% (find-angg "LATEX/2008sdg.tex")
% (find-dn4ex "edrx08.sty")
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2008sdg.tex && latex    2008sdg.tex"))
% (defun c () (interactive) (find-zsh "cd ~/LATEX/ && ~/dednat4/dednat41 2008sdg.tex && pdflatex 2008sdg.tex"))
% (find-dvipage "~/LATEX/2008sdg.dvi")
% (find-pspage  "~/LATEX/2008sdg.pdf")

% (find-twupfile "LATEX/")
% (find-twusfile "LATEX/")
% http://angg.twu.net/LATEX/
% http://angg.twu.net/LATEX/2008sdg.pdf
% (ee-cp "~/LATEX/2008sdg.pdf" (ee-twupfile "/LATEX/2008sdg.pdf") 'over)
% (ee-cp "~/LATEX/2008sdg.pdf" (ee-twusfile "/LATEX/2008sdg.pdf") 'over)

% «.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")


\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 2008sdg.dnt

%
% (eedn4-51-bounded)

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
%:     -----  ---                  ---  ---          ---  ---
%:     _1    _2    \id           _1  _3          _2  _3
%:  ---------------  ---           --------          --------
%:  \ang{_1,_2};+  _3           \ang{_1,_3};·   \ang{_2,_3};·
%:  ----------------------------   ---------------------------------
%:  \ang{(\ang{_1,_2};+),_3};·  \ang{(\ang{_1,_3};·),(\ang{_2,_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{_1,_2};+),_3};·
%D    \R^3 \R -> sl_ .plabel= b \ang{(\ang{_1,_3};·),(\ang{_2,_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         +120
%D 2Dx         +13
%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 $\cc := (\aa^{-1};_2)$ 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["<.|"] = 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	   			          _1            _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 _1
%D     @ 2 @ 3 -> .plabel= b _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 _1
%D     @ 2 @ 3 |-> .plabel= b _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 $D := \sst{dx Ý R}{dx^2=0}$

(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 $(\aa;\bb)=_1$.

If $\aa^{-1}$ exists then $(\aa;\bb)=_1$ iff $\bb = (\aa^{-1};_1)$.

%

\end{document}

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