Borsuk-Ulam in real-cohesive homotopy type theory Daniel Cicala, - PowerPoint PPT Presentation

Borsuk-Ulam in real-cohesive homotopy type theory Daniel Cicala, University of New Haven Amelia Tebbe, Indiana University Kokomo Chandrika Sadanand, University of Illinois Urbana Champaign 1

Thanks to 2017 MRC in HoTT program Mike Shulman for his guidance and patience with three non -experts Univalence, which I’ll be recklessly using without mentioning I’m doing so 2

What’s this talk about? Borsuk-Ulam is a result in classical algebraic topology . We want to import it into HoTT. 3

Outline of this talk 1. real-cohesive homotopy type theory 2. Borsuk-Ulam: algebraic topology vs. HoTT 3. proof sketch 4

real-cohesive homotopy type theory 5

Algebraic topology has many proofs that involve discontinuous constructions For instance, Brouwer fixed-pt theorem � � � � � � � � � � f ( x ) = x � � � � � � � � � � � � f : D 2 → D 2 x : D 2 0 No continuous way to pick x 6

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However, we don’t merely have HoTT... we have real-cohesive HoTT 8

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For spaces X and Y , how can we make a discontinuous map X → Y into a continuous map? Retopologize! discrete ( X ) → Y or X → codiscrete ( Y ) There’s a ready-made theory for this... Lawvere’s cohesive topoi 10

For spaces X and Y , how can we make a discontinuous map X → Y into a continuous map? Retopologize! discrete ( X ) → Y or X → codiscrete ( Y ) There’s a ready-made theory for this... Lawvere’s cohesive topoi 10

For spaces X and Y , how can we make a discontinuous map X → Y into a continuous map? Retopologize! discrete ( X ) → Y or X → codiscrete ( Y ) There’s a ready-made theory for this... Lawvere’s cohesive topoi 10

Discontinuity via cohesive topoi SPACES SPACES ♯ := codiscretize ♭ := discretize codiscrete discrete forget ⊣ ⊣ ⊣ POINTS SPACES ♭ X → X → ♯ X Interpret ♭ X → Y or X → ♯ Y as not necessarily continuous maps from X → Y . 11

Discontinuity via cohesive topoi SPACES SPACES ♯ := codiscretize ♭ := discretize codiscrete discrete forget ⊣ ⊣ ⊣ POINTS SPACES ♭ X → X → ♯ X Interpret ♭ X → Y or X → ♯ Y as not necessarily continuous maps from X → Y . 11

Two concerns importing this to HoTT: 1) Algebraic topology trades in spaces not higher inductive types . 2) How can we retopologize when HoTT doesn’t have topologies (open sets)? 12

higher inductive types vs. spaces in HoTT 13

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Lawvere’s theory of cohesive topoi has more to offer! SPACES connected components codiscrete discrete forget ⊣ ⊣ ⊣ POINTS � gives modality : SPACES → SPACES 15

� Cohesive topos: is connected components � HoTT: is fundamental ∞ -groupoid � ⊣ ♭ ⊣ ♯ 16

� Cohesive topos: is connected components � HoTT: is fundamental ∞ -groupoid � ⊣ ♭ ⊣ ♯ 16

HoTT � + ⊣ ♭ ⊣ ♯ (suitably defined) cohesive homotopy type theory Notation S 1 := { ( x , y ) : x 2 + y 2 = 1 } S 1 := higher inductive type S 1 = S 1 , but we’re not there yet. � We want 17

HoTT � + ⊣ ♭ ⊣ ♯ (suitably defined) cohesive homotopy type theory Notation S 1 := { ( x , y ) : x 2 + y 2 = 1 } S 1 := higher inductive type S 1 = S 1 , but we’re not there yet. � We want 17

HoTT � + ⊣ ♭ ⊣ ♯ (suitably defined) cohesive homotopy type theory Notation S 1 := { ( x , y ) : x 2 + y 2 = 1 } S 1 := higher inductive type S 1 = S 1 , but we’re not there yet. � We want 17

incorporating topology into HoTT 18

The topology of a type X is encoded in the type of “continuous paths” R → X . � Needed to define : � An axiom ensuring that is constructed from continuous paths indexed by intervals in R . Axiom R ♭ : A type X is discrete iff const : X → ( R → X ) is an equivalence. 19

The topology of a type X is encoded in the type of “continuous paths” R → X . � Needed to define : � An axiom ensuring that is constructed from continuous paths indexed by intervals in R . Axiom R ♭ : A type X is discrete iff const : X → ( R → X ) is an equivalence. 19

The topology of a type X is encoded in the type of “continuous paths” R → X . � Needed to define : � An axiom ensuring that is constructed from continuous paths indexed by intervals in R . Axiom R ♭ : A type X is discrete iff const : X → ( R → X ) is an equivalence. 19

TYPES Axiom R ♭ : ∞ -groupoid codiscrete discrete forget X is discrete ⊣ ⊣ ⊣ + iff const : X = − → ( R → X ) TYPES —equals— S 1 = S 1 . � real-cohesive homotopy type theory , a place where 20

Borsuk-Ulam 21

Three related statements in classical algebraic topology: For any continuous map f : S n → R n , there exists BU-classic an x ∈ S n such that f ( x ) = f ( − x ). For any continuous map f : S n → R n with the property that f ( − x ) = − f ( x ), there is an x ∈ S n BU-odd such that f ( x ) = 0 There is no continuous map f : S n → S n − 1 with the property that there exists an x ∈ S n such that BU-retract f ( − x ) = − f ( x ). Proof of BU-classic involves proving that 1. show (BU-classic) ≃ (BU-odd) 2. show ¬ (BU-odd) ⇒ ¬ (BU-retract) 3. hence (BU-retract) ⇒ (BU-classic). 4. prove (BU-retract) 5. conclude (BU-classic) 22

Three related statements in classical algebraic topology: For any continuous map f : S n → R n , there exists BU-classic an x ∈ S n such that f ( x ) = f ( − x ). For any continuous map f : S n → R n with the property that f ( − x ) = − f ( x ), there is an x ∈ S n BU-odd such that f ( x ) = 0 There is no continuous map f : S n → S n − 1 with the property that there exists an x ∈ S n such that BU-retract f ( − x ) = − f ( x ). Proof of BU-classic involves proving that 1. show (BU-classic) ≃ (BU-odd) 2. show ¬ (BU-odd) ⇒ ¬ (BU-retract) 3. hence (BU-retract) ⇒ (BU-classic). 4. prove (BU-retract) 5. conclude (BU-classic) 22

Three related statements in classical algebraic topology: For any continuous map f : S n → R n , there exists BU-classic an x ∈ S n such that f ( x ) = f ( − x ). For any continuous map f : S n → R n with the property that f ( − x ) = − f ( x ), there is an x ∈ S n BU-odd such that f ( x ) = 0 There is no continuous map f : S n → S n − 1 with the property that there exists an x ∈ S n such that BU-retract f ( − x ) = − f ( x ). Proof of BU-classic involves proving that 1. show (BU-classic) ≃ (BU-odd) 2. show ¬ (BU-odd) ⇒ ¬ (BU-retract) 3. hence (BU-retract) ⇒ (BU-classic). 4. prove (BU-retract) 5. conclude (BU-classic) 22

Three related statements in classical algebraic topology: For any continuous map f : S n → R n , there exists BU-classic an x ∈ S n such that f ( x ) = f ( − x ). For any continuous map f : S n → R n with the property that f ( − x ) = − f ( x ), there is an x ∈ S n BU-odd such that f ( x ) = 0 There is no continuous map f : S n → S n − 1 with the property that there exists an x ∈ S n such that BU-retract f ( − x ) = − f ( x ). Proof of BU-classic involves proving that 1. show (BU-classic) ≃ (BU-odd) 2. show ¬ (BU-odd) ⇒ ¬ (BU-retract) 3. hence (BU-retract) ⇒ (BU-classic). 4. prove (BU-retract) 5. conclude (BU-classic) 22

Three related statements in classical algebraic topology: For any continuous map f : S n → R n , there exists BU-classic an x ∈ S n such that f ( x ) = f ( − x ). For any continuous map f : S n → R n with the property that f ( − x ) = − f ( x ), there is an x ∈ S n BU-odd such that f ( x ) = 0 There is no continuous map f : S n → S n − 1 with the property that there exists an x ∈ S n such that BU-retract f ( − x ) = − f ( x ). Proof of BU-classic involves proving that 1. show (BU-classic) ≃ (BU-odd) 2. show ¬ (BU-odd) ⇒ ¬ (BU-retract) 3. hence (BU-retract) ⇒ (BU-classic). 4. prove (BU-retract) 5. conclude (BU-classic) 22

BU-* in real-cohesive homotopy type theory: � � � � � � � � BU-classic � x : S n f ( − x ) = f ( x ) � � � � � � � f : S n → R n � � � � � � � � � � � x : S n f ( − x ) = − f ( x ) → � � BU-odd x : S n f ( x ) = 0 � � � � � � f : S n → R n � � � � � � � � � � BU-retract � x : S n f ( − x ) = − f ( x ) → 0 � � � � � � � � � f : S n → S n − 1 � � � � 23

Proof of BU-classic, strategy: 1. show (BU-classic) ≃ (BU-odd) 2. show ¬ (BU-odd) ⇒ ¬ (BU-retract) 3. hence (BU-retract) ⇒ ¬¬ (BU-odd) 4. prove (BU-retract) 5. conclude ¬¬ (BU-classic) ¬¬ (BU-classic) � = (BU-classic) continuously but ¬¬ (BU-classic) = (BU-classic) discontinuously Lemma: (Shulman) For P a proposition, ♯ P = ¬¬ P 24

Proof of BU-classic, strategy: 1. show (BU-classic) ≃ (BU-odd) 2. show ¬ (BU-odd) ⇒ ¬ (BU-retract) 3. hence (BU-retract) ⇒ ¬¬ (BU-odd) 4. prove (BU-retract) 5. conclude ¬¬ (BU-classic) ¬¬ (BU-classic) � = (BU-classic) continuously but ¬¬ (BU-classic) = (BU-classic) discontinuously Lemma: (Shulman) For P a proposition, ♯ P = ¬¬ P 24