L L V E E f S n A (n-sphere) functions from S n to some type - - PowerPoint PPT Presentation

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Feb 26, 2023 312 likes •565 views

H L L V E E f S n A (n-sphere) functions from S n to some type +1 boundaries of a line -1 S 0 S 1 S 2 S 3 points in R n+1 of distance 1 from the origin S n functions from S n f(S n ) are foldings of S n truncation level n no interesting

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L E V E L

H

SLIDE 2

A

Sn

f

functions from Sn to some type

(n-sphere)

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S0 S1 S2 S3

points in Rn+1 of distance 1 from the origin

boundaries

f a line
1

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Sn

f(Sn)

functions from Sn are foldings of Sn

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no interesting folding of Sm>n

truncation level n

no interesting homotopy above dimension n

[Voevodsky] expressible in type theory!

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wear your 4d glasses

+ + + + + + + + +

use your telepathy

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+ + +

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Πx,y:A Πp,q:Id(x;y) Id(p;q) Πx,y:A Id(x;y) Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Id(r;s)

1

no interesting foldings of no interesting foldings of no …

… or above! (due to continuity) … or above! … or above!

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Πx,y:A Πp,q:Id(x;y) Id(p;q) Πx,y:A Id(x;y) Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Id(r;s) Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Πu,v:Id(r;s) Id(u;v) Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Πu,v:Id(r;s) Πa,b:Id(u;v) Id(a;b) Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Πu,v:Id(r;s) Πa,b:Id(u;v) Πm,n:Id(a;b) Id

has-level-2(A) := is-contr(A)* has-leveln+1(A) := Πx,y:Ahas-leveln(IdA(x; y))

level -1 level 0 level 1 level 2

*is-contr(A) instead of just A to match level -2

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has-level-2(A) := is-contr(A) has-leveln+1(A) := Πx,y:Ahas-leveln(IdA(x; y))

homotopies above n trivial

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homotopies in dim 0 trivial

possibly not continuously

Πx,y:A Id(x;y)

[HoTT 7.3]

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possibly not continuously

Πx,y:A Πp,q:Id(x;y) Id(p;q)

homotopies in dim 1 trivial

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Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Id(r;s)

possibly not continuously

homotopies in dim 2 trivial

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possibly not continuously

Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Πu,v:Id(r;s) Id(u;v)

homotopies in dim 3 trivial

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Πx,y:A Πp,q:Id(x;y) Id(p;q) Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Id(r;s) Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Πu,v:Id(r;s) Id(u;v) Πx,y:A Id(x;y) A

[Whitehead]

All good spaces with trivial homotopies are contractible

(in classical theory, probably not in HoTT) (special case)

is-contr(A)

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Πx,y:A Πp,q:Id(x;y) Id(p;q) Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Id(r;s) Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Πu,v:Id(r;s) Id(u;v) Πx,y:A Id(x;y) A

Πx,y:A

[Whitehead]

All good spaces with trivial homotopies are contractible

(in classical theory, probably not in HoTT) (special case)

is-contr(Id(x;y))

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Πx,y:A Πp,q:Id(x;y) Id(p;q) Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Id(r;s) Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Πu,v:Id(r;s) Id(u;v) Πx,y:A Id(x;y) A

Πx,y:A Πp,q:Id(x;y)

[Whitehead]

All good spaces with trivial homotopies are contractible

(in classical theory, probably not in HoTT) (special case)

is-contr(Id(p;q))

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Πx,y:A Πp,q:Id(x;y) Id(p;q) Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Id(r;s) Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Πu,v:Id(r;s) Id(u;v) Πx,y:A Id(x;y) A

Πx,y:A Πp,q:Id(x;y)

[Whitehead]

All good spaces with trivial homotopies are contractible

Πr,s:Id(p;q)

(in classical theory, probably not in HoTT)

is-contr(Id(r;s))

(special case)

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Πx,y:A is-contr(Id(x;y))

homotopies above -1 trivial

in classical theory

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Πx,y:A Πp,q:Id(x;y) is-contr(Id(p;q)) homotopies above 0 trivial

in classical theory

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Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) is-contr(Id(r;s))

homotopies above 1 trivial

in classical theory

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homotopies above 2 trivial

in classical theory Πx,y:A Πp,q:Id(x;y) Πr,s:Id(p;q) Πu,v:Id(r;s) is-contr(Id(u;v))

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homotopies above n trivial

in classical theory

has-level-2(A) := is-contr(A) has-leveln+1(A) := Πx,y:Ahas-leveln(IdA(x; y))

in HoTT (and classical theory)