Cellular Cohomology In Homotopy Type Theory 20180709 Ulrik - PowerPoint PPT Presentation

Cellular Cohomology In Homotopy Type Theory 20180709 Ulrik Buchholtz Favonia TU Darmstadt U of Minnesota 1

Cohomology Groups { mappings from holes in a space } 2

Cohomology Groups { mappings from holes in a space } Cellular Axiomatic Eilenberg-Steenrod cohomology for CW complexes cohomology Goal: prove they are the same! 2

CW complexes inductively-de � ined spaces 3

CW complexes inductively-de � ined spaces points 3

CW complexes inductively-de � ined spaces points lines 3

CW complexes inductively-de � ined spaces points lines faces (and more...) Data: cells and how they a � ach 3

CW complexes Sets of cell indices: A n A � aching: α n+1 : A n+1 × S n → X n X n is the construction up to dim. n X 0 := A 0 X n+1 := A n+1 ×S n A n+1 α n+1 X n X n+1 4

CW complexes Sets of cell indices: A n A � aching: α n+1 : A n+1 × S n → X n X n is the construction up to dim. n X 0 := A 0 X n+1 := A n+1 ×S n A n+1 α n+1 X n X n X n+1 4

CW complexes Sets of cell indices: A n A � aching: α n+1 : A n+1 × S n → X n X n is the construction up to dim. n X 0 := A 0 X n+1 := a : A n+1 A n+1 ×S n A n+1 α n+1 X n X n X n+1 4

CW complexes Sets of cell indices: A n A � aching: α n+1 : A n+1 × S n → X n X n is the construction up to dim. n X 0 := A 0 X n+1 := a : A n+1 A n+1 ×S n A n+1 S n α n+1 (a,-) α n+1 X n X n X n+1 4

Cellular Cohomology { mappings from holes in a space } dualize Cellular Homology { holes in a space } 5

One-Dimensional Holes* { elements of Z[A 1 ] forming cycles } holes a d a + b + c c b - a - b - c e f a + b + c + e + g + f … g *Holes are cycles in the classical homology theory 6

One-Dimensional Holes { elements of Z[A 1 ] forming cycles } x a boundary function ∂ y a ∂ ( ) = y - x c b x y ∂ (a+b+c) = (y - x) + (z - y) z + (x - z) = 0 set of holes = kernel of ∂ 7

First Homology Groups { un � illed one-dimensional holes } a 2-dim. boundary function ∂ 2 p a c b ∂ 2 ( ) = a + b + c p c b � illed holes = image of ∂ 2 8

First Homology Groups { un � illed one-dimensional holes } a 2-dim. boundary function ∂ 2 p a c b ∂ 2 ( ) = a + b + c p c b � illed holes = image of ∂ 2 H 1 (X) := kernel of ∂ 1 / image of ∂ 2 (un � illed (all holes) ( � illed holes) holes) 8

Homology Groups { un � illed holes } C n := Z[A n ] formal sums of cells (chains) ∂ n+2 ∂ n+1 ∂ n ∂ n-1 ⋯ → C n+2 → C n+1 → C n → C n-1 → C n-2 → ⋯ H n (X) := kernel of ∂ n / image of ∂ n+1 9

Cohomology Groups ∂ n+2 ∂ n+1 ∂ n ∂ n-1 ⋯ → C n+2 → C n+1 → C n → C n-1 → C n-2 → ⋯ Dualize by Hom(—, G). Let C n = Hom(C n , G). δ n+2 δ n+1 δ n δ n-1 ⋯ ← C n+2 ← C n+1 ← C n ← C n-1 ← C n-2 ← ⋯ H n (X; G) := kernel of δ n+1 / image of δ n 10

2-Dimensional Boundary a p c b a ∂ 2 ( ) = a + b + c p c b How to compute the coe � icients from α 2 ? 11

2-Dimensional Boundary a p c b a a a α 2 (p,—) identify squash points other loops coe � icient = winding number of this map (can be generalized to higher dimensions) 12

Cohomology Groups { mappings from holes in a space } Cellular Axiomatic Eilenberg-Steenrod cohomology for H n (X; G) CW-complexes cohomology Prove they are the same! 13

Eilenberg-Steenrod* cohomology A family of functors h n (—): pointed spaces → abelian groups *ordinary, reduced 14

Eilenberg-Steenrod* cohomology A family of functors h n (—): pointed spaces → abelian groups 1. h n+1 (susp(X)) ≃ h n (X), natural in X *ordinary, reduced 14

Eilenberg-Steenrod* cohomology A family of functors h n (—): pointed spaces → abelian groups 1. h n+1 (susp(X)) ≃ h n (X), natural in X 2. f A B 1 Cof f *ordinary, reduced 14

Eilenberg-Steenrod* cohomology A family of functors h n (—): pointed spaces → abelian groups 1. h n+1 (susp(X)) ≃ h n (X), natural in X h n (A) h n (A) h n (B) 2. f A B exact! Cof f h n (Cof f ) 1 *ordinary, reduced 14

Eilenberg-Steenrod* cohomology A family of functors h n (—): pointed spaces → abelian groups 1. h n+1 (susp(X)) ≃ h n (X), natural in X 3. h n ( ⋁ i X i ) ≃ ∏ i h n (X i ) h n (A) h n (A) h n (B) 2. f if the index type A B exact! is nice enough** Cof f h n (Cof f ) 1 *ordinary, reduced **see our paper 14

Eilenberg-Steenrod* cohomology A family of functors h n (—): pointed spaces → abelian groups 1. h n+1 (susp(X)) ≃ h n (X), natural in X 3. h n ( ⋁ i X i ) ≃ ∏ i h n (X i ) h n (A) h n (A) h n (B) 2. f if the index type A B exact! is nice enough** Cof f h n (Cof f ) 4. h n (2) trivial for n ≠ 0 1 *ordinary, reduced **see our paper 14

Cohomology Groups { mappings from holes in a space } Cellular Axiomatic Eilenberg-Steenrod cohomology for H n (X; G) h n (X) CW-complexes cohomology Prove they are the same! 15

Proof Plan H n (X; h 0 (2)) ≃ h n (X) ? 16

Proof Plan H n (X; h 0 (2)) ≃ h n (X) ? ≃ := ker( δ n+1 )/im( δ n ) ker( δ ' n+1 )/im( δ ' n ) 1. Find δ ' such that h n (X) ≃ ker( δ ' n+1 )/im( δ ' n ) 16

Proof Plan H n (X; h 0 (2)) ≃ h n (X) ? ≃ := ker( δ n+1 )/im( δ n ) ker( δ ' n+1 )/im( δ ' n ) δ ≃ δ ' 1. Find δ ' such that h n (X) ≃ ker( δ ' n+1 )/im( δ ' n ) 2. Show δ and δ ' are equivalent 16

Proof Plan H n (X; h 0 (2)) ≃ h n (X) ? ≃ := ker( δ n+1 )/im( δ n ) ker( δ ' n+1 )/im( δ ' n ) δ ≃ δ ' 1. Find δ ' such that h n (X) ≃ ker( δ ' n+1 )/im( δ ' n ) 2. Show δ and δ ' are equivalent As usual, fully mechanized in Agda! 16

Step 1: Reverse Engineering For any theory h, � inite pointed CW-complex X, there exist homomorphisms δ ' δ ' n+2 δ ' n+1 δ ' n δ ' n-1 ⋯ ← D n+2 ← D n+1 ← D n ← D n-1 ← D n-2 ← ⋯ such that h n (X) ≃ kernel of δ ' n+1 / image of δ ' n 17

Important Lemmas for Step 1 Long exact sequenses f A B 1 Cof f 18

Important Lemmas for Step 1 Long exact sequenses h n (A) h n (B) f A B n++ Cof f h n (Cof f ) 1 h n (Cof f ) h n (B) h n (A) h n+1 (Cof f ) h n+1 (B) h n+1 (A) 18

Important Lemmas for Step 1 Long exact sequenses h n (A) h n (B) f A B n++ Cof f h n (Cof f ) 1 Wedges of cells h m (X n /X n-1 ) ≃ hom(Z[A n ], h 0 (2)) trivial if when m = n or trivial otherwise m ≠ n h m (X 0 ) ≃ hom(Z[A 0 \{pt}], h 0 (2)) when m = 0 or trivial otherwise 19

Ultimate Co � iber Diagram X 0 X n-1 X n X n+1 1 X n-1/0 X n/0 X n+1/0 X n/m := X n /X m 1 X n/n-1 X n+1/n-1 1 X n+1/n 20

Ultimate Co � iber Diagram X 0 X n-1 X n X n+1 1 X n-1/0 X n/0 X n+1/0 X n/m := X n /X m Plan: 1 X n/n-1 X n+1/n-1 Obtain long exact sequences and use group-theoretic magic 1 X n+1/n 20

Ultimate Co � iber Diagram X 0 X n-1 X n X n+1 1 X n-1/0 X n/0 X n+1/0 X n/m := X n /X m Plan: 1 X n/n-1 X n+1/n-1 Obtain long exact sequences and use group-theoretic magic 1 X n+1/n 20

group + theory h n (X) ≃ ker( δ ' n+1 )/im( δ ' n ) 21

Proof Plan (updated) H n (X; h 0 (2)) ≃ h n (X) ? ≃ := ker( δ n+1 )/im( δ n ) ker( δ ' n+1 )/im( δ ' n ) δ ≃ δ ' 1. Find δ ' such that h n (X) ≃ ker( δ ' n+1 )/im( δ ' n ) 2. Show δ and δ ' are equivalent 22

Step 2: Calculation δ n+2 δ n+1 δ n δ n-1 ⋯ ← C n+2 ← C n+1 ← C n ← C n-1 ← C n-2 ← ⋯ ≃ ≃ ≃ ≃ ≃ ⋯ ← D n+2 ← D n+1 ← D n ← D n-1 ← D n-2 ← ⋯ δ ' n+2 δ ' n+1 δ ' n δ ' n-1 The n=0 case (C 0 ≃ D 0 ) is interesting 23

Summary Cellular Ordinary reduced ≃ cohomology cohomology ( � inite) groups groups 24

Summary Cellular Ordinary reduced ≃ cohomology cohomology ( � inite) groups groups To-Do In � inity: colimits Homology ⇨ Poincaré duality, ... Parametrization ⇨ non-orientability, ... 24

Higher-Dim. Boundary α 2 (p,—) X n /X n-1 ≃⋁ S n S n S n X n α n+1 (p,—) identify squash lower structs. coe � icient = degree of this map 25

Higher-Dim. Boundary α 2 (p,—) X n /X n-1 ≃⋁ S n S n S n X n α n+1 (p,—) identify squash lower structs. coe � icient = degree of this map - squashing needs decidable equality - linear sum needs closure- � initeness (free for � inite cases) 25

Higher-Dim. Boundary S n A n ×S n-1 A n+1 ×S n A n A n+1 X n-1 X n X n+1 X n /X n-1 ≃⋁ S n 1 S n 26

X n/n-1 X n+1/n-1 1 X n+1/n 27

X n/n-1 X n+1/n-1 1 X n+1/n h n (X n+1/n ) h n (X n+1/n-1 ) h n (X n/n-1 ) h n+1 (X n+1/n ) h n+1 (X n+1/n-1 ) h n+1 (X n/n-1 ) 27

X n/n-1 X n+1/n-1 1 X n+1/n h n (X n+1/n ) h n (X n+1/n-1 ) h n (X n/n-1 ) our choice of δ ' h n+1 (X n+1/n ) h n+1 (X n+1/n-1 ) h n+1 (X n/n-1 ) 27

X n/n-1 X n+1/n-1 1 X n+1/n ker( δ ') trivial ≃ inj h n (X n+1/n ) h n (X n+1/n-1 ) h n (X n/n-1 ) our choice of δ ' h n+1 (X n+1/n ) h n+1 (X n+1/n-1 ) h n+1 (X n/n-1 ) surj ≃ trivial 27 coker( δ ')

X m X m+1 1 X m+1/m h n (X m+1/m ) h n (X m+1 ) h n (X m ) h n+1 (X m+1/m ) If n ≠ m, m+1, both ends trivial, h n (X m+1 ) ≃ h n (X m ) h n (X n-1 ) ≃ h n (X n-2 ) ≃ ⋯ ≃ h n (X 0 ), trivial three h n (X n ) possible values h n (X n+1 ) ≃ h n (X n+2 ) ≃ ⋯ ≃ h n (X) 28