Transcript Concepts locaux et globaux. Deuxième partie: Théorie ‚fonctorielle‘ Guerino Mazzola U & ETH Zürich Internet Institute for Music Science [email protected] www.encyclospace.org.

Concepts locaux et globaux.
Deuxième partie:
Théorie ‚fonctorielle‘
Guerino Mazzola
U & ETH Zürich
Internet Institute for Music Science
[email protected]
www.encyclospace.org
Contents
• Where we are...
• What is a topos?
• The topos of presheaves
• Functorial local compositions
• Concept modeling over topoi
Where we are...
C Ÿ12
(chords)
M — 2
(motives)
Ambient space
Ÿ12= finite ->enumeration, Pólya & de Bruijn
—2 = infinite -> ??
Where we are...
K B
B
set
module
B @0Ÿ@B
A Ÿ@B
K 0
• A = Ÿn: sequences
(b0,b1,…,bn)
• A = B: self-addressed tones
Need general addresses A
Where we are...
B
M A@B
B
A@B = eB.Lin(A,B)
A=R
R@B = eB.Lin(R,B)
 B2
Where we are...
Ÿ12
S
A@B = eB.Lin(A,B)
R = Ÿ, A = Ÿ11, B= Ÿ12
Series: S  Ÿ11 @ Ÿ12 = e Ÿ12.Lin(Ÿ11, Ÿ12)
 Ÿ12 12
Where we are...
I
II
III
IV
V
VI
VII
Where we are...
The class nerve cn(K) of global composition
is not classifying
2
5
II
2
10
5
10
2
10
6
2
V
VII
10
2
VI
IV
2
6
2
2
5
15
2
2
10
I
2
6
5
2
2
2
10
III
Where we are...
Motivic strip of Zig-Zag
(15)
5
6
4
(19)
(19)
7
3
(2)
1
(11)
8
(20)
2
9
(10)
(15)
(16)
Where we are...
M Ÿ@B
B = „EH“  —2
H
E
E
Where we are...
Have universal construction of a „resolution of KI“
res: ADn* KI
It is determined only by the KI address A and the
nerve n* of the covering atlas I.
res
ADn*
KI
Where we are...
3
6
1
4
5
0Dn*
res
2
4
6
2
2
1
5
1
4
d
c
a
b
4
5
KI
6
3
1
3
6
3
2
5
Where we are...
The category ObLocomA of local objective A-addressed
compositions has
as objects the couples (K, A@C) of sets K of
affine morphisms in A@C
and as morphisms f: (K, A@C) L, A@D)
set maps f: K L which are naturally induced by
affine morphism F in C@D
The category ObGlocomA of global objective A-addressed
compositions has
as objects KI coverings of sets K by atlases I of local objective
A-addressed compositions with manifold gluing conditions
and manifold morphisms ff: KI LJ, including and
compatible with atlas morphisms f: I  J
What is a Topos?
Sets
cartesian products X x Y
disjoint sums X Y
powersets XY
characteristic maps c:X —> 2
no „algebra“
Mod@
F: Mod —> Sets
presheaves
have all these
properties
Mod
direct products A≈B
has „algebra“
no powersets
no characteristic maps
What is a Topos?
A category E is a topos iff it
• has terminal object 1 and products A  B
• has initial object 0 and coproducts A +B
• has exponentials XY
• has a subobject classifier 1  W
Our examples:
1) E = Sets
2) E = Mod@
sets
presheaves over the
category Mod of modules
What is a Topos?
Example: E = Sets
A  B = cartesian product
A¥B
B
b
(a,b)
A
a
1 = {} is terminal:
There is a unique
!:X 1:
What is a Topos?
Example: E = Sets
A +B = disjoint union, 0 = 
A
A+B
B
0 =  is initial:
There is a unique
!:0 X:??
What is a Topos?
Example: E = Sets
XY = {maps f:Y  X}
Hom(Z, XY) ≈ Hom(Z  Y, X)
g: Z  XY ~> g*: Z  Y  X
g*(z,y) = g(z)(y)
What is a Topos?
Example: E = Sets
subobject classifier 1 W
X
W = 2 = {0,1}
1  2: 0 ~> 0
c)=0
iff
X
Y
c
!
1
2
X
Y
Subobjects(Y) ≈ Hom(Y,W)
{0,1}
What is a Topos?
Counterexample: E = ModR with R-linear maps
There is no subobject classifier here!
X
Y
c
!
W
0-module
X = Ker(c)
Y/X ≈ Im(c)  W
absurd!
What is a Topos?
More generally, take the category Mod of modules over any
rings, together with (di)affine morphism „A@B“.
This is not only not a topos, it has other not very agreable
properties:
• Have no module M+N for the property
k  A@M or k  A@N iff k  A@(M+N)
• Have no module P(M) for the property
K  A@M iff K  A@P(M)
Presheaves
Problem:
When replacing M by the set A@M, we loose
all information about M.
Solution:
Replace a module M by the system of sets
@M: Mod  Sets: A ~> A@M
„set of all perspectives of M, as viewed from A“
B
u
A
g
1A@M = 1A@M
g.u
M
@M = presheaf of M
u@M: A@M  B@M
u.v:C  B  A
u.v@M = [email protected]@M
Presheaves
Presheaves:
A@F
F: Mod  Sets: A ~> F(A)
Together with the transition maps
u@F : A@F  B@F for u:B  A
with the properties
1A@F = 1A@F
u.v: C  B  A
u.v@F = [email protected]@F
Mod@ = category of presheaves on Mod
Presheaves
Example 1
S = set,
@S: Mod  Sets: A ~> A@S = S
Transition maps, u: B  A,
u@S = 1S : S  S
Sets
„small topos within a
large topos“
@Sets
Mod@
Presheaves
Example 2
M = module,
2@M: Mod  Sets: A ~> A@ 2@M = 2A@M
Transition maps, u: B  A,
u@2@M : A@2@M  B@2@M
u@M: A@M  B@M
K  A@M
K
~>
u@M(K)  B@M
u@M(K)
Presheaves
Example 2*
F = presheaf,
2F: Mod  Sets: A ~> A@2F = 2A@F
Transition maps, u: B  A,
u@2F : A@2F  B@2F
u@F: A@F  B@F
K  A@F
K
~>
u@F(K)  B@F
u@F(K)
Presheaves
Example 3
M, N = modules,
@M+@N: Mod  Sets: A ~> A@M + A@N
Transition maps, u: B  A
A@M
A@N
B@M
B@N
Presheaves
Example 3*
F, G = presheaves,
F+G: Mod  Sets: A ~> A@F + A@G
Transition maps, u: B  A
A@F
A@G
B@F
B@G
Presheaves
Why are presheaves a solution?
Yoneda Lemma
The functorial map @: Mod  Mod@ is fully faithfull.
M  @M
• M@N ≈ Hom(@M,@N)
• M ≈ N iff @M ≈ @N
• M@F ≈ Hom(@M,F)
Mod@
@Mod
Mod
Presheaves
Example: E = Mod@
F  G = pointwise cartesian product
A@(F
G
G) = A@F  A@ G
F¥
A@G
G
g
(f,g)
A@F
F
f
1 = {} is terminal:
A@1
Unique
A@!: A@X
!:X 1:
Presheaves
Example: E = Mod@
F +G = pointwise disjoint union
A@(G
H) = A@G + A@ H
G +H
A@G
G
H
A@H
0 =  is initial:
A@0
Unique
!:0 X:??
A@!: A@0 A@X:
Presheaves
Example: E = Mod@
A@XY
 Hom(@A, XY)
 Hom(@A  Y, X)
(Yoneda!)
(axiom)
Define:
A@XY = Hom(@A  Y, X)
Presheaves
Example: E = Mod@
subobject classifier 1 W
A@W= {subpresheaves of @A} = {sieves in @A}
1  W : 0 ~> @A
X
Y
c
!
Subpresheaves(Y)  Hom(Y,W)
1
W
Functorial Locs
In Mod@ replace 2F by WF
Understand the musical meaning of the difference!
A@2F = 2A@F ={subsets of A@F}
= {A-addressed local objective compositions
in F}
? ObLocomA, but F = presheaf, not only module!
A@ WF ≈ Hom(@A,WF)
≈ Hom(@A  F,W)
≈ Subpresheaves(@A  F)
= {A-addressed local functorial compositions
in F}
Functorial Locs
^: A@2F  A@WF
K  A@F ~> K^  @A  F
X@K^  X@A  X@F
= {(f,x.f), f:X  A, x  K}
f@F: A@F  X@F: x ~> x.f
F
K
f@K^
@A
1A
f:X  A
Functorial Locs
K Ÿ @F
F = @EH  @—2
H
E
f10: 0Ÿ  Ÿ: 0 ~> 10
Functorial Locs
Ÿ12
S
series
S  Ÿ11 @ Ÿ12
K = {S}
More general: set of k sequences of pitch classes of length t+1
K = {S1,S2,...,Sk}
This is a „polyphonic“ local composition
Ÿ12
K  Ÿt @ Ÿ12
S1
Sk
Functorial Locs
Ÿ12
S1
Sk
s ≤ t, define morphism f: Ÿs  Ÿt
e0 ~> ei(0)
e1 ~> ei(1)
.................
es ~> ei(s)
f@K^
Ÿ12
S1.f
Sk.f
es
e0
e1
Ÿs
Functorial Locs
The „functorial“ change K ~> K^ has dramatic consequences
for the global theory!
II
A = 0Ÿ
I
II
III
IV
V
VI
V
VII
VI
I
VII
A = Ÿ12
IV
X  Ÿ12
~>
III
X* = End*(X)  Ÿ12@Ÿ12
Functorial Locs
IIII*
VI
VI*
VV*
IV*
IV
II*
VII*
VII
ToM, ch. 25
III*
III
I*
I*  II* = 
Ÿ12@Ÿ12
II*
Functorial Locs
X*  Ÿ12@Ÿ12
X*^  (Ÿ12@Ÿ12)^  @Ÿ12  @Ÿ12
@Ÿ12  @Ÿ12
I*^
I*
I*^I*II*^
II* 
=
Ÿ12@Ÿ12)^
(Ÿ
II*^
II*
Functorial Locs
@Ÿ12
I*
f@I*^
e0.4
I*^
II*
f@I*^f@II*^ e8.0
e11.3
II*^
f@II*^
@Ÿ12
1Ÿ12
f = e11.0: Ÿ12  Ÿ12
e0.4.e11.0 = e11.3.e11.0 = e8.0
Functorial Locs
@Ÿ12
I*
I*^
I*^  II*^
II*
II*^
@Ÿ12
0 = (I*  II*)^  I*^  II*^
Functorial Locs
Consequences for sheaves of functions
(Xi)
Xi
Z
(Xj)
Xj
(Xij) ¿ ≈ ? (Xji)
Functorial Locs
Grothendieck topology of finite covering families
Xi
Z
(Xi)
Xj
( Xi Z Xj)
Xi Z Xj
(Xj)
concept modeling
unity
completeness
discourse
infinite recursion
universal ramification
ordered combinatorics
concept
concept
concept
concept
concept modeling
AnchorNote
Pause
Note
Onset
Duration
Onset
Pitch
Loudness
Duration
–
–
–
Ÿ
STRG
–
concept modeling
MakroNote
• Ornaments
• Schenker Analysis
Satellites
AnchorNote
MakroNote
Pause
Onset
–
Note
Duration
–
Onset
Pitch
–
Ÿ
Loudness Duration
STRG
–
concept modeling
• FM-Synthesis
concept modeling
• FM-Synthesis
FM-Object
Knot
Support
Modulator
Amplitude
Frequency
Phase
–
–
–
FM-Object
concept modeling
F = form name
one of five „space“ types
Forms
a name diagram √ in Mod@
an identifier
monomorphism in Mod@
id: Functor(F) >Frame(√)

>
Functor(F)
F:id.type(√)
Frame(√)
concept modeling
Frame(√)-space for type:
synonyme √ = „G“ Functor(G)
synonyme(√) = Functor(G)
simple
√ = „“@B
simple(√) = @B
limit
√ = name diagram Mod@
limit(√) = lim(n. diagram Mod@)
colimit √ = name diagram Mod@
colimit(√) = colim(n. diagram Mod@)
power
√ = „G“ Functor(G)
power(√) = WFunctor(G)
renaming
representation
conjunction
disjunction
collection
Denotators
concept modeling
D = denotator name
address A
A
K
K  A@ Functor(F)
„A-valued point“

>
Form F
Functor(F)
D:A@F(K)
Frame(√)
concept modeling
concept modeling
ModE@ = Topos
R 
E @
Mod
 Mod
Forms
Names
F
S
Dia(Formsº, Mod@)
S(F) = (typeF,idF, √F)
Mono(Mod@)
Types
Sema(Forms, Mod@) = Types x Mono(Mod@) x Dia(Formsº, Mod@)
concept modeling
E = Topos
R E
Forms
Names
F
S
Dia(Formsº,E )
S(F) = (typeF,idF, √F)
Mono(E )
Types
Sema(Forms,E ) = Types x Mono(E ) x Dia(Formsº,E )
concept modeling
Names
F
√G
Forms
typeF
√F
H
typeG
typeH
√H
G
E -Denotators
concept modeling
D = denotator name
R  E
„address“ A R
A
K
K:A  Topor(F)
Topor(F) E
Form F:id.type(√)
id: Topor(F)
D:A@F(K)
 Frame(√)
>
concept modeling
Galois Theory
Form Semiotic
Defining equation
Defining diagram
fS(X) = 0
x1
x2
x3
Field S
xn
√F
F2
F1
Fr
Form Semiotic S