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give last step of associativity of smash

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there are still unproven lemma's
This commit is contained in:
Floris van Doorn 2017-01-25 12:14:47 -05:00
parent 216a25af4f
commit d6de922d1f
1 changed files with 24 additions and 7 deletions
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31
homotopy/smash_adjoint.hlean
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@ -1,5 +1,5 @@
-- Authors: Floris van Doorn
-- in collaboration with Egbert, Stefano, Robin
-- in collaboration with Egbert, Stefano, Robin, Ulrik
import .smash
@ -479,6 +479,12 @@ namespace smash
refine !smash_adjoint_pmap_natural_pt
end
definition smash_assoc_elim_inv_natural_pt {A B C X X' : Type*} (f : X →* X') (g : (A ∧ B) ∧ C →* X) :
f ∘* (smash_assoc_elim_equiv A B C X)⁻¹ᵉ* g ~* (smash_assoc_elim_equiv A B C X')⁻¹ᵉ* (f ∘* g) :=
begin
exact sorry
end
definition smash_assoc_elim_natural {A B C X X' : Type*} (f : X →* X') :
ppcompose_left f ∘* smash_assoc_elim_equiv A B C X ~*
smash_assoc_elim_equiv A B C X' ∘* ppcompose_left f :=
@ -495,14 +501,25 @@ namespace smash
-- refine !smash_adjoint_pmap_natural_pt
end
-- definition smash_assoc (A B C : Type*) : A ∧ (B ∧ C) ≃* (A ∧ B) ∧ C :=
-- definition smash_assoc_to (A B C : Type*) : A ∧ (B ∧ C) →* (A ∧ B) ∧ C :=
-- begin
-- fapply pequiv.MK,
-- { exact !smash_assoc_elim_equiv⁻¹ᵉ* !pid },
-- { exact !smash_assoc_elim_equiv !pid },
-- { },
-- { }
-- end
-- TODO: remove pap
definition smash_assoc (A B C : Type*) : A ∧ (B ∧ C) ≃* (A ∧ B) ∧ C :=
begin
fapply pequiv.MK2,
{ exact !smash_assoc_elim_equiv⁻¹ᵉ* !pid },
{ exact !smash_assoc_elim_equiv !pid },
{ refine !smash_assoc_elim_inv_natural_pt ⬝* _,
refine pap !smash_assoc_elim_equiv⁻¹ᵉ* !pcompose_pid ⬝* _,
apply phomotopy_of_eq, apply to_left_inv !smash_assoc_elim_equiv },
{ refine !smash_assoc_elim_natural_pt ⬝* _,
refine pap !smash_assoc_elim_equiv !pcompose_pid ⬝* _,
apply phomotopy_of_eq, apply to_right_inv !smash_assoc_elim_equiv }
end
end smash