db8402e1af
The cup product on Eilenberg Maclane spaces is now defined, but no properties are proven yet
123 lines
5 KiB
Text
123 lines
5 KiB
Text
-- Authors: Floris van Doorn
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import .EM .smash_adjoint ..algebra.ring ..algebra.arrow_group
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open algebra eq EM is_equiv equiv is_trunc is_conn pointed trunc susp smash group nat function
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namespace EM
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definition EM1product_adj {R : Ring} :
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EM1 (AbGroup_of_Ring R) →* ppmap (EM1 (AbGroup_of_Ring R)) (EMadd1 (AbGroup_of_Ring R) 1) :=
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begin
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have is_trunc 1 (ppmap (EM1 (AbGroup_of_Ring R)) (EMadd1 (AbGroup_of_Ring R) 1)),
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from is_trunc_pmap_of_is_conn _ _ _ _ _ _ (le.refl 2) !is_trunc_EMadd1,
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apply EM1_pmap, fapply inf_homomorphism.mk,
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{ intro r, refine pfunext _ _, exact !loop_EM2⁻¹ᵉ* ∘* EM1_functor (ring_right_action r), },
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{ intro r r', exact sorry }
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end
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definition EMproduct_map {A B C : AbGroup} (φ : A → B →g C) (n m : ℕ) (a : A) :
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EMadd1 B n →* EMadd1 C n :=
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begin
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fapply EMadd1_functor (φ a) n
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end
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definition EM0EMadd1product {A B C : AbGroup} (φ : A →g B →gg C) (n : ℕ) :
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A →* EMadd1 B n →** EMadd1 C n :=
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EMadd1_pfunctor B C n ∘* pmap_of_homomorphism φ
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-- TODO: simplify
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definition EMadd1product {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
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EMadd1 A n →* EMadd1 B m →** EMadd1 C (m + succ n) :=
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begin
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assert H1 : is_trunc n.+1 (EMadd1 B m →** EMadd1 C (m + succ n)),
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{ refine is_trunc_pmap_of_is_conn _ (m.-1) !is_conn_EMadd1 _ _ _ _ !is_trunc_EMadd1,
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exact le_of_eq (trunc_index.of_nat_add_plus_two_of_nat m n)⁻¹ᵖ },
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apply EMadd1_pmap,
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refine (gloopn_pmap_isomorphism (succ n) _ _)⁻¹ᵍ⁸ ∘∞g
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gpmap_loop_homomorphism_right (EMadd1 B m) (loopn_EMadd1_add_of_eq C !succ_add)⁻¹ᵉ* ∘∞g
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gloop_pmap_isomorphism _ _ ∘∞g
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(deloop_isomorphism _)⁻¹ᵍ⁸ ∘∞g
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EM_ehomomorphism B C (succ m) ∘∞g
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inf_homomorphism_of_homomorphism φ
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end
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definition EMproduct1 {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
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EM A n →* EM B m →** EM C (m + n) :=
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begin
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cases n with n,
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{ cases m with m,
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{ exact pmap_of_homomorphism2 φ },
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{ exact EM0EMadd1product φ m }},
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{ cases m with m,
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{ exact ppcompose_left (ptransport (EMadd1 C) (zero_add n)⁻¹) ∘*
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pmap_swap_map (EM0EMadd1product (homomorphism_swap φ) n) },
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{ exact ppcompose_left (ptransport (EMadd1 C) !succ_add⁻¹) ∘* EMadd1product φ n m }}
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end
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definition EMproduct2 {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
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EM A n →* (EM B m →** EM C (m + n)) :=
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begin
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assert H1 : is_trunc n (gpmap_loop' (EM B m) (loop_EM C (m + n))),
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{ exact is_trunc_pmap_of_is_conn_nat _ m !is_conn_EM _ _ _ !le.refl !is_trunc_EM },
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apply EM_pmap (gpmap_loop' (EM B m) (loop_EM C (m + n))) n,
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exact sorry
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-- exact _ /- (loopn_ppmap_pequiv _ _ _)⁻¹ᵉ* -/ ∘∞g _ /-ppcompose_left !loopn_EMadd1_add⁻¹ᵉ*-/ ∘∞g
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-- _ ∘∞g inf_homomorphism_of_homomorphism φ
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end
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definition EMproduct3' {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
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gEM A n →∞g gpmap_loop' (EM B m) (loop_EM C (m + n)) :=
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begin
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assert H1 : is_trunc n (gpmap_loop' (EM B m) (loop_EM C (m + n))),
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{ exact is_trunc_pmap_of_is_conn_nat _ m !is_conn_EM _ _ _ !le.refl !is_trunc_EM },
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-- refine EM_homomorphism _ _ _,
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-- --(gmap_loop' (EM B m) (loop_EM C (m + n))) n,
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-- exact _ /- (loopn_ppmap_pequiv _ _ _)⁻¹ᵉ* -/ ∘∞g _ /-ppcompose_left !loopn_EMadd1_add⁻¹ᵉ*-/ ∘∞g
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-- _ ∘∞g inf_homomorphism_of_homomorphism φ
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exact sorry
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end
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definition EMproduct4 {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
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gEM A n →∞g Ωg (EM B m →** EM C (m + n + 1)) :=
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begin
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assert H1 : is_trunc (n+1) (EM B m →** EM C (m + n + 1)),
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{ exact is_trunc_pmap_of_is_conn_nat _ m !is_conn_EM _ _ _ !le.refl !is_trunc_EM },
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apply EM_homomorphism_gloop,
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refine (gloopn_pmap_isomorphism _ _ _)⁻¹ᵍ⁸ ∘∞g _ ∘∞g inf_homomorphism_of_homomorphism φ,
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-- exact _ /- (loopn_ppmap_pequiv _ _ _)⁻¹ᵉ* -/ ∘∞g _ /-ppcompose_left !loopn_EMadd1_add⁻¹ᵉ*-/ ∘∞g
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-- _ ∘∞g inf_homomorphism_of_homomorphism φ
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exact sorry
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end
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definition EMproduct5 {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
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InfGroup_of_deloopable (EM A n) →∞g InfGroup_of_deloopable (EM B m →** EM C (m + n)) :=
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begin
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assert H1 : is_trunc (n + 1) (deloop (EM B m →** EM C (m + n))),
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{ exact is_trunc_pmap_of_is_conn_nat _ m !is_conn_EM _ _ _ !le.refl !is_trunc_EM },
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refine EM_homomorphism_deloopable _ _ _ _ _,
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-- exact _ /- (loopn_ppmap_pequiv _ _ _)⁻¹ᵉ* -/ ∘∞g _ /-ppcompose_left !loopn_EMadd1_add⁻¹ᵉ*-/ ∘∞g
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-- _ ∘∞g inf_homomorphism_of_homomorphism φ
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exact sorry
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end
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definition EMadd1product2 {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
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gEM A (n+1) →∞g Ωg[succ n] (EMadd1 B m →** EMadd1 C m) :=
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begin
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assert H1 : is_trunc (n+1) (Ω[n] (EMadd1 B m →** EMadd1 C m)),
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{ apply is_trunc_loopn, exact sorry },
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-- refine EM_homomorphism_gloop (Ω[n] (EMadd1 B m →** EMadd1 C m)) _ _,
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/- the underlying pointed map is: -/
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-- exact sorry
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-- refine (loopn_ppmap_pequiv _ _ _)⁻¹ᵉ* ∘* ppcompose_left !loopn_EMadd1_add⁻¹ᵉ* ∘*
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-- EM0EMadd1product φ m
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exact sorry
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end
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end EM
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