^ .plabel= a f^{-1}
%D B0 B1 -> sl_ .plabel= b f_*
%D C0 C1 -> .plabel= a f
%D
%D newnode: A3' at: @A3+v(60,0) .TeX= =\bigcup\setofst{V∈\Opens(T)}{f^{-1}V⊆U} place
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
% (find-maclanemoerdijkpage (+ 11 472) "(Locales)")
(Page 472):
%
%D diagram ??
%D 2Dx 100 +65 +30 +30 +40
%D 2D 100 A0 B0 - B1 C0 - C1
%D 2D | | | | |
%D 2D +25 A1 B2 - B3 C2 - C3
%D 2D | | | | |
%D 2D +25 A2 B4 - B5 C4 - C5
%D 2D
%D ren A0 A1 A2 ==> (\Locales) (\Frames)^\op (\Spaces)
%D ren B0 B1 C0 C1 ==> S T (\Opens(S),⊆) (\Opens(T),⊆)
%D ren B2 B3 C2 C3 ==> \Opens(S) \Opens(T) (\Opens(S),⊆) (\Opens(T),⊆)
%D ren B4 B5 C4 C5 ==> S T (S,\Opens(S)) (T,\Opens(T))
%D
%D
%D (( A0 A1 <->
%D A1 A2 <-
%D
%D B0 B1 -> .plabel= a f
%D B0 B2 <->
%D B1 B3 <->
%D B0 B3 varrownodes nil 17 nil <->
%D B2 B3 <- .plabel= a f^{-1}
%D B2 B4 <->
%D B3 B5 <->
%D B2 B5 varrownodes nil 17 nil <-|
%D B4 B5 -> .plabel= a f
%D
%D C0 C1 <-
%D C0 C2 <->
%D C1 C3 <->
%D C0 C3 varrownodes nil 17 nil <->
%D C2 C3 <-
%D C2 C4 <->
%D C3 C5 <->
%D C2 C5 varrownodes nil 17 nil <-|
%D C4 C5 ->
%D
%D newnode: A1' at: @A1+v(35,0) .TeX= (\Frames) place
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
\newpage
% (find-maclanemoerdijkpage (+ 11 472) "(Locales)")
(Page 474):
%
Lemma 1, archetypal case:
%
%D diagram ??
%D 2Dx 100 +100 +30
%D 2D 100 B0 - B1
%D 2D
%D 2D +15 A0 B2 - B3
%D 2D |
%D 2D +20 A1 B4 - B5
%D 2D
%D ren A0 A1 ==> (\Frames) (\Spaces)^\op
%D ren B0 B1 ==> p^{-1}U U
%D ren B2 B3 ==> \{0,1\} \Opens(X)
%D ren B4 B5 ==> 1 X
%D
%D (( A0 A1 <-
%D newnode: A1' at: @A1+v(35,0) .TeX= (\Spaces) place
%D
%D B0 B1 <-|
%D B2 B3 <- .plabel= a p^{-1}
%D B4 B5 -> .plabel= a p
%D
%D newnode: B0' at: @B0+v(-45,0) .TeX= \setofst{*∈1}{p(*)\not∈U}= place
%D newnode: B2' at: @B2+v(-30,0) .TeX= \{∅,\{*\}\}= place
%D newnode: B4' at: @B4+v(-22,0) .TeX= \{*\}= place
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
For each $p:1→X$ we define $K$ and $P$ as:
%
$$\begin{array}{rcl}
\Opens(X) \;\; ⊇ \;\;
K &:=& \Ker p^{-1} \\
&=& \setofst{U∈\Opens(X)}{p^{-1}U=0} \\
&=& \setofst{U∈\Opens(X)}{p^{-1}U=∅} \\
&=& \setofst{U∈\Opens(X)}{*\not∈p^{-1}U} \\
&=& \setofst{U∈\Opens(X)}{p(*)\not∈U} \\
\Opens(X) \;\; \ni \;\;
P &:=& \bigcup K \\
&=& \bigcup\setofst{U∈\Opens(X)}{p(*)\not∈U} \\
&=& \Int(X-\{p(*)\}) \\
\end{array}
$$
This $K$ is a {\sl kernel} on $\Opens(X)$. The definition is: a kernel
on $\Opens(X)$ is a subset $K⊆\Opens(X)$ that is closed downwards,
closed by taking arbitrary unions, and it obeys $1\not∈K$ and, for all
$U,V∈\Opens(X)$:
%
$$U∧V∈K \text{ implies } U∈K \text{ or } V∈K.$$
This $P$ is a {\sl proper prime element} of $\Opens(X)$. The
definition is: a $P∈\Opens(X)$ is a proper prime element iff $1≠P$,
and, for all $U,V∈\Opens(X)$:
%
$$U∩V⊆P \text{ implies } U⊆P \text{ or } V⊆P.$$
\newpage
% _ _ _
% | | ___ ___ | | _ __ | |_
% | | / _ \ / __| _____| | | '_ \| __|
% | |__| (_) | (__ |_____| | | |_) | |_
% |_____\___/ \___| | | | .__/ \__|
% |_| |_|
%
% «spaces-from-locales» (to ".spaces-from-locales")
% (mmop 14 "spaces-from-locales")
% (mmoa "spaces-from-locales")
% (find-maclanemoerdijkpage (+ 11 473) "There is an obvious functor Loc")
(Page 473):
There is an obvious functor $\Loc$ from spaces to locales:
%D diagram ??
%D 2Dx 100 +45 +40 +40
%D 2D 100 A0 - A1 C0 - C1
%D 2D | | | |
%D 2D +20 A2 - A3 C2 - C3
%D 2D
%D 2D +15 B0 - B1
%D 2D
%D ren A0 A1 ==> S \Loc(S)
%D ren A2 A3 ==> T \Loc(T)
%D ren B0 B1 ==> (\Spaces) (\Locales)
%D ren C0 C1 ==> (S,\Opens(S)) (\Opens(S),⊆)
%D ren C2 C3 ==> (T,\Opens(S)) (\Opens(T),⊆)
%D
%D (( A0 A1 |->
%D A0 A2 -> .plabel= l f
%D A1 A3 -> .plabel= r \Loc(f)
%D A0 A3 harrownodes nil 20 nil |->
%D A2 A3 |->
%D B0 B1 -> .plabel= a \Loc
%D
%D C0 C1 |->
%D C0 C2 -> .plabel= l f
%D C1 C3 <- .plabel= r f^{-1}
%D C0 C3 harrownodes nil 20 nil |->
%D C2 C3 |->
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
\bsk
(Page 475):
IX.3: Spaces from locales
% (find-maclanemoerdijkpage (+ 11 475) "3. Spaces from Locales")
% (find-maclanemoerdijkpage (+ 11 477) "Proposition 2")
On locales that come from a topological spaces we define the functor
$\Pt: (\Locales)→(\Spaces)$ as this. The functor takes each locale
$X≡(\Opens(X),⊆)$ to a topological space $\Pt(X) ≡
(\Pt(X),\Opens(\Pt(X))$, where:
%
$$\begin{array}{rcl}
\Pt(X) &:=& \setofst{p}{p:1→X} \\
\text{for $U∈\Opens(X)$, } \;\;\;
\Pt(U) &:=& \setofst{p:1→X}{p(*)∈U} \\
&=& \setofst{p:1→X}{*∈p^{-1}(U)} \\
&=& \setofst{p:1→X}{p^{-1}U = 1} \\
\Opens(\Pt(X)) &:=& \setofst{\Pt(U)}{U∈\Opens(X)} \\
\end{array}
$$
On locales $X≡(X,≤)$ we define the functor $\Pt: (\Locales)→(\Spaces)$
as this generalization of the idea above:
%
$$\begin{array}{rcl}
\Pt(X) &:=& \setofst{p}{p:1→X} \\
\text{for $U∈X$, } \;\;\;
\Pt(U) &:=& \setofst{p:1→X}{p^{-1}U = 1} \\
\Opens(\Pt(X)) &:=& \setofst{\Pt(U)}{U∈X} \\
\end{array}
$$
We draw that functor as:
%
%D diagram ??
%D 2Dx 100 +40 +30 +45 +40
%D 2D 100 A0 - A1 C0 - C1 D0
%D 2D | | | | |
%D 2D +20 A2 - A3 C2 - C3 D1
%D 2D
%D 2D +15 B0 - B1
%D 2D
%D ren A0 A1 A2 A3 ==> X \Pt(X) Y \Pt(Y)
%D ren B0 B1 ==> (\Locales) (\Spaces)
%D ren C0 C1 ==> (X,≤) (\Pt(X),\Opens(\Pt(X))
%D ren C2 C3 ==> (Y,≤) (\Pt(Y),\Opens(\Pt(Y))
%D ren D0 D1 ==> p f∘p
%D
%D (( A0 A1 |->
%D A0 A2 -> .plabel= l f
%D A1 A3 -> .plabel= r \Pt(f)
%D A0 A3 harrownodes nil 20 nil |->
%D A2 A3 |->
%D B0 B1 -> .plabel= a \Pt
%D
%D C0 C1 |->
%D C0 C2 <-
%D C1 C3 ->
%D C0 C3 harrownodes nil 20 nil |->
%D C2 C3 |->
%D
%D D0 D1 |->
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
\newpage
(Page 476):
Theorem 1: The functor $\Pt: (\Locales) → (\Spaces)$ is right adjoint
to the functor $\Loc: (\Spaces) → (\Locales)$.
% (find-maclanemoerdijkpage (+ 11 476) "Theorem 1. Loc -| pt")
%D diagram ??
%D 2Dx 100 +20 +40 +30 +50
%D 2D 100 A0 B0 - B1 C0 - C1
%D 2D | | | | |
%D 2D +20 A1 B2 - B3 C2 - C3
%D 2D
%D 2D +15 D0 - D1
%D 2D
%D ren D0 D1 ==> (\Locales) (\Spaces)
%D ren B0 B1 B2 B3 ==> \Loc(S) S X \Pt(X)
%D ren C0 C1 C2 C3 ==> (\Opens(S),⊆) (S,\Opens(S)) (X,≤) (\Pt(X),\Opens(\Pt(X))
%D
%D (( D0 D1 <- sl^ .plabel= a \Loc
%D D0 D1 -> sl_ .plabel= b \Pt
%D
%D B0 B1 <-|
%D B0 B2 -> .plabel= l f
%D B1 B3 -> .plabel= r g
%D B0 B3 harrownodes nil 20 nil <->
%D B2 B3 |->
%D
%D C0 C1 <-|
%D C0 C2 <- .plabel= l f^{-1}
%D C1 C3 -> .plabel= r g
%D C0 C3 harrownodes nil 20 nil <->
%D C2 C3 |->
%D
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
% (find-maclanemoerdijkpage (+ 11 477) "Proof")
Proof:
%
$$\begin{array}{rcl}
f^{-1} &:=& \\
\end{array}
$$
% ____ _ _ _ _ _
% | ___| | | ___ ___ __ _| (_) ___ | |_ ___ _ __ ___ (_)
% |___ \ | | / _ \ / __/ _` | | |/ __| | __/ _ \| '_ \ / _ \| |
% ___) | | |__| (_) | (_| (_| | | | (__ | || (_) | |_) | (_) | |
% |____(_) |_____\___/ \___\__,_|_|_|\___| \__\___/| .__/ \___/|_|
% |_|
%
% «5._localic_topoi» (to ".5._localic_topoi")
% (find-maclanemoerdijkpage (+ 11 487) "Proof")
% (find-maclanemoerdijkpage (+ 11 488) "Theorem 1")
\subsubsection*{IX.5. Localic topoi}
(Page 487):
(Page 488):
Theorem 1: For a Grothendieck topos the following are equivalent:
(i) $\calE$ is localic,
(ii) there exists a site for $\calE$ with a poset as underlying category,
(iii) $\calE$ is generated by the subobjects of its terminal object 1.
Proof. Since a frame is a poset, (i) trivially implies (ii).
(ii) $⇒$ (iii) Suppose that $\calE = \Sh(\catP,J)$, where $J$ is a
Grothendieck topology on a poset $\catP$, and write $𝐛a𝐛y: \catP →
\calE$ for the process of sheafification $𝐛a$ followed by the Yoneda
embedding. Now for each $p∈\catP$ the map is necessarily monic in
presheaves, while sheafification $𝐛a$ is left exact, hence preserves
monics. Thus every map $𝐛a𝐛y(p)→1$ is monic, hence gives a subobject
of 1. But III.6(17) showed that the images of the $𝐛a𝐛y$ generate the
topos $\calE$.
%D diagram ??
%D 2Dx 100 +30 +35 +25
%D 2D 100 A0 - A1 - A2
%D 2D
%D 2D +20 A4 - A5
%D 2D
%D 2D +15 B0 - B1 - B2 = B3
%D 2D
%D ren A0 A1 A2 ==> p 𝐛yp 𝐛a𝐛yp
%D ren A4 A5 ==> 1 1
%D ren B0 B1 B2 B3 ==> \catP \Sets^{\catP^\op} \Sh(\catP,J) \calE
%D
%D (( A0 A1 |->
%D A1 A2 |->
%D A1 A4 >->
%D A2 A5 >->
%D A4 A5 |->
%D B0 B1 -> .plabel= a 𝐛y
%D B1 B2 -> .plabel= a 𝐛a
%D B2 B3 =
%D # newnode: B2' at: @B2+v(20,0) .TeX= =\calE place
%D B0 B3 -> .slide= -10pt .plabel= b 𝐛a𝐛y
%D
%D ))
%D enddiagram
%D
$$\pu
\diag{??}
$$
% (find-maclanemoerdijkpage (+ 11 487) "5. Localic Topoi")
% (find-maclanemoerdijkpage (+ 11 488) "Theorem 1. For a Grothendieck topos")
% (find-maclanemoerdijkpage (+ 11 488) "canonical")
% (find-maclanemoerdijkpage (+ 11 488) "following the Yoneda embedding")
\newpage
$\acz
\psm{
& · & & & \\
& & · & & \\
& · & & · & \\
20 & & · & & 02 \\
& 10 & & 01 & \\
& & 00 & & \\
}
$
is a covering sieve for 22
% (find-books "__cats/__cats.el" "godement")
% (find-books "__cats/__cats.el" "grothendieck-tohoku")
% (find-grothtohokutpage (+ 9 36) "3 Cohomology with coefficients in a sheaf")
\printbibliography
\GenericWarning{Success:}{Success!!!} % Used by `M-x cv'
\end{document}
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make -f 2019.mk STEM=2020maclane-moerdijk pdf
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