H I T H I T I H H T T I T T T I T T H I T H T - - PowerPoint PPT Presentation

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Feb 15, 2023 134 likes •420 views

H H H I T H I T I H H T T I T T T I T T H I T H T I T H I H H I I H I I pushout suspension join sphere circle Higher Inductive Types PUSHOUT g(c) f(c) c C A B g C f B A data Pushout {A B C : U}

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H

I

T H

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T H

I

T H

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T H

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T H

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T H

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T H

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T H

I

T H

I

T H

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T H

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T

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Higher Inductive Types

circle sphere suspension join pushout

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PUSHOUT

A B

C

f(c)

g(c)

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data Pushout {A B C : U} (f : C → A) (g : C → B) : U where inl : A → Pushout f g inr : B → Pushout f g push : (c : C) → inl (f c) ≡ inr (g c)

A B

f g

SLIDE 5

f g

push

inl inr

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circle sphere suspension join pushout

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COEQUALIZER

A

f(b) g(b)

B

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A

f(b) g(b)

data Coequalizer {A B : U} (f g : B → A) : U where inc : A → Coequalizer f g eq : (b : B) → inc (f b) ≡ inc (g b)

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f g

inc

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A B A A+B

C C B

B+B

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SUSPENSION

A

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A

north south

data Susp (A : U) : U where north : Susp A south : Susp A merid : (a : A) → north ≡ south

merid

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=

Sn := Suspn(2)

See the lecture on truncation levels

A

north south

merid

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A B

WEDGE

sum types for pointed types

a b

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A B

a b

data Wedge (A B : U) (a : A) (b : B) : U where inl : A → Wedge A B a b inr : A → Wedge A B a b glue : inl a ≡ inr b

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A ∧ B := A × B / A ∨ B

smash wedge sum

SMASH

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SMASH

A B A×B/

a b

wedge sum

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A B

SMASH

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A B

data Smash (A B : U) (a : A) (b : B) : U where pair : A → B → Smash A B a b basel : Smash A B a b baser : Smash A B a b gluel : (a' : A) → inc a' b ≡ basel gluer : (b' : B) → inc a b' ≡ baser

basel baser

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JOIN

Paths between all pairs

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data Join (A B : U) : U where inl : A → Join A B inr : B → Join A B join : (a : A) (b : B) → inl a ≡ inr b

A B

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X ★ Y ≃ Susp (X ∧ Y) Susp (X ∧ Y) ≃ (Susp X) ∧ Y ≃ X ∧ (Susp Y) X ∧ Y ∙→ Z ≃ X ∙→ (Y ∙→ Z) A × B → C ≃ A → (B → C) Sn ∧ Sm ≃ Sn+m Sn ★ Sm ≃ Sn+m+1

point- preserving functons

CURRYING

join smash

X, Y and Z are pointed types

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n-TRUNCATION

Best n-type approximation

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n-TRUNCATION

A

Fill every image of Sn+1 with a cone

See the lecture on truncation levels

also [HoTT, 7.3]

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data Trunc n (A : U) : U where inc : A → Trunc n A hub : (Sn+1 → A) → Trunc n A spoke : (l : Sn+1 → A) (x : Sn+1) → hub f ≡ f x

hub l

spoke l x

effectively has-level n (Trunc n A) [HoTT, 7.3]

SLIDE 26

inc

f

any n-type n-truncation

Any function to an n-type factors through n-truncation

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More2: sequential colimits

e.g. define S∞ as lim Sn

All definable using pushouts More: set quotients

coequalizer + 0-truncation

More3: textbooks or ask Favonia