Concepts locaux et globaux. Deuxième partie: Théorie ‚fonctorielle‘ Guerino Mazzola U & ETH Zürich Internet Institute for Music Science guerino@mazzola.ch www.encyclospace.org

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

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

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

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

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

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

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...

Where we are... Motivic strip of Zig-Zag (15) 5 6 4 (16) (19) (19) 7 3 8 2 5 3 (15) Where we are... (16) (19) (19) (2) (11) (20) (10) (15)

B = „EH“ ª —2 M Í Ÿ@B E H Where we are... 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. Where we are... ADn* KI res

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

„Classified“ 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 Where we are... „Classified“

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

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 = Mod@ presheaves over the category Mod of modules

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

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: ? ~> ?

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?

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}

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!

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) What is a Topos?

Presheaves @M = presheaf of M 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“ Presheaves B u A M u@M: A@M  B@M 1A@M = 1A@M u.v:C  B  A u.v@M = v@M.u@M g g.u @M = presheaf of M

Presheaves Mod@ = category of presheaves on Mod Presheaves: F: Mod  Sets: A ~> F(A) A@F Together with the transition maps u@F : A@F  B@F for u:B  A with the properties Presheaves 1A@F = 1A@F u.v: C  B  A u.v@F = v@F.u@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 Presheaves „small topos within a large topos“ Sets @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 ~> u@M(K)  B@M Presheaves K 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 ~> u@F(K)  B@F Presheaves K u@F(K)

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

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

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

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

Presheaves A@XY ª Hom(@A, XY) (Yoneda!) ª Hom(@A ¥ Y, X) (axiom) Example: E = Mod@ A@XY ª Hom(@A, XY) (Yoneda!) ª Hom(@A ¥ Y, X) (axiom) Define: Presheaves 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 Presheaves 1 W X Y c ! Subpresheaves(Y) ª Hom(Y,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

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 Functorial Locs K F f@K^ 1A f:X  A @A

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

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  Ÿt @ Ÿ12 Ÿ12 S1 Sk

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 f@K^ S1.f Sk.f Ÿ12

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  Ÿ12 ~> X* = End*(X)  Ÿ12@Ÿ12 A = Ÿ12

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

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

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

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

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

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

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

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

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

FM-Synthesis concept modeling

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

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

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

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

concept modeling

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, Mod@) Types Mono(Mod@) Sema(Forms, Mod@) = Types x Mono(Mod@) x Dia(Formsº, Mod@)

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

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

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) D:A@F(K)

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

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