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

Feb 26, 2023 •312 likes •558 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

  1. H L L V E E

  2. f S n A (n-sphere) functions from S n to some type

  3. +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

  4. S n functions from S n f(S n ) are foldings of S n

  5. truncation level n no interesting folding of S m>n no interesting homotopy above dimension n [Voevodsky] expressible in type theory!

  6. wear your 4d glasses use your + + + + + + + telepathy + +

  7. + + + + + +

  8. -1 no interesting foldings of Π x,y:A Id(x;y) … or above! (due to continuity) 0 no interesting foldings of Π x,y:A Π p,q:Id(x;y) Id(p;q) … or above! 1 no … of … or Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Id(r;s) above!

  9. Π x,y:A Id(x;y) level -1 Π x,y:A Π p,q:Id(x;y) Id(p;q) level 0 Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Id(r;s) level 1 Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Π u,v:Id(r;s) Id(u;v) level 2 Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Π u,v:Id(r;s) Π a,b:Id(u;v) Id(a;b) Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Π u,v:Id(r;s) Π a,b:Id(u;v) Π m,n:Id(a;b) Id has-level n+1 (A) := Π x,y:A has-level n (Id A (x; y)) has-level -2 (A) := is-contr(A)* *is-contr(A) instead of just A to match level -2

  10. has-level n+1 (A) := Π x,y:A has-level n (Id A (x; y)) has-level -2 (A) := is-contr(A) homotopies above n trivial

  11. homotopies in dim 0 trivial Π x,y:A Id(x;y) possibly not continuously [HoTT 7.3]

  12. homotopies in dim 1 trivial Π x,y:A Π p,q:Id(x;y) Id(p;q) possibly not continuously

  13. homotopies in dim 2 trivial Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Id(r;s) possibly not continuously

  14. homotopies in dim 3 trivial Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Π u,v:Id(r;s) Id(u;v) possibly not continuously

  15. Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Π u,v:Id(r;s) Id(u;v) is-contr(A) Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Id(r;s) Π x,y:A Π p,q:Id(x;y) Id(p;q) [Whitehead] (special case) All good spaces with Π x,y:A Id(x;y) trivial homotopies are contractible A (in classical theory, probably not in HoTT)

  16. Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Π u,v:Id(r;s) Id(u;v) is-contr(Id(x;y)) Π x,y:A Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Id(r;s) Π x,y:A Π p,q:Id(x;y) Id(p;q) [Whitehead] (special case) All good spaces with Π x,y:A Id(x;y) trivial homotopies are contractible A (in classical theory, probably not in HoTT)

  17. Π p,q:Id(x;y) Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Π u,v:Id(r;s) Id(u;v) is-contr(Id(p;q)) Π x,y:A Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Id(r;s) Π x,y:A Π p,q:Id(x;y) Id(p;q) [Whitehead] (special case) All good spaces with Π x,y:A Id(x;y) trivial homotopies are contractible A (in classical theory, probably not in HoTT)

  18. is-contr(Id(r;s)) Π p,q:Id(x;y) Π r,s:Id(p;q) Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Π u,v:Id(r;s) Id(u;v) Π x,y:A Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Id(r;s) Π x,y:A Π p,q:Id(x;y) Id(p;q) [Whitehead] (special case) All good spaces with Π x,y:A Id(x;y) trivial homotopies are contractible A (in classical theory, probably not in HoTT)

  19. homotopies above -1 trivial in classical theory Π x,y:A is-contr(Id(x;y))

  20. homotopies above 0 trivial in classical theory Π x,y:A Π p,q:Id(x;y) is-contr(Id(p;q))

  21. homotopies above 1 trivial in classical theory Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) is-contr(Id(r;s))

  22. homotopies above 2 trivial in classical theory Π x,y:A Π p,q:Id(x;y) Π r,s:Id(p;q) Π u,v:Id(r;s) is-contr(Id(u;v))

  23. homotopies above n trivial in classical theory in HoTT (and classical theory) has-level n+1 (A) := Π x,y:A has-level n (Id A (x; y)) has-level -2 (A) := is-contr(A)

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