higher_groups: finish adjunction between loop and deloop

higher_groups.hlean

@@ -180,7 +180,7 @@ Grp.mk (ptrunctype.mk (Ω G) !is_trunc_loop_nat pt)
 
 definition Deloop_adjoint_Loop (G : [n;k+1]Grp) (H : [n+1;k]Grp) :
   ppmap (B (Deloop G)) (B H) ≃* ppmap (B G) (B (Loop H)) :=
-(connect_intro_pequiv _ !is_conn_pconntype)⁻¹ᵉ* /- still a sorry here -/
+(connect_intro_pequiv _ !is_conn_pconntype)⁻¹ᵉ*
 
 /- to do: naturality -/
 
@@ -263,8 +263,21 @@ definition ωStabilize_of_le (H : k ≥ n + 2) (G : [n;k]Grp) : [n;ω]Grp :=
 definition ωStabilize (G : [n;k]Grp) : [n;ω]Grp :=
 ωStabilize_of_le !le_max_left (nStabilize !le_max_right G)
 
+theorem ωstabilization (H : k ≥ n + 2) : is_equiv (@ωStabilize n k) :=
+sorry
+
 /- to do: adjunction (and ωStabilize ∘ ωForget =?= id) -/
 
+definition Grp_hom (G H : [n;k]Grp) : Type :=
+B G →* B H
+
+definition is_trunc_Grp_hom (G H : [n;k]Grp) : is_trunc n (Grp_hom G H) :=
+is_trunc_pmap_of_is_conn _ (k.-2) _ (k + n) _ (le_of_eq (sub_one_add_plus_two_sub_one k n)⁻¹)
+  (is_trunc_B' H)
+
+definition is_set_Grp_hom (G H : [0;k]Grp) : is_set (Grp_hom G H) :=
+is_trunc_Grp_hom G H
+
 definition is_trunc_Grp (n k : ℕ) : is_trunc (n + 1) [n;k]Grp :=
 begin
   apply @is_trunc_equiv_closed_rev _ _ (n + 1) (Grp_equiv n k),
@@ -272,18 +285,9 @@ begin
   apply @is_trunc_equiv_closed_rev _ _ _ (ptruncconntype_eq_equiv X Y),
   apply @is_trunc_equiv_closed_rev _ _ _ (pequiv.sigma_char_pmap X Y),
   apply @is_trunc_subtype (X →* Y) (λ f, trunctype.mk' -1 (is_equiv f)),
-  apply is_trunc_pmap_of_is_conn X k.-2 (n.-1) (n + k) Y,
-  { apply le_of_eq, exact (sub_one_add_plus_two_sub_one n k)⁻¹ ⬝ !add_plus_two_comm },
-  { exact _ }
+  exact is_trunc_Grp_hom ((Grp_equiv n k)⁻¹ᵉ X) ((Grp_equiv n k)⁻¹ᵉ Y)
 end
 
-definition Grp_hom (G H : [n;k]Grp) : Type :=
-B G →* B H
-
-definition is_set_Grp_hom (G H : [0;k]Grp) : is_set (Grp_hom G H) :=
-is_trunc_pmap_of_is_conn _ (k.-2) _ k _ (le_of_eq (sub_one_add_plus_two_sub_one k 0)⁻¹)
-  (is_trunc_B' H)
-
 local attribute [instance] is_set_Grp_hom
 
 definition Grp_precategory [constructor] (k : ℕ) : precategory [0;k]Grp :=
@@ -298,10 +302,8 @@ Precategory.mk _ (Grp_precategory k)
 definition cGrp_equivalence_cType [constructor] (k : ℕ) : cGrp k ≃c cType*[k.-1] :=
 sorry
 
--- print category.Grp
--- set_option pp.universes true
--- print equivalence.symm
--- print equivalence.trans
+definition cGrp_equivalence_Grp [constructor] : cGrp 1 ≃c category.Grp :=
+sorry
 
 -- set_option pp.all true
 -- definition cGrp_equivalence_Grp [constructor] : cGrp 1 ≃c category.Grp :=
@@ -320,5 +322,16 @@ sorry
 --   (equivalence.symm Grp_equivalence_cptruncconntype')
 
 
+--has sorry
+print axioms ωstabilization
+print axioms Decat_adjoint_Disc_natural
+print axioms cGrp_equivalence_Grp
+
+
+-- no sorry's
+print axioms Decat_adjoint_Disc
+print axioms Deloop_adjoint_Loop
+print axioms stabilization
+print axioms is_trunc_Grp
 
 end higher_group

move_to_lib.hlean

@@ -24,6 +24,15 @@ end algebra
 
 namespace eq
 
+  -- this should maybe replace whisker_left_idp and whisker_left_idp_con
+  definition whisker_left_idp_square {A : Type} {a a' : A} {p q : a = a'} (r : p = q) :
+    square (whisker_left idp r) r (idp_con p) (idp_con q) :=
+  begin induction r, exact hrfl end
+
+  definition ap_con_idp_left {A B : Type} (f : A → B) {a a' : A} (p : a = a') :
+    square (ap_con f idp p) idp (ap02 f (idp_con p)) (idp_con (ap f p)) :=
+  begin induction p, exact ids end
+
   definition pathover_tr_pathover_idp_of_eq {A : Type} {B : A → Type} {a a' : A} {b : B a} {b' : B a'} {p : a = a'}
     (q : b =[p] b') :
     pathover_tr p b ⬝o pathover_idp_of_eq (tr_eq_of_pathover q) = q :=
@@ -1321,29 +1330,23 @@ definition connect_intro_ppoint [constructor] {k : ℕ} {X : Type*} {Y : Type*}
   (f : X →* connect k Y) : connect_intro H (ppoint (ptr k Y) ∘* f) ~* f :=
 begin
   cases f with f f₀,
-  -- revert f₀, refine equiv_rect (fiber_eq_equiv' _ _)⁻¹ᵉ _ _,
-  -- revert f,
-  -- refine equiv_rect (!sigma_pi_equiv_pi_sigma ⬝e arrow_equiv_arrow_right _ !fiber.sigma_char⁻¹ᵉ) _ _,
-  -- intro fg pq, induction pq with p q, induction fg with f g,
-  -- induction Y with Y y₀, esimp at *, esimp [connect] at (f, p, q), induction p,
   fapply phomotopy.mk,
   { intro x, fapply fiber_eq, reflexivity,
     refine @is_conn.elim (k.-1) _ _ _ (λx', !is_trunc_eq) _ x,
     refine !is_conn.elim_β ⬝ _,
     refine _ ⬝ !idp_con⁻¹,
-    refine !ap_compose⁻¹ ⬝ _,
-    exact ap_is_constant point_eq f₀ },
-  { esimp,
-    refine whisker_left _ !fiber_eq_eta ⬝ !fiber_eq_con ⬝ apd011 fiber_eq !idp_con _,
-    esimp,
+    symmetry, refine _ ⬝ !con_idp, exact fiber_eq_pr2 f₀ },
+  { esimp, refine whisker_left _ !fiber_eq_eta ⬝ !fiber_eq_con ⬝ apd011 fiber_eq !idp_con _, esimp,
     apply eq_pathover_constant_left,
     refine whisker_right _ (whisker_right _ (whisker_right _ !is_conn.elim_β)) ⬝pv _,
-    exact sorry
-    --apply move_bot_of_left,
-
---    refine whisker_right _ _ ⬝ _,
---    refine !is_conn.elim_β ⬝ _,
-    }
+    esimp [connect], refine _ ⬝vp !con_idp,
+    apply move_bot_of_left, refine !idp_con ⬝ !con_idp⁻¹ ⬝ph _,
+    refine !con.assoc ⬝ !con.assoc ⬝pv _, apply whisker_tl,
+    note r := eq_bot_of_square (transpose (whisker_left_idp_square (fiber_eq_pr2 f₀))⁻¹ᵛ),
+    refine !con.assoc⁻¹ ⬝ whisker_right _ r⁻¹ ⬝pv _, clear r,
+    apply move_top_of_left,
+    refine whisker_right_idp (ap_con tr idp (ap point f₀))⁻¹ᵖ ⬝pv _,
+    exact (ap_con_idp_left tr (ap point f₀))⁻¹ʰ }
 end
 
 definition connect_intro_equiv [constructor] {k : ℕ} {X : Type*} (Y : Type*) (H : is_conn k X) :