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% (find-LATEX "2008sdg-utf8.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2008sdg-utf8.tex" :end))
% (defun d () (interactive) (find-pdf-page "~/LATEX/2008sdg-utf8.pdf"))
% (defun e () (interactive) (find-LATEX "2008sdg-utf8.tex"))
% (defun u () (interactive) (find-latex-upload-links "2008sdg-utf8"))
% (find-pdf-page "~/LATEX/2008sdg-utf8.pdf")
% (find-sh0 "cp -v ~/LATEX/2008sdg-utf8.pdf /tmp/")
% (find-sh0 "cp -v ~/LATEX/2008sdg-utf8.pdf /tmp/pen/")
% file:///home/edrx/LATEX/2008sdg-utf8.pdf
% file:///tmp/2008sdg-utf8.pdf
% file:///tmp/pen/2008sdg-utf8.pdf
% http://angg.twu.net/LATEX/2008sdg-utf8.pdf
% (find-LATEX "2019.mk")
\documentclass[oneside]{book}
\usepackage[colorlinks,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\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")
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\input edrxgac2.tex % (find-LATEX "edrxgac2.tex")
%
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\begin{document}
\catcode`\^^J=10
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% (find-dednat6 "dednat6/block.lua" "TexLines")
\directlua{tf:processuntil(texlines:nlines())}
% «.ring-objects» (to "ring-objects")
% «.ring-object-tan-space» (to "ring-object-tan-space")
% «.ring-object-functions» (to "ring-object-functions")
% «.ring-object-morphism» (to "ring-object-morphism")
% «.ring-object-of-lt» (to "ring-object-of-lt")
% «.beta-is-known» (to "beta-is-known")
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 {Ring objects} {2}
\tocline {A ring object: the tangent space} {3}
\tocline {Another ring object: a ring of functions} {4}
\tocline {A homomorphism between ring objects} {5}
\tocline {Ring objects of line type} {6}
\tocline {Lemma: the map $\beta $ is known} {7}
\newpage
% --------------------
% «ring-objects» (to ".ring-objects")
% (s "Ring objects" "ring-objects")
\myslide {Ring objects} {ring-objects}
$(\R,0,1,+,·,-)$ can be seen as a ``ring object'' in $\Set$,
that is, as five functions from powers of $\R$ to $\R$,
one for each operation:
%D diagram R-as-ring-object
%D 2Dx 100 +30 +30
%D 2D 100 1 =====> \R <===== \R^2
%D 2D
%D 2D +20 * |----> 0
%D 2D +5 {}* |----> 1{}
%D 2D +5 a+b <-----| a,b
%D 2D +5 ab <-----| a,b{}
%D 2D
%D (( 1 \R -> sl^ .plabel= a 0
%D 1 \R -> sl_ .plabel= b 1
%D \R \R^2 <-| sl^ .plabel= a +
%D \R \R^2 <-| sl_ .plabel= b ·
%D ))
%D (( * 0 |->
%D {}* 1{} |->
%D a+b a,b <-|
%D ab a,b{} <-|
%D ))
%D enddiagram
%D
$$\diag{R-as-ring-object}$$
(we will never draw the additive inverse $-:\R \to \R$).
\ssk
These arrows must obey some equations ---
for example, $(a+b)c = ac+bc$, that becomes:
%:
%: a,b,c a,b,c a,b,c a,b,c a,b,c a,b,c
%: ----- ----- ----- ----- ----- -----
%: a b a,b,c a c b c
%: -------- ----- -------- --------
%: a+b c ac bc
%: ------------ ----------------
%: (a+b)c = ac+bc
%:
%: ^distr-eq-1 ^distr-eq-2
%:
$$\ded{distr-eq-1} \quad = \quad \ded{distr-eq-2}$$
%:
%: \id \id \id \id \id \id
%: ----- --- --- --- --- ---
%: π_1 π_2 \id π_1 π_3 π_2 π_3
%: --------------- --- -------- --------
%: \ang{π_1,π_2};+ π_3 \ang{π_1,π_3};· \ang{π_2,π_3};·
%: ---------------------------- ---------------------------------
%: \ang{(\ang{π_1,π_2};+),π_3};· \ang{(\ang{π_1,π_3};·),(\ang{π_2,π_3};·)};+
%:
%: ^distr-eq-1b ^distr-eq-2b
%:
$$\ded{distr-eq-1b} \quad = \quad \ded{distr-eq-2b}$$
%D diagram R-associativity
%D 2Dx 100 +75
%D 2D 100 \R^3 =====> \R
%D 2D
%D 2D +15 a,b,c |----> (a+b)c
%D 2D +5 {}a,b,c |----> ac+bc
%D 2D
%D (( \R^3 \R -> sl^ .plabel= a \ang{(\ang{π_1,π_2};+),π_3};·
%D \R^3 \R -> sl_ .plabel= b \ang{(\ang{π_1,π_3};·),(\ang{π_2,π_3};·)};+
%D ))
%D (( a,b,c (a+b)c |->
%D {}a,b,c ac+bc |->
%D ))
%D enddiagram
%D
$$\diag{R-associativity}$$
As $\Set$ has finite products every
$(\R,0,1,+,·,-)$-polynomial in $n$ variables
can be represented as a morphism $\R^n \to \R$;
each of the ring axioms becomes the statement
that two ``$(\R,0,1,+,·,-)$-polynomials'' are equal.
\bsk
Note: the part $(\R,0,1,+,·,-)$ of the definition of a ring object
is sometimes called the ``structure'' of the ring object; the uple
with one equality between $(\R,0,1,+,·,-)$-polynomials for each of
the ring axioms is called ``properties''. We will not spell out in
detail the ``properties'' part here, and we will write just
``$(\R,0,1,+,·,-)$'' --- or ``$\R$'' --- for everything.
\newpage
% --------------------
% «ring-object-tan-space» (to ".ring-object-tan-space")
% (s "A ring object: the tangent space" "ring-objects-tan-space")
\myslide {A ring object: the tangent space} {ring-objects-tan-space}
The tangent space of $\R$, $T\R$, has the same points as $\R^2$,
and a ring structure, with special definitions for `1' and `$·$'.
We will denote its points as $(a,a_x), (b,b_x), \ldots$
Here is its ring structure:
%D diagram TR-as-ring-object
%D 2Dx 100 +35 +60
%D 2D 100 1 =====> T\R <============== (T\R)^2
%D 2D
%D 2D +20 * |----> (0,0)
%D 2D +6 {}* |----> (1,0)
%D 2D +6 (a+b,a_x+b_x) <-----| (a,a_x),(b,b_x)
%D 2D +6 (ab,a_xb+b_xa) <----| (a,a_x),(b,b_x){}
%D 2D
%D (( 1 T\R -> sl^ .plabel= a 0
%D 1 T\R -> sl_ .plabel= b 1
%D T\R (T\R)^2 <- sl^ .plabel= a +
%D T\R (T\R)^2 <- sl_ .plabel= b ·
%D ))
%D (( * (0,0) |->
%D {}* (1,0) |->
%D (a+b,a_x+b_x) (a,a_x),(b,b_x) <-|
%D (ab,a_xb+b_xa) (a,a_x),(b,b_x){} <-|
%D ))
%D enddiagram
%D
$$\diag{TR-as-ring-object}$$
\newpage
% --------------------
% «ring-object-functions» (to ".ring-object-functions")
% (s "Another ring object: a ring of functions" "ring-objects-functions")
\myslide {Another ring object: a ring of functions} {ring-objects-functions}
\def\AffLin{\mathrm{AffLin}}
\widemtos
For any set $S$ the space of functions $S \to \R$ (a.k.a. ``$\R^S$'')
has a natural ring structure:
%D diagram SR-as-ring-object
%D 2Dx 100 +35 +70
%D 2D 100 1 ========> (S->\R) <============ (S->\R)^2
%D 2D
%D 2D +20 * |-------> (s|->0)
%D 2D +6 {}* |-------> (s|->1)
%D 2D +6 (s|->a[s]+b[s]) <-----| (s|->a[s],(s|->b[s])
%D 2D +7 (s|->a[s]b[s]) <------| (s|->a[s],(s|->b[s]){}
%D 2D
%D (( 1 (S->\R) -> sl^ .plabel= a 0
%D 1 (S->\R) -> sl_ .plabel= b 1
%D (S->\R) (S->\R)^2 <- sl^ .plabel= a +
%D (S->\R) (S->\R)^2 <- sl_ .plabel= b ·
%D ))
%D (( * (s|->0) |->
%D {}* (s|->1) |->
%D (s|->a[s]+b[s]) (s|->a[s],(s|->b[s]) <-|
%D (s|->a[s]b[s]) (s|->a[s],(s|->b[s]){} <-|
%D ))
%D enddiagram
%D
$$\diag{SR-as-ring-object}$$
% (s \mto a[s]):
If $S \subseteq \R$ then some functions $S \to \R$ are ``affine
linear'',
in the sense that they can be characterized by two
reals ---
a ``constant part'' (`$a$') and a ``slope'' (`$a_x$').
\msk
Let's write these functions as $s \mto a + a_x s$.
\msk
Then the set of affine linear functions in $S \to \R$ is \und{almost}
closed by the ring operations --- the only problem is the
second-order term in the result of the multiplication
(underlined below):
%D diagram SRL-as-ring-object
%D 2Dx 100 +50 +12 +80
%D 2D 100 * |---> (s|->0)
%D 2D +6 {}* |---> (s|->1)
%D 2D +6 a+b <-------------------| a,b
%D 2D +7 ab <-------------| a,b{}
%D 2D
%D (( a+b .tex= (s|->a+b+(a_x+b_x)s)
%D ab .tex= (s|->ab+(a_xb+ab_x)s+\und{a_xb_xs^2})
%D a,b .tex= (\ldots),(\ldots)
%D a,b{} .tex= (\ldots),(\ldots)
%D * (s|->0) |->
%D {}* (s|->1) |->
%D a+b a,b <-|
%D ab a,b{} <-|
%D ))
%D enddiagram
%D
$$\diag{SRL-as-ring-object}$$
However, if $S \subseteq \sst{x \in \R}{x^2 = 0}$ then the
second-order term
disappears, and the set of affine linear functions
$$\AffLin(S \to R)
:= \sst{f:S \to \R}{f \text{ is affine linear}}
\subseteq (S \to \R)
$$
is a subring of $S \to \R$, and, furthermore, there is a ring
homeomorphism $\phi: T\R \to (S \to \R)$...
\newpage
% --------------------
% «ring-object-morphism» (to ".ring-object-morphism")
% (s "A homomorphism between ring objects" "ring-object-morphism")
\myslide {A homomorphism between ring objects} {ring-object-morphism}
``$\phi: T\R \to (S \to \R)$ is a ring homomorphism'' means that
for each of the five operations, $0, 1, +, ·, -$, a certain square
commutes...
%D diagram romorphism
%D 2Dx 100 +45 +65
%D 2D 100 1 =======> T\R <============ T\R×T\R
%D 2D | | |
%D 2D | | |
%D 2D | v v
%D 2D +30 {}1 ====> (S->\R) <====== (S->\R)×(S->\R)
%D 2D
%D (( 1 T\R T\R×T\R
%D {}1 (S->\R) (S->\R)×(S->\R)
%D @ 0 @ 1 -> sl^ .plabel= a 0
%D @ 0 @ 1 -> sl_ .plabel= b 1
%D @ 1 @ 2 <- sl^ .plabel= a +
%D @ 1 @ 2 <- sl_ .plabel= b ·
%D @ 3 @ 4 -> sl^ .plabel= a 0
%D @ 3 @ 4 -> sl_ .plabel= b 1
%D @ 4 @ 5 <- sl^ .plabel= a +
%D @ 4 @ 5 <- sl_ .plabel= b ·
%D @ 0 @ 3 -> .plabel= l \id
%D @ 1 @ 4 -> .plabel= r \phi
%D @ 2 @ 5 -> .plabel= r \phi×\phi
%D ))
%D enddiagram
%D
$$\diag{romorphism}$$
(We do not draw the `$-$' arrows).
The less trivial case is the square for `$·$':
%D diagram rmomult
%D 2Dx 100 +13 +120
%D 2D 100 ab1 <-----| a,b1
%D 2D - -
%D 2D | |
%D 2D v |
%D 2D +30 ab2 |
%D 2D v
%D 2D +10 ab3 <-| a,b3
%D 2D
%D (( ab1 .tex= (ab,a_xb+ab_x)
%D a,b1 .tex= (a,a_x),(b,b_x)
%D ab2 .tex= (s|->ab+(a_xb+ab_x)s)
%D ab3 .tex= (s|->ab+(a_xb+ab_x)s+\und{a_xb_xs^2})
%D a,b3 .tex= (s|->a+a_xs),(s|->b+b_xs)
%D @ 0 @ 1 <-|
%D @ 0 @ 2 |-> @ 1 @ 4 |-> # @ 2 @ 3 =
%D @ 3 @ 4 <-|
%D ))
%D enddiagram
%D
$$\diag{rmomult}$$
As we are supposing that $S \subseteq \sst{x \in \R}{x^2 = 0}$,
the term $a_x b_x s^2$ is zero, and that square commutes.
\bsk
In $\R$ the set of square-zero elements, $\sst{x \in \R}{x^2=0}$,
is too small for this to be interesting --- {\sl but the same
constructions work for any ring $R$.}
\msk
Example: $R := \R[X, Y]/\ang{X^2,Y^2}$ --- the ring of polynomials
on two variables, `$X$' and `$Y$', with coefficients on $\R$,
divided by an ideal to force $X^2=0$ and $Y^2=0$.
\msk
{\bf Notational convention:} $\ee^2=0$ and $\dd^2=0$.
Then, using `$\ee$' and `$\dd$' as variables, we can write
just ``$\R[\ee, \dd]$'' instead of ``$\R[\ee,
\dd]/\ang{\ee^2,\dd^2}$''.
\msk
Note that $(\ee+\dd)^2 = \ee^2 + 2\ee\dd + \dd^2 = 2\ee\dd \neq 0$ ---
so $\ee + \dd$ is not a square-zero element in $\R[\ee, \dd]$.
\newpage
% --------------------
% «ring-object-of-lt» (to ".ring-object-of-lt")
% (s "Ring objects of line type" "ring-object-of-lt")
\myslide {Ring objects of line type} {ring-object-of-lt}
\ssk
Fact (a.k.a. ``Main Theorem'', proved in the next slides):
When the arrow $\aa$ below is invertible we can use
the composite $\cc := (\aa^{-1};π_2)$ to define, for any
$f: R \to R$, its derivative $f': R \to R$,
and these derivatives behave as expected:
\ssk
$\begin{array}{rcl}
(kf)' & = & kf' \\
(f+g)' & = & f'+g', \\
(fg)' & = & f'g + fg', \\
(f∘g)' & = & (f'∘g)\,g'. \\
\end{array}
$
%L -- forths["sl_/2"] = macro(".slide= -1.25pt")
%L -- forths["sl^/2"] = macro(".slide= 1.25pt")
%L forths["sl_/2"] = function () ds:pick(0).slide = "-1.25pt" end
%L forths["sl^/2"] = function () ds:pick(0).slide = "1.25pt" end
%L forths["<.|"] = function () pusharrow("<.|") end
%L forths["|.>"] = function () pusharrow("|.>") end
\msk
\widemtos
%D diagram aabbcc
%D 2Dx 100 +35 +35 +20 +35 +35
%D 2D 100 --| R^D |-- dx|->a+a_xdx
%D 2D / ^ \ \ ^ /.
%D 2D \bb / | \ \bb / | `.
%D 2D v - v v - v
%D 2D +30 {}R <----- R×R -----> R{} a <------| a,a_x |----> a_x
%D 2D π_1 π_2
%D (( R^D {}R R×R R{}
%D @ 0 @ 1 -> .plabel= l \bb
%D @ 0 @ 2 <- sl_ .plabel= l \aa
%D @ 0 @ 2 .> sl^/2 .plabel= r \aa^{-1}
%D @ 0 @ 3 .> .plabel= r \cc
%D @ 1 @ 2 <- .plabel= b π_1
%D @ 2 @ 3 -> .plabel= b π_2
%D ))
%D (( dx|->a+a_xdx .tex= (dx|->a+a_xdx)
%D a a,a_x a_x
%D @ 0 @ 1 |-> .plabel= l \bb
%D @ 0 @ 2 <-| sl_ .plabel= l \aa
%D @ 0 @ 2 |.> sl^ .plabel= r \aa^{-1}
%D @ 0 @ 3 |.> .plabel= r \cc
%D @ 1 @ 2 <-| .plabel= b π_1
%D @ 2 @ 3 |-> .plabel= b π_2
%D ))
%D enddiagram
%D
$\diag{aabbcc}$
\bsk
The hypotheses are just these:
$\catC$ is a category with finite limits,
$(R, 0, 1, +, ·, -)$ is a ring object in $\catC$,
and $D := \sst{dx ∈ R}{dx^2=0}$
(that is definable as an equalizer)
is exponentiable.
\msk
({\sl Stronger hypotheses, simpler to understand:}
$\catC$ is cartesian closed and has pullbacks,
$(R, 0, 1, +, ·, -)$ is a ring object in $\catC$.)
\msk
Then if the (definable) map $\aa: R×R \to R^D$ is
invertible, we have a notion of ``derivative'' for
functions $R \to R$, that behaves as expected.
\bsk
A ring $(R, 0, 1, +, ·, -)$ for which
$\aa: R×R \to R^D$ is invertible
is said to be ``of line type''.
\newpage
% --------------------
% «beta-is-known» (to ".beta-is-known")
% (s "Lemma: the map $beta$ is known" "beta-is-known")
\myslide {Lemma: the map $\beta$ is known} {beta-is-known}
Lemma: even when $\aa^{-1}$ does not exist $\bb$ is known...
More precisely: {\sl define} $\bb$ as ``evaluate $dx \mto a + a_x dx$
at $dx:=0$''; then $(\aa;\bb)=π_1$.
If $\aa^{-1}$ exists then $(\aa;\bb)=π_1$ iff $\bb = (\aa^{-1};π_1)$.
\end{document}
% (find-angg "LATEX/2008sdg.tex")
* (eepitch-shell)
* (eepitch-kill)
* (eepitch-shell)
cp -v ~/LATEX/2008sdg.tex /tmp/o
~/LUA/texcatcodes.lua -trans /tmp/o /tmp/o2
# (find-fline "/tmp/o2")
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