85 lines
3.3 KiB
Text
85 lines
3.3 KiB
Text
import homotopy.susp types.pointed2
|
||
|
||
open susp eq pointed function is_equiv
|
||
|
||
namespace susp
|
||
variables {X X' Y Y' Z : Type*}
|
||
definition susp_functor_pconst_homotopy [unfold 3] {X Y : Type*} (x : psusp X) :
|
||
psusp_functor (pconst X Y) x = pt :=
|
||
begin
|
||
induction x,
|
||
{ reflexivity },
|
||
{ exact (merid pt)⁻¹ },
|
||
{ apply eq_pathover, refine !elim_merid ⬝ph _ ⬝hp !ap_constant⁻¹, exact square_of_eq !con.right_inv⁻¹ }
|
||
end
|
||
|
||
definition susp_functor_pconst [constructor] (X Y : Type*) : psusp_functor (pconst X Y) ~* pconst (psusp X) (psusp Y) :=
|
||
begin
|
||
fapply phomotopy.mk,
|
||
{ exact susp_functor_pconst_homotopy },
|
||
{ reflexivity }
|
||
end
|
||
|
||
definition psusp_pfunctor [constructor] (X Y : Type*) : ppmap X Y →* ppmap (psusp X) (psusp Y) :=
|
||
pmap.mk psusp_functor (eq_of_phomotopy !susp_functor_pconst)
|
||
|
||
definition psusp_pelim [constructor] (X Y : Type*) : ppmap X (Ω Y) →* ppmap (psusp X) Y :=
|
||
ppcompose_left (loop_psusp_counit Y) ∘* psusp_pfunctor X (Ω Y)
|
||
|
||
definition loop_psusp_pintro [constructor] (X Y : Type*) : ppmap (psusp X) Y →* ppmap X (Ω Y) :=
|
||
ppcompose_right (loop_psusp_unit X) ∘* pap1 (psusp X) Y
|
||
|
||
definition loop_psusp_pintro_natural_left (f : X' →* X) :
|
||
psquare (loop_psusp_pintro X Y) (loop_psusp_pintro X' Y)
|
||
(ppcompose_right (psusp_functor f)) (ppcompose_right f) :=
|
||
!pap1_natural_left ⬝h* ppcompose_right_psquare (loop_psusp_unit_natural f)⁻¹*
|
||
|
||
definition loop_psusp_pintro_natural_right (f : Y →* Y') :
|
||
psquare (loop_psusp_pintro X Y) (loop_psusp_pintro X Y')
|
||
(ppcompose_left f) (ppcompose_left (Ω→ f)) :=
|
||
!pap1_natural_right ⬝h* !ppcompose_left_ppcompose_right⁻¹*
|
||
|
||
definition is_equiv_loop_psusp_pintro [constructor] (X Y : Type*) :
|
||
is_equiv (loop_psusp_pintro X Y) :=
|
||
begin
|
||
fapply adjointify,
|
||
{ exact psusp_pelim X Y },
|
||
{ intro g, apply eq_of_phomotopy, exact psusp_adjoint_loop_right_inv g },
|
||
{ intro f, apply eq_of_phomotopy, exact psusp_adjoint_loop_left_inv f }
|
||
end
|
||
|
||
definition psusp_adjoint_loop' [constructor] (X Y : Type*) : ppmap (psusp X) Y ≃* ppmap X (Ω Y) :=
|
||
pequiv_of_pmap (loop_psusp_pintro X Y) (is_equiv_loop_psusp_pintro X Y)
|
||
|
||
definition psusp_adjoint_loop_natural_right (f : Y →* Y') :
|
||
psquare (psusp_adjoint_loop' X Y) (psusp_adjoint_loop' X Y')
|
||
(ppcompose_left f) (ppcompose_left (Ω→ f)) :=
|
||
loop_psusp_pintro_natural_right f
|
||
|
||
definition psusp_adjoint_loop_natural_left (f : X' →* X) :
|
||
psquare (psusp_adjoint_loop' X Y) (psusp_adjoint_loop' X' Y)
|
||
(ppcompose_right (psusp_functor f)) (ppcompose_right f) :=
|
||
loop_psusp_pintro_natural_left f
|
||
|
||
definition iterate_psusp_iterate_psusp_rev (n m : ℕ) (A : Type*) :
|
||
iterate_psusp n (iterate_psusp m A) ≃* iterate_psusp (m + n) A :=
|
||
begin
|
||
induction n with n e,
|
||
{ reflexivity },
|
||
{ exact psusp_pequiv e }
|
||
end
|
||
|
||
definition iterate_psusp_pequiv [constructor] (n : ℕ) {X Y : Type*} (f : X ≃* Y) :
|
||
iterate_psusp n X ≃* iterate_psusp n Y :=
|
||
begin
|
||
induction n with n e,
|
||
{ exact f },
|
||
{ exact psusp_pequiv e }
|
||
end
|
||
|
||
open algebra nat
|
||
definition iterate_psusp_iterate_psusp (n m : ℕ) (A : Type*) :
|
||
iterate_psusp n (iterate_psusp m A) ≃* iterate_psusp (n + m) A :=
|
||
iterate_psusp_iterate_psusp_rev n m A ⬝e* pequiv_of_eq (ap (λk, iterate_psusp k A) (add.comm m n))
|
||
|
||
end susp
|