Comparative Smootheology

Transcription

1 Comparative Smootheology Workshop on Smooth Structures in Logic, Category Theory, and Physics, Ottawa Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim 3rd May 2009

2 Contents 1. Smootheology Aim: Frölicher spaces are obvious generalisation of manifolds. 2. Comparative Aim: how the various categories of smooth objects fit together.

3 Smootheology There now follows a party political broadcast for the Frölicher Party

4 Goal Build a category extending smooth manifolds.

5 Goal Build a category extending smooth manifolds. To contain: Orbifolds: quotients

6 Goal Build a category extending smooth manifolds. To contain: Orbifolds: quotients Stratifolds: push-outs

7 Goal Build a category extending smooth manifolds. To contain: Orbifolds: quotients Stratifolds: push-outs Loop spaces: mapping objects

8 Goal Build a category extending smooth manifolds. To contain: Orbifolds: quotients Stratifolds: push-outs Loop spaces: mapping objects P 1 (M): mapping objects AND quotients

9 Goal Build a category extending smooth manifolds. To contain: Orbifolds: quotients Stratifolds: push-outs Loop spaces: mapping objects P 1 (M): mapping objects AND quotients Embedding spaces: mapping objects AND subobjects

10 Goal Build a category extending smooth manifolds. To contain: Orbifolds: quotients Stratifolds: push-outs Loop spaces: mapping objects P 1 (M): mapping objects AND quotients Embedding spaces: mapping objects AND subobjects And more...

11 A Sinclair Category

12 Goal Build a category extending smooth manifolds. To contain: Orbifolds: quotients Stratifolds: push-outs Loop spaces: mapping objects P 1 (M): mapping objects AND quotients Embedding spaces: mapping objects AND subobjects And more... C 5 : complete, co-complete, cartesian closed category.

13 Goal Build a category extending smooth manifolds. To contain: Orbifolds: quotients Stratifolds: push-outs Loop spaces: mapping objects P 1 (M): mapping objects AND quotients Embedding spaces: mapping objects AND subobjects And more... C 5 : complete, co-complete, cartesian closed category. Guiding principle: The Smooth,

14 Goal Build a category extending smooth manifolds. To contain: Orbifolds: quotients Stratifolds: push-outs Loop spaces: mapping objects P 1 (M): mapping objects AND quotients Embedding spaces: mapping objects AND subobjects And more... C 5 : complete, co-complete, cartesian closed category. Guiding principle: The Smooth, the Whole Smooth,

15 Goal Build a category extending smooth manifolds. To contain: Orbifolds: quotients Stratifolds: push-outs Loop spaces: mapping objects P 1 (M): mapping objects AND quotients Embedding spaces: mapping objects AND subobjects And more... C 5 : complete, co-complete, cartesian closed category. Guiding principle: The Smooth, the Whole Smooth, and Nothing But the Smooth.

16 The Smooth

17 Building a Category A category has Objects and Morphisms

18 Building a Category A category has Objects and Morphisms

19 Morphism of Manifolds f ψ φ 1 f : M N smooth if φ 1 fψ is C

20 Charts Charts control smooth structure

21 Charts Charts control smooth structure Domains are standard spaces

22 Charts Charts control smooth structure Domains are standard spaces Use both charts and inverses

23 Charts Charts control smooth structure Domains are standard spaces Use both charts and inverses Charts are homeomorphisms

24 Charts Charts control smooth structure Domains are standard spaces Use both charts and inverses Charts are homeomorphisms Lose homeomorphism: too fragile.

25 Charts Charts control smooth structure Domains are standard spaces Use both charts and inverses Charts are homeomorphisms Lose homeomorphism: too fragile. Retain: I(U) = {ψ: R m U M} O(V; R m ) = {φ: M V R m } input test functions output test functions

26 First Candidate Definition (First Attempt) A smooth space is a triple (X, I, O) where: X is a topological space I(U) Top(U, X), U R m open, O(V; R m ) Top(V, R m ), V X open.

27 First Candidate Definition (First Attempt) A smooth space is a triple (X, I, O) where: X is a topological space I(U) Top(U, X), U R m open, O(V; R m ) Top(V, R m ), V X open. A morphism is a continuous map f : X Y such that φfψ is C for ψ I(U), φ O(V; R m )

28 First Candidate Definition (First Attempt) A smooth space is a triple (X, I, O) where: X is a topological space I(U) Top(U, X), U R m open, O(V; R m ) Top(V, R m ), V X open. A morphism is a continuous map f : X Y such that φfψ C for ψ I, φ O Notation: smooth map = morphism ψ I, φ O, θ C

29 Morphisms φ f ψ

30 Categorical Construct? 1. Identities: 1 X : X X must be smooth. ψ I, φ O then φψ = φ1 X ψ C

31 Categorical Construct? 1. Identities: 1 X : X X must be smooth. ψ I, φ O then φψ = φ1 X ψ C ψ φ

32 Categorical Construct? 1. Identities: 1 X : X X must be smooth. ψ I, φ O then φψ = φ1 X ψ C ψ φ Compatibility Condition

33 Categorical Construct? 2. Composition: cts R const R const R cts

34 Categorical Construct? 2. Composition: cts R const smooth R const R cts

35 Categorical Construct? 2. Composition: cts R const smooth R const smooth R cts

36 Categorical Construct? 2. Composition: cts R const smooth R const not smooth smooth R cts

37 Composition Definition The completed inputs and outputs are 1. I(U) = {ψ: U X : φψ C, φ O} 2. O(V; R m ) = {φ: V R m : φψ C, ψ I}

38 Composition Definition The completed inputs and outputs are 1. I(U) = {ψ: U X : φψ C, φ O} 2. O(V; R m ) = {φ: V R m : φψ C, ψ I} ψ ψ φ

39 Composition Proposition Composition is well-defined for smooth objects where (I, O) satisfies compatibility.

40 Composition Proposition Composition is well-defined for smooth objects where (I, O) satisfies compatibility. ψ ψ φ φ

41 Composition Proposition Composition is well-defined for smooth objects where (I, O) satisfies compatibility. 1 ψ ψ φ φ 1

42 First Candidate Definition (First Attempt) A smooth space is a triple (X, I, O) where: X is a topological space I(U) Top(U, X), U R m open, O(V; R m ) Top(V, R m ), V X open. A morphism is a continuous map f : X Y such that φfψ C for ψ I, φ O

43 Second Candidate Definition (Second Attempt) A smooth space is a triple (X, I, O) where: X is a topological space I(U) Top(U, X), U R m open, O(V; R m ) Top(V, R m ), V X open. such that I and O are compatible, I and O are also compatible. A morphism is a continuous map f : X Y such that φfψ C for ψ I, φ O

44 The Whole Smooth

45 Too Many Smooth Spaces Spoil The Category Lemma (X, I, O) smooth space then (X, I, O) also smooth space and moreover is an isomorphism. 1 X : (X, I, O) (X, I, O)

46 Too Many Smooth Spaces Spoil The Category Lemma (X, I, O) smooth space then (X, I, O) also smooth space and moreover is an isomorphism. Proof. 1. I = I and O = O 1 X : (X, I, O) (X, I, O)

47 Too Many Smooth Spaces Spoil The Category Lemma (X, I, O) smooth space then (X, I, O) also smooth space and moreover is an isomorphism. Proof. 1. I = I and O = O 1 X : (X, I, O) (X, I, O) 2. φψ C for φ O, ψ I and for φ O, ψ I

48 Second Candidate Definition (Second Attempt) A smooth space is a triple (X, I, O) where: X is a topological space I(U) Top(U, X), U R m open, O(V; R m ) Top(V, R m ), V X open. such that I and O are compatible, I and O are also compatible. A morphism is a continuous map f : X Y such that φfψ C for ψ I, φ O

49 Third Candidate Definition (Third Attempt) A smooth space is a triple (X, I, O) where: X is a topological space I(U) Top(U, X), U R m open, O(V; R m ) Top(V, R m ), V X open. such that I and O are compatible, I and O are saturated: I = I, O = O. A morphism is a continuous map f : X Y such that φfψ C for ψ I, φ O

50 Comments Non-smoothness is detectable: If φ: X V R m is not in O(V; R m ) then there is ψ I(U) such that φψ is not C.

51 Comments Non-smoothness is detectable: If φ: X V R m is not in O(V; R m ) then there is ψ I(U) such that φψ is not C. Analogy: 2nd Definition versus 3rd Definition smooth atlas versus maximal smooth atlas

52 Detox Lemma (X, I, O) is completely determined by O(V; R) I(R)

53 Detox Lemma (X, I, O) is completely determined by O(V; R) I(R) Proof. 1. φ: U R m is C if and only if each p i φ: U R is C, p i projection

54 Detox Lemma (X, I, O) is completely determined by O(V; R) I(R) Proof. 1. φ: U R m is C if and only if each p i φ: U R is C, p i projection 2. ψ: U R m is C if and only if each ψγ: R R m is C, γ C (R, U) [Boman s Theorem]

55 Third Candidate Definition (Third Attempt) A smooth space is a triple (X, I, O) where: X is a topological space I(U) Top(U, X), U R m open, O(V; R m ) Top(V, R m ), V X open. such that I and O are compatible, I and O are saturated: I = I, O = O. A morphism is a continuous map f : X Y such that φfψ C for ψ I, φ O

56 Fourth Candidate Definition (Fourth Attempt) A smooth space is a triple (X, I, O) where: X is a topological space I Top(R, X), O(V) Top(V, R), V X open. such that I and O are compatible, I and O are saturated: I = I, O = O. A morphism is a continuous map f : X Y such that φfψ C for ψ I, φ O

57 Nothing But The Smooth

58 Chicken and Egg Question Is Smooth built on top of Continuous or alongside?

59 Egg and Chicken Question Is Smooth built on top of Continuous or alongside? Topology only used for local functions.

60 Egg and Chicken Question Is Smooth built on top of Continuous or alongside? Topology only used for local functions. Can get topology from I and O.

61 Egg and Chicken Question Is Smooth built on top of Continuous or alongside? Topology only used for local functions. Can get topology from I and O. Topology from O has bump functions.

62 Egg and Chicken Question Is Smooth built on top of Continuous or alongside? Topology only used for local functions. Can get topology from I and O. Topology from O has bump functions. Local functions extend globally.

63 Fourth Candidate Definition (Fourth Attempt) A smooth space is a triple (X, I, O) where: X is a topological space I Top(R, X), O(V) Top(V, R), V X open. such that I and O are compatible, I and O are saturated: I = I, O = O. A morphism is a continuous map f : X Y such that φfψ C for ψ I, φ O

64 Fifth Candidate Definition (Fifth Attempt) A smooth space is a triple (X, I, O) where: X is a set I Set(R, X), O Set(X, R). such that I and O are compatible, I and O are saturated: I = I, O = O. A morphism is a map f : X Y such that φfψ C for ψ I, φ O

65 Fifth Candidate: Frölicher Space Definition (Frölicher Space) A Frölicher space is a triple (X, C, F ) where: X is a set C Set(R, X), F Set(X, R). such that C = {ψ: R X : φψ C (R, R), φ F } F = {φ: X R : φψ C (R, R), ψ C} A morphism is a map f : X Y such that φfψ C for ψ C, φ F

66 Smootheology Theorem (Frölicher) The category of Frölicher spaces is a complete, co-complete, cartesian closed category. Which was what we wanted!

67 Smootheology Theorem (Frölicher) The category of Frölicher spaces is a C 5. W 5!

68 Smootheology Theorem (Frölicher) The category of Frölicher spaces is a C 5. Conclusion W 5! By focussing on morphisms and looking for simplicity we found a natural path to Frölicher spaces.

69 Comparative Frölicher spaces are not the only candidate.

70 Comparative Frölicher spaces are not the only candidate. Other versions due to K. T. Chen (4 versions!) J. M. Souriau (diffeological spaces) J. W. Smith R. Sikorski More?

71 Comparative Frölicher spaces are not the only candidate. Other versions due to K. T. Chen (4 versions!) J. M. Souriau (diffeological spaces) J. W. Smith R. Sikorski More? Goal: extend category of smooth manifolds to contain more things.

72 Comparative Frölicher spaces are not the only candidate. Other versions due to K. T. Chen (4 versions!) J. M. Souriau (diffeological spaces) J. W. Smith R. Sikorski More? Goal: extend category of smooth manifolds to contain more things. Motivation: particular examples and problems.

73 Countdown Categories All involve probing using test objects. Minor differences: underlying objects, test objects, direction. Key difference: Closure condition on probing functions.

74 Countdown Categories All involve probing using test objects. Minor differences: underlying objects, test objects, direction. Key difference: Closure condition on probing functions. Closure: Things that ought to be the same are the same.

75 Countdown Categories All involve probing using test objects. Minor differences: underlying objects, test objects, direction. Key difference: Closure condition on probing functions. Closure: Things that ought to be the same are the same. Examples Saturation: Frölicher, Smith. Sheaf: Chen, Souriau, Sikorski. Determined: Chen.

76 Examples: Souriau, Sikorski Definition (Diffeological (Souriau) space) Set X, diffeology D(U) Set(U, X) φ: U X in D if it is locally in D.

77 Examples: Souriau, Sikorski Definition (Diffeological (Souriau) space) Set X, diffeology D(U) Set(U, X) φ: U X in D if it is locally in D. Definition (Sikorski space) Top space X, functions F Top(X, R) algebra, locality, g(f 1,..., f n ) F for g C (R n, R)

78 Schematic Chen 1973 Chen 1975a Chen 1975b Smith Sikorski Chen 1977 Souriau Frölicher denotes adjunction

79 Distinctions: Souriau and Frölicher R 2

80 Distinctions: Souriau and Frölicher R 2 R 2 /R

81 Distinctions: Souriau and Frölicher R 2 R 2 /R Souriau

82 Distinctions: Souriau and Frölicher R 2 R 2 /R Souriau Frölicher

83 Distinctions: Sikorski and Frölicher X, Y R 2 as (t, t 2 ) Y X

84 Distinctions: Sikorski and Frölicher X, Y R 2 as (t, t 2 ) Y X Sikorski: X Y, t t smooth on Y, not on X,

85 Distinctions: Sikorski and Frölicher X, Y R 2 as (t, t 2 ) Y X Sikorski: X Y, t t smooth on Y, not on X, Frölicher: X Y, t t smooth on both.

86 Conclusion Frölicher spaces arise by taking seriously the notion of a morphism of smooth manifolds.

87 Conclusion Frölicher spaces arise by taking seriously the notion of a morphism of smooth manifolds. The various categories fit into a neat setting but are distinct. So may have distinct behaviour.