1. Concepts locaux et globaux. Deuxième partie: Théorie ‚fonctorielle‘
Guerino Mazzola
U & ETH Zürich
Internet Institute for Music Science

2. Where we are...
What is a topos?
The topos of presheaves
Functorial local compositions
Concept modeling over topoi
Contents

3. Where we are... C Í Ÿ12 (chords) M Í — 2 (motives) Ambient space
Ÿ12 = finite -> enumeration, Pólya & de Bruijn
—2 = infinite -> ??

4. Where we are... B K Í B set module B @ 0Ÿ@B K Í 0Ÿ@B A
A = Ÿn: sequences (b0,b1,…,bn)
A = B: self-addressed tones
Need general addresses A

5. Where we are... B M Í A@B M Í B A@B = eB.Lin(A,B) A = R
= eB.Lin(R,B)
ª B2

6. Where we are... Ÿ12 S A@B = eB.Lin(A,B) R = Ÿ, A = Ÿ11, B= Ÿ12
Series: S Î Ÿ12 = e Ÿ12.Lin(Ÿ11, Ÿ12)
ª Ÿ12 12

7. I II
III IV V VI
VII
Where we are...

8. Where we are... The class nerve cn(K) of global composition
is not classifying I IV II VI V
III
VII 10 15 5 6 2
Where we are...

9. Where we are... Motivic strip of Zig-Zag (15) 5 6 4 (16) (19) (19) 7 38 2 5 3
(15)
Where we are...
(16)
(19)
(19)
(2)
(11)
(20)
(10)
(15)

10. B = „EH“ ª —2
M Í E H
Where we are... E

11. 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.
Where we are...
ADn* KI
res

12. Where we are... 0Dn* res KI 6 5 2 3 4 1 a d b c 1 2 3 4 6 5 5 6 3 4 1

13. „Classified“ Where we are...
The category ObLocomA of local objective A-addressed compositions has
as objects the couples (K, of sets K of affine morphisms in
and as morphisms f: (K, ® (L, set maps f: K ® L which are naturally induced by affine morphism F in
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
Where we are...
„Classified“

14. What is a Topos? Sets cartesian products X x Y Mod@
F: Mod —> Sets
presheaves
have all these properties
Sets
cartesian products X x Y
disjoint sums X È Y
powersets XY
characteristic maps c: X —> 2
no „algebra“
What is a Topos?
Mod
direct products A≈B
has „algebra“
no powersets
no characteristic maps

15. 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
What is a Topos?
Our examples:
1) E = Sets sets
2) E = presheaves over the category Mod of modules

16. What is a Topos? A ¥ B = cartesian product 1 = {Æ} is terminal:
Example: E = Sets
A ¥ B = cartesian product B
A ¥ B
What is a Topos?
(a,b) b a A
1 = {Æ} is terminal:
There is a unique !:X ® 1: x ~> Æ

17. What is a Topos? A + B = disjoint union, 0 = Æ A B 0 = Æ is initial:
Example: E = Sets
A + B = disjoint union, 0 = Æ
A + B
What is a Topos? A B
0 = Æ is initial:
There is a unique !:0 ® X: ? ~> ?

18. What is a Topos? XY = {maps f:Y ® X} Hom(Z, XY) ª Hom(Z ¥ Y, X)
Example: E = Sets
XY = {maps f:Y ® X}
Hom(Z, XY) ª Hom(Z ¥ Y, X)
g: Z ® XY ~> g*: Z ¥ Y ® X g*(z,y)=g(z)(y)
What is a Topos?

19. What is a Topos? Example: E = Sets subobject classifier 1 ® W
= 2 = {0,1}
1 ® 2: 0 ~> 0 X Y ! c
What is a Topos?
c(x) = 0
iff
x Î X 1 2 Y X
Subobjects(Y) ª Hom(Y,W)
{0,1}

20. What is a Topos? Counterexample: E = ModR with R-linear maps
There is no subobject classifier here! 
0-module X Y c !
What is a Topos?
X = Ker(c)
Y/X ≈ Im(c)   absurd!

21. More generally, take the category Mod of modules over any rings, together with (di)affine morphism
This is not only not a topos, it has other not very agreable properties:
Have no module M+N for the property k  or k  iff k 
Have no module P(M) for the property K  iff K 
What is a Topos?

22. Presheaves @M = presheaf of M
Problem: When replacing M by the set we loose all information about M.
Solution: Replace a module M by the system of sets @M: Mod  Sets: A ~>
„set of all perspectives of M, as viewed from A“
Presheaves B u A M  =
u.v:C  B  A = g
g.u
@M = presheaf of M

23. Presheaves Mod@ = category of presheaves on Mod Presheaves:
F: Mod  Sets: A ~> F(A)
Together with the transition maps
:  for u:B  A
with the properties
Presheaves =
u.v: C  B  A =
= category of presheaves on Mod

24. Presheaves Example 1 S = set, @S: Mod  Sets: A ~> A@S = S
Transition maps, u: B  A,
= 1S : S  S
Presheaves
„small topos within a
large topos“
Sets
@Sets

25. Presheaves Example 2 M = module,
Mod  Sets: A ~> =
Transition maps, u: B  A,
:  
K  ~> 
Presheaves K

26. Presheaves Example 2* F = presheaf,
2F: Mod  Sets: A ~> =
Transition maps, u: B  A,
:  
K  ~> 
Presheaves K

27. Presheaves Example 3 M, N = modules,
Mod  Sets: A ~> +
Transition maps, u: B  A
Presheaves

28. Presheaves Example 3* F, G = presheaves,
F+G: Mod  Sets: A ~> +
Transition maps, u: B  A
Presheaves

29. Presheaves Why are presheaves a solution? Yoneda Lemma
The functorial map @: Mod ® is fully faithfull M ≈
M ≈ N ≈
Presheaves
@Mod
Mod

30. Presheaves F ¥ G = pointwise cartesian product A@1 = {Æ}
Example: E =
F ¥ G = pointwise cartesian product G
F ¥ G
¥ G) = ¥ G
Presheaves
(f,g) g f F
= {Æ}
1 = {Æ} is terminal:
Unique !:X ® 1: x ~> Æ

31. Presheaves F + G = pointwise disjoint union A@G G H A@H A@0
Example: E =
F + G = pointwise disjoint union
Presheaves
G + H
+ H) = + H G H
0 = Æ is initial:
Unique !:0 ® X: ? ~> ? ®

32. Presheaves A@XY ª Hom(@A, XY) (Yoneda!) ª Hom(@A ¥ Y, X) (axiom)
Example: E =
ª XY) (Yoneda!)
ª ¥ Y, X) (axiom)
Define:
Presheaves
= ¥ Y, X)

33. Presheaves Example: E = Mod@ subobject classifier 1 ® W
= {subpresheaves = {sieves
1 ® W : 0
Presheaves 1 W X Y c !
Subpresheaves(Y) ª Hom(Y,W)

34. ? Functorial Locs In Mod@ replace 2F by WF
Understand the musical meaning of the difference!
= ={subsets of
= {A-addressed local objective compositions in F}
ObLocomA, but F = presheaf, not only module!
WF ≈
≈ ¥ F,W)
≈ ¥ F)
= {A-addressed local functorial compositions in F} ?
Functorial Locs

35. Functorial Locs ^: A@2F  A@WF K  A@F ~> K^  @A  F
 
= {(f,x.f), f:X  A, x  K}
 x ~> x.f
Functorial Locs K F 1A
f:X  A @A

36. H Functorial Locs E K Í Ÿ @F F = @EH ª @—2 f1: 0Ÿ  Ÿ: 0 ~> 1

37. Functorial Locs series S Î Ÿ11 @ Ÿ12 K = {S} S
More general: set of k sequences of pitch classes of length t+1
K = {S1,S2,...,Sk}
This is a „polyphonic“ local composition K  Ÿ12
Ÿ12 S1 Sk

38. Functorial Locs s ≤ t, define morphism f: Ÿs  Ÿt e0 ~> ei(0)Sk
Ÿ12
Functorial Locs
s ≤ t, define morphism f: Ÿs  Ÿt
e0 ~> ei(0)
e1 ~> ei(1)
es ~> ei(s) e0 e1 es Ÿs
S1.f
Sk.f
Ÿ12

39. The „functorial“ change K ~> K^ has dramatic consequences
for the global theory! I IV V II
III VI
VII I IV II VI V
III
VII
Functorial Locs
A = 0Ÿ
X  Ÿ ~> X* = End*(X) 
A = Ÿ12

40. Functorial Locs ToM, ch. 25 II* I* Ÿ12@Ÿ12 I*  II* =  I* IV* II* VI*

41. Functorial Locs X*  Ÿ12@Ÿ12 X*^  (Ÿ12@Ÿ12)^  @Ÿ12  @Ÿ12 (Ÿ12@Ÿ12)^
I*  II* = 
II*
I*^  II*^  
II*^

42. Functorial Locs @Ÿ12 I*  e0.4 I*^ II*^ f@I*^f@II*^  e8.0 II* 
1Ÿ12
II*^
Functorial Locs 
e8.0
II* 
e11.3
f = e11.0: Ÿ12  Ÿ12
@Ÿ12
e0.4.e11.0 = e11.3.e11.0 = e8.0

43. Functorial Locs I* I*^ I*^  II*^ II* II*^ @Ÿ12

44. Functorial Locs Consequences for sheaves of functions Z Xi Xj (Xi)
(Xij)
(Xj)
(Xji)
¿ ≈ ?

45. Functorial Locs Grothendieck topology of finite covering families Xi ZXj
( Xi ¥Z Xj)
Xi ¥Z Xj
(Xj)

46. concept modeling unity infinite recursion completeness discourse
universal ramification
ordered combinatorics
concept modeling
concept
concept

47. concept modeling AnchorNote Pause Note Onset Duration Onset Loudness
Pitch – – – Ÿ
STRG –

48. concept modeling MakroNote Satellites AnchorNote MakroNote Ornaments
Schenker Analysis
Satellites
AnchorNote –
Onset
Loudness
Duration
Pitch
Note
STRG Ÿ
Pause
concept modeling
MakroNote

49. FM-Synthesis
concept modeling

50. concept modeling FM-Object Knot Support Modulator Amplitude Phase
FM-Synthesis
FM-Object
Knot
concept modeling
Support
Modulator
Amplitude
Phase
Frequency
FM-Object – – –

51. concept modeling Forms F = form name one of five „space“ types
a name diagram √ in
Forms
an identifier monomorphism in id: Functor(F) >® Frame(√)
concept modeling
Frame(√)
>®
Functor(F)
F:id.type(√)

52. concept modeling renaming representation conjunction disjunction
Frame(√)-space for type:
synonyme √ = „G“ ~> Functor(G)
synonyme(√) = Functor(G)
renaming
simple √ =
simple(√)
representation
concept modeling
limit √ = name diagram ®
limit(√) = lim(n. diagram ®
conjunction
colimit √ = name diagram ®
colimit(√) = colim(n. diagram ®
disjunction
power √ = „G“ ~> Functor(G)
power(√) = WFunctor(G)
collection

53. concept modeling Denotators D = denotator name A address A K
Frame(√)
K Î Functor(F)
„A-valued point“
>®
Functor(F)
Form F

54. concept modeling

55. concept modeling E = Topos Mod@ = Topos R Í E S Mod Í Mod@ Names F
Forms S
S(F) = (typeF,idF, √F) F
concept modeling
Dia(Formsº,
Types
Sema(Forms, = Types x x Dia(Formsº,

56. concept modeling E = Topos R Í E S Names F S(F) = (typeF,idF, √F)
Forms
Sema(Forms,E ) = Types x Mono(E ) x Dia(Formsº,E )
Types
Mono(E )
Dia(Formsº,E ) S
S(F) = (typeF,idF, √F) F
concept modeling

57. Names F √G
Forms
typeF
concept modeling √F H
typeG
typeH √H G

58. concept modeling E -Denotators R Í E D = denotator name A
„address“ A Î R
K: A ® Topor(F)
Topor(F) Î E K
concept modeling
Form F:id.type(√)
Frame(√)
>®
id: Topor(F)

59. concept modeling Galois Theory Form Semiotic Defining equation
Defining diagram
fS(X) = 0
√ F x2 x1 xn x3 F2 Fr F1
concept modeling
Field S
Form Semiotic S

60. Local Techniques Qwertzuiopü¨$äölkjhgfdsayxcvbnm,.-
As¥≈©◊˙ASDFGHJKLéà£_:;MNBVCXYQWERTZUIOPè!?`=)(/&%ç*“
Local Techniques