use psquare for naturality squares consistently
this commit renames some definitions and swaps some arguments around for consistency
This commit is contained in:
Floris van Doorn
parent
a7b69aeb60
commit
2b722b3e34
4 changed files with 68 additions and 63 deletions
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@ -161,22 +161,21 @@ namespace eq
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@(is_equiv_trunc_functor 0 _) (is_equiv_apn n f H)
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definition homotopy_group_succ_in_natural (n : ℕ) {A B : Type*} (f : A →* B) :
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homotopy_group_succ_in n B ∘* π→[n + 1] f ~*
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π→[n] (Ω→ f) ∘* homotopy_group_succ_in n A :=
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psquare (homotopy_group_succ_in n A) (homotopy_group_succ_in n B)
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(π→[n + 1] f) (π→[n] (Ω→ f)) :=
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begin
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refine !ptrunc_functor_pcompose⁻¹* ⬝* _ ⬝* !ptrunc_functor_pcompose,
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exact ptrunc_functor_phomotopy 0 (apn_succ_phomotopy_in n f)
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exact ptrunc_functor_psquare 0 (loopn_succ_in_natural n f),
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end
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definition homotopy_group_succ_in_natural_unpointed (n : ℕ) {A B : Type*} (f : A →* B) :
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hsquare (homotopy_group_succ_in n A) (homotopy_group_succ_in n B) (π→[n+1] f) (π→[n] (Ω→ f)) :=
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(homotopy_group_succ_in_natural n f)⁻¹*
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homotopy_group_succ_in_natural n f
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definition is_equiv_homotopy_group_functor_ap1 (n : ℕ) {A B : Type*} (f : A →* B)
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[is_equiv (π→[n + 1] f)] : is_equiv (π→[n] (Ω→ f)) :=
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have is_equiv (π→[n] (Ω→ f) ∘ homotopy_group_succ_in n A),
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from is_equiv_of_equiv_of_homotopy (equiv.mk (π→[n+1] f) _ ⬝e homotopy_group_succ_in n B)
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(homotopy_group_succ_in_natural n f),
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(homotopy_group_succ_in_natural n f)⁻¹*,
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is_equiv.cancel_right (homotopy_group_succ_in n A) _
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definition tinverse [constructor] {X : Type*} : π[1] X →* π[1] X :=
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@ -272,7 +272,7 @@ namespace chain_complex
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definition pid_or_pinverse_add4_rev (n : ℕ) :
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pid_or_pinverse (n + 4) ~* !pinverse ∘* Ω→(pid_or_pinverse (n + 1)) :=
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!ap1_pcompose_pinverse
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!pinverse_natural
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theorem fiber_sequence_phomotopy_loop_spaces : Π(n : ℕ),
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fiber_sequence_pequiv_loop_spaces n ∘* fiber_sequence_fun n ~*
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@ -300,7 +300,7 @@ namespace chain_complex
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xrewrite [loop_spaces_fun_add3, pid_or_pinverse_add4, to_pmap_pequiv_trans],
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refine _ ⬝* !passoc⁻¹*,
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refine _ ⬝* pwhisker_left _ !passoc⁻¹*,
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refine _ ⬝* pwhisker_left _ (pwhisker_left _ !ap1_pcompose_pinverse),
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refine _ ⬝* pwhisker_left _ (pwhisker_left _ !pinverse_natural),
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refine !passoc⁻¹* ⬝* _ ⬝* !passoc ⬝* !passoc,
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apply pwhisker_right,
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refine !ap1_pcompose⁻¹* ⬝* _ ⬝* !ap1_pcompose ⬝* pwhisker_right _ !ap1_pcompose,
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@ -345,7 +345,7 @@ namespace chain_complex
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apply pwhisker_left,
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refine !ap1_pcompose ⬝* _ ⬝* !passoc ⬝* !passoc,
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apply pwhisker_right,
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refine _ ⬝* pwhisker_right _ !ap1_pcompose_pinverse,
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refine _ ⬝* pwhisker_right _ !pinverse_natural,
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refine _ ⬝* !passoc⁻¹*,
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refine !pcompose_pid⁻¹* ⬝* pwhisker_left _ _,
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symmetry, apply pinverse_pinverse
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@ -364,7 +364,7 @@ namespace chain_complex
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replace (k + 4) with (k + 1 + 3),
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rewrite [loop_spaces_fun_add3],
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refine !passoc⁻¹* ⬝* _ ⬝* !passoc⁻¹*,
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refine _ ⬝* pwhisker_left _ !ap1_pcompose_pinverse,
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refine _ ⬝* pwhisker_left _ !pinverse_natural,
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refine _ ⬝* !passoc,
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apply pwhisker_right,
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refine !ap1_pcompose⁻¹* ⬝* _ ⬝* !ap1_pcompose,
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@ -213,6 +213,8 @@ namespace susp
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{ reflexivity }
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end
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notation `⅀→`:(max+5) := susp_functor
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definition is_equiv_susp_functor [constructor] (f : X →* Y) [Hf : is_equiv f]
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: is_equiv (susp_functor f) :=
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susp.is_equiv_functor f
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@ -245,6 +247,8 @@ namespace susp
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{ reflexivity },
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end
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notation `⅀⇒`:(max+5) := susp_functor_phomotopy
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definition susp_functor_pid (A : Type*) : susp_functor (pid A) ~* pid (susp A) :=
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begin
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fapply phomotopy.mk,
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@ -266,8 +270,9 @@ namespace susp
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end
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definition loop_susp_unit_natural (f : X →* Y)
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: loop_susp_unit Y ∘* f ~* Ω→ (susp_functor f) ∘* loop_susp_unit X :=
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: psquare (loop_susp_unit X) (loop_susp_unit Y) f (Ω→ (susp_functor f)) :=
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begin
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apply ptranspose,
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induction X with X x, induction Y with Y y, induction f with f pf, esimp at *, induction pf,
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fapply phomotopy.mk,
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{ intro x', symmetry,
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@ -294,7 +299,7 @@ namespace susp
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end
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definition loop_susp_counit_natural (f : X →* Y)
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: f ∘* loop_susp_counit X ~* loop_susp_counit Y ∘* (susp_functor (ap1 f)) :=
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: psquare (loop_susp_counit X) (loop_susp_counit Y) (⅀→ (Ω→ f)) f :=
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begin
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induction X with X x, induction Y with Y y, induction f with f pf, esimp at *, induction pf,
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fconstructor,
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@ -360,7 +365,7 @@ namespace susp
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definition loop_susp_intro_natural {X Y Z : Type*} (g : susp Y →* Z) (f : X →* Y) :
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loop_susp_intro (g ∘* susp_functor f) ~* loop_susp_intro g ∘* f :=
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pwhisker_right _ !ap1_pcompose ⬝* !passoc ⬝* pwhisker_left _ !loop_susp_unit_natural⁻¹* ⬝*
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pwhisker_right _ !ap1_pcompose ⬝* !passoc ⬝* pwhisker_left _ !loop_susp_unit_natural ⬝*
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!passoc⁻¹*
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definition susp_adjoint_loop_right_inv {X Y : Type*} (g : X →* Ω Y) :
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@ -368,7 +373,7 @@ namespace susp
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begin
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refine !pwhisker_right !ap1_pcompose ⬝* _,
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refine !passoc ⬝* _,
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refine !pwhisker_left !loop_susp_unit_natural⁻¹* ⬝* _,
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refine !pwhisker_left !loop_susp_unit_natural ⬝* _,
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refine !passoc⁻¹* ⬝* _,
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refine !pwhisker_right !loop_susp_counit_unit ⬝* _,
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apply pid_pcompose
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@ -403,7 +408,8 @@ namespace susp
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{ apply eq_pathover, refine !elim_merid ⬝ph _ ⬝hp !ap_constant⁻¹, exact square_of_eq !con.right_inv⁻¹ }
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end
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definition susp_functor_pconst [constructor] (X Y : Type*) : susp_functor (pconst X Y) ~* pconst (susp X) (susp Y) :=
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definition susp_functor_pconst [constructor] (X Y : Type*) :
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susp_functor (pconst X Y) ~* pconst (susp X) (susp Y) :=
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begin
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fapply phomotopy.mk,
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{ exact susp_functor_pconst_homotopy },
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@ -422,7 +428,7 @@ namespace susp
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definition loop_susp_pintro_natural_left (f : X' →* X) :
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psquare (loop_susp_pintro X Y) (loop_susp_pintro X' Y)
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(ppcompose_right (susp_functor f)) (ppcompose_right f) :=
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!pap1_natural_left ⬝h* ppcompose_right_psquare (loop_susp_unit_natural f)⁻¹*
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!pap1_natural_left ⬝h* ppcompose_right_psquare (loop_susp_unit_natural f)
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definition loop_susp_pintro_natural_right (f : Y →* Y') :
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psquare (loop_susp_pintro X Y) (loop_susp_pintro X Y')
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@ -309,13 +309,6 @@ namespace pointed
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: pinverse X p⁻¹ = (pinverse X p)⁻¹ :=
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idp
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definition ap1_pcompose_pinverse [constructor] {X Y : Type*} (f : X →* Y) :
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Ω→ f ∘* pinverse X ~* pinverse Y ∘* Ω→ f :=
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phomotopy.mk (ap1_gen_inv f (respect_pt f) (respect_pt f))
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abstract begin
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induction Y with Y y₀, induction f with f f₀, esimp at * ⊢, induction f₀, reflexivity
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end end
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definition is_equiv_pcast [instance] {A B : Type*} (p : A = B) : is_equiv (pcast p) :=
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!is_equiv_cast
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@ -580,6 +573,8 @@ namespace pointed
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definition ap1_phomotopy {f g : A →* B} (p : f ~* g) : Ω→ f ~* Ω→ g :=
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pap Ω→ p
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notation `Ω⇒`:(max+5) := ap1_phomotopy
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definition ap1_phomotopy_refl {X Y : Type*} (f : X →* Y) :
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ap1_phomotopy (phomotopy.refl f) = phomotopy.refl (Ω→ f) :=
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!pap_refl
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@ -676,11 +671,6 @@ namespace pointed
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(r : p = q) : ptransport B p ~* ptransport B q :=
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phomotopy.mk (λb, ap (λp, transport B p b) r) begin induction r, apply idp_con end
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definition pnatural_square {A B : Type} (X : B → Type*) {f g : A → B}
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(h : Πa, X (f a) →* X (g a)) {a a' : A} (p : a = a') :
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h a' ∘* ptransport X (ap f p) ~* ptransport X (ap g p) ∘* h a :=
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by induction p; exact !pcompose_pid ⬝* !pid_pcompose⁻¹*
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definition apn_pid [constructor] {A : Type*} (n : ℕ) : apn n (pid A) ~* pid (Ω[n] A) :=
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begin
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induction n with n IH,
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@ -730,12 +720,6 @@ namespace pointed
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{ reflexivity}
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end
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definition pcast_commute [constructor] {A : Type} {B C : A → Type*} (f : Πa, B a →* C a)
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{a₁ a₂ : A} (p : a₁ = a₂) : pcast (ap C p) ∘* f a₁ ~* f a₂ ∘* pcast (ap B p) :=
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phomotopy.mk
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begin induction p, reflexivity end
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begin induction p, esimp, refine !idp_con ⬝ !idp_con ⬝ !ap_id⁻¹ end
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/- pointed equivalences -/
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structure pequiv (A B : Type*) :=
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@ -868,11 +852,6 @@ namespace pointed
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definition peap {A B : Type*} (F : Type* → Type*) (p : A ≃* B) : F A ≃* F B :=
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pequiv_of_pmap (pcast (ap F (eq_of_pequiv p))) begin cases eq_of_pequiv p, apply is_equiv_id end
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-- rename pequiv_of_eq_natural
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definition pequiv_of_eq_commute [constructor] {A : Type} {B C : A → Type*} (f : Πa, B a →* C a)
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{a₁ a₂ : A} (p : a₁ = a₂) : pequiv_of_eq (ap C p) ∘* f a₁ ~* f a₂ ∘* pequiv_of_eq (ap B p) :=
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pcast_commute f p
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-- definition pequiv.eta_expand [constructor] {A B : Type*} (f : A ≃* B) : A ≃* B :=
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-- pequiv.mk' f (to_pinv f) (pequiv.to_pinv2 f) (pright_inv f) _
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@ -1178,27 +1157,6 @@ namespace pointed
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... = Ω[2+n] A : eq_of_pequiv !loopn_add
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... = Ω[n+2] A : by rewrite [algebra.add.comm]
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definition apn_succ_phomotopy_in (n : ℕ) (f : A →* B) :
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loopn_succ_in n B ∘* Ω→[n + 1] f ~* Ω→[n] (Ω→ f) ∘* loopn_succ_in n A :=
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begin
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induction n with n IH,
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{ reflexivity},
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{ exact !ap1_pcompose⁻¹* ⬝* ap1_phomotopy IH ⬝* !ap1_pcompose}
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end
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definition loopn_succ_in_natural {A B : Type*} (n : ℕ) (f : A →* B) :
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loopn_succ_in n B ∘* Ω→[n+1] f ~* Ω→[n] (Ω→ f) ∘* loopn_succ_in n A :=
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!apn_succ_phomotopy_in
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definition loopn_succ_in_inv_natural {A B : Type*} (n : ℕ) (f : A →* B) :
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Ω→[n + 1] f ∘* (loopn_succ_in n A)⁻¹ᵉ* ~* (loopn_succ_in n B)⁻¹ᵉ* ∘* Ω→[n] (Ω→ f):=
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begin
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apply pinv_right_phomotopy_of_phomotopy,
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refine _ ⬝* !passoc⁻¹*,
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apply phomotopy_pinv_left_of_phomotopy,
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apply apn_succ_phomotopy_in
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end
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section psquare
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/-
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Squares of pointed maps
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@ -1206,7 +1164,7 @@ namespace pointed
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We treat expressions of the form
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psquare f g h k :≡ k ∘* f ~* g ∘* h
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as squares, where f is the top, g is the bottom, h is the left face and k is the right face.
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Then the following are operations on squares
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These squares are very useful for naturality squares
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-/
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variables {A' A₀₀ A₂₀ A₄₀ A₀₂ A₂₂ A₄₂ A₀₄ A₂₄ A₄₄ : Type*}
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@ -1226,6 +1184,9 @@ namespace pointed
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definition phomotopy_of_psquare (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : f₂₁ ∘* f₁₀ ~* f₁₂ ∘* f₀₁ :=
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p
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definition hsquare_of_psquare (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : hsquare f₁₀ f₁₂ f₀₁ f₂₁ :=
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to_homotopy p
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definition phdeg_square {f f' : A →* A'} (p : f ~* f') : psquare !pid !pid f f' :=
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!pcompose_pid ⬝* p⁻¹* ⬝* !pid_pcompose⁻¹*
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definition pvdeg_square {f f' : A →* A'} (p : f ~* f') : psquare f f' !pid !pid :=
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@ -1233,9 +1194,9 @@ namespace pointed
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variables (f₁₀ f₁₂ f₀₁ f₂₁)
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definition hpconst_square : psquare !pconst !pconst f₀₁ f₂₁ :=
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definition phconst_square : psquare !pconst !pconst f₀₁ f₂₁ :=
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!pcompose_pconst ⬝* !pconst_pcompose⁻¹*
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definition vpconst_square : psquare f₁₀ f₁₂ !pconst !pconst :=
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definition pvconst_square : psquare f₁₀ f₁₂ !pconst !pconst :=
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!pconst_pcompose ⬝* !pcompose_pconst⁻¹*
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definition phrefl : psquare !pid !pid f₀₁ f₀₁ := phdeg_square phomotopy.rfl
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definition pvrefl : psquare f₁₀ f₁₀ !pid !pid := pvdeg_square phomotopy.rfl
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@ -1243,6 +1204,9 @@ namespace pointed
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definition phrfl : psquare !pid !pid f₀₁ f₀₁ := phrefl f₀₁
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definition pvrfl : psquare f₁₀ f₁₀ !pid !pid := pvrefl f₁₀
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definition ptranspose (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : psquare f₀₁ f₂₁ f₁₀ f₁₂ :=
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p⁻¹*
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definition phconcat (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) (q : psquare f₃₀ f₃₂ f₂₁ f₄₁) :
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psquare (f₃₀ ∘* f₁₀) (f₃₂ ∘* f₁₂) f₀₁ f₄₁ :=
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!passoc⁻¹* ⬝* pwhisker_right f₁₀ q ⬝* !passoc ⬝* pwhisker_left f₃₂ p ⬝* !passoc⁻¹*
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@ -1300,4 +1264,40 @@ namespace pointed
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end psquare
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definition pinverse_natural [constructor] {X Y : Type*} (f : X →* Y) :
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psquare (pinverse X) (pinverse Y) (Ω→ f) (Ω→ f) :=
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phomotopy.mk (ap1_gen_inv f (respect_pt f) (respect_pt f))
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abstract begin
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induction Y with Y y₀, induction f with f f₀, esimp at * ⊢, induction f₀, reflexivity
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end end
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definition pcast_natural [constructor] {A : Type} {B C : A → Type*} (f : Πa, B a →* C a)
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{a₁ a₂ : A} (p : a₁ = a₂) : psquare (pcast (ap B p)) (pcast (ap C p)) (f a₁) (f a₂) :=
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phomotopy.mk
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begin induction p, reflexivity end
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begin induction p, exact whisker_left idp !ap_id end
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definition pequiv_of_eq_natural [constructor] {A : Type} {B C : A → Type*} (f : Πa, B a →* C a)
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{a₁ a₂ : A} (p : a₁ = a₂) :
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psquare (pequiv_of_eq (ap B p)) (pequiv_of_eq (ap C p)) (f a₁) (f a₂) :=
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pcast_natural f p
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definition loopn_succ_in_natural {A B : Type*} (n : ℕ) (f : A →* B) :
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psquare (loopn_succ_in n A) (loopn_succ_in n B) (Ω→[n+1] f) (Ω→[n] (Ω→ f)) :=
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begin
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induction n with n IH,
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{ exact phomotopy.rfl },
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{ exact ap1_psquare IH }
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end
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definition loopn_succ_in_inv_natural {A B : Type*} (n : ℕ) (f : A →* B) :
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psquare (loopn_succ_in n A)⁻¹ᵉ* (loopn_succ_in n B)⁻¹ᵉ* (Ω→[n] (Ω→ f)) (Ω→[n + 1] f) :=
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(loopn_succ_in_natural n f)⁻¹ʰ*
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definition pnatural_square {A B : Type} (X : B → Type*) {f g : A → B}
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(h : Πa, X (f a) →* X (g a)) {a a' : A} (p : a = a') :
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psquare (ptransport X (ap f p)) (ptransport X (ap g p)) (h a) (h a') :=
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by induction p; exact !pcompose_pid ⬝* !pid_pcompose⁻¹*
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end pointed
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Reference in a new issue