feat(connectedness): show that if f is n-connected, then trunc_functor k f is so, too

This commit is contained in:
Floris van Doorn 2016-03-28 12:46:17 -04:00 committed by Leonardo de Moura
parent 54da5bcbda
commit 4895726c54
4 changed files with 110 additions and 21 deletions

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@ -40,8 +40,20 @@ end
structure linear_weak_order [class] (A : Type) extends weak_order A :=
(le_total : Πa b, le a b ⊎ le b a)
definition le.total [s : linear_weak_order A] (a b : A) : a ≤ b ⊎ b ≤ a :=
!linear_weak_order.le_total
section
variables [linear_weak_order A]
theorem le.total (a b : A) : a ≤ b ⊎ b ≤ a := !linear_weak_order.le_total
theorem le_of_not_ge {a b : A} (H : ¬ a ≥ b) : a ≤ b := sum.resolve_left !le.total H
definition le_by_cases (a b : A) {P : Type} (H1 : a ≤ b → P) (H2 : b ≤ a → P) : P :=
begin
cases (le.total a b) with H H,
{ exact H1 H},
{ exact H2 H}
end
end
/- strict orders -/

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@ -27,7 +27,7 @@ namespace is_conn
theorem is_conn_of_le (A : Type) {n k : ℕ₋₂} (H : n ≤ k) [is_conn k A] : is_conn n A :=
begin
apply is_contr_equiv_closed,
apply trunc_trunc_equiv_left _ n k H
apply trunc_trunc_equiv_left _ H
end
theorem is_conn_fun_of_le {A B : Type} (f : A → B) {n k : ℕ₋₂} (H : n ≤ k)
@ -244,14 +244,21 @@ namespace is_conn
-- Corollary 7.5.5
definition is_conn_homotopy (n : ℕ₋₂) {A B : Type} {f g : A → B}
(p : f ~ g) (H : is_conn_fun n f) : is_conn_fun n g :=
@retract_of_conn_is_conn _ _ (arrow.arrow_hom_of_homotopy p) (arrow.is_retraction_arrow_hom_of_homotopy p) n H
@retract_of_conn_is_conn _ _
(arrow.arrow_hom_of_homotopy p) (arrow.is_retraction_arrow_hom_of_homotopy p) n H
-- all types are -2-connected
definition is_conn_minus_two (A : Type) : is_conn -2 A :=
is_trunc_trunc -2 A
_
-- the following trivial cases are solved by type class inference
definition is_conn_of_is_contr (k : ℕ₋₂) (A : Type) [is_contr A] : is_conn k A := _
definition is_conn_fun_of_is_equiv (k : ℕ₋₂) {A B : Type} (f : A → B) [is_equiv f] :
is_conn_fun k f :=
_
-- Lemma 7.5.14
theorem is_equiv_trunc_functor_of_is_conn_fun {A B : Type} (n : ℕ₋₂) (f : A → B)
theorem is_equiv_trunc_functor_of_is_conn_fun [instance] {A B : Type} (n : ℕ₋₂) (f : A → B)
[H : is_conn_fun n f] : is_equiv (trunc_functor n f) :=
begin
fapply adjointify,
@ -265,7 +272,7 @@ namespace is_conn
[H : is_conn_fun n f] : trunc n A ≃ trunc n B :=
equiv.mk (trunc_functor n f) (is_equiv_trunc_functor_of_is_conn_fun n f)
definition is_conn_fun_trunc_functor {n k : ℕ₋₂} {A B : Type} (f : A → B) (H : k ≤ n)
definition is_conn_fun_trunc_functor_of_le {n k : ℕ₋₂} {A B : Type} (f : A → B) (H : k ≤ n)
[H2 : is_conn_fun k f] : is_conn_fun k (trunc_functor n f) :=
begin
apply is_conn_fun.intro,
@ -278,4 +285,20 @@ namespace is_conn
induction a with a, esimp, rewrite [is_conn_fun.elim_β]}
end
definition is_conn_fun_trunc_functor_of_ge {n k : ℕ₋₂} {A B : Type} (f : A → B) (H : n ≤ k)
[H2 : is_conn_fun k f] : is_conn_fun k (trunc_functor n f) :=
begin
apply is_conn_fun_of_is_equiv,
apply is_equiv_trunc_functor_of_le f H
end
-- Exercise 7.18
definition is_conn_fun_trunc_functor {n k : ℕ₋₂} {A B : Type} (f : A → B)
[H2 : is_conn_fun k f] : is_conn_fun k (trunc_functor n f) :=
begin
eapply algebra.le_by_cases k n: intro H,
{ exact is_conn_fun_trunc_functor_of_le f H},
{ exact is_conn_fun_trunc_functor_of_ge f H}
end
end is_conn

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@ -11,7 +11,7 @@ Properties of trunc_index, is_trunc, trunctype, trunc, and the pointed versions
import .pointed2 ..function algebra.order types.nat.order
open eq sigma sigma.ops pi function equiv trunctype
is_equiv prod pointed nat is_trunc algebra
is_equiv prod pointed nat is_trunc algebra sum
/- basic computation with ℕ₋₂, its operations and its order -/
namespace trunc_index
@ -64,13 +64,30 @@ namespace trunc_index
{ exfalso, apply @not_succ_le_self n, exact trunc_index.le_trans H1 H2}
end
protected definition le_succ {n m : ℕ₋₂} (H1 : n ≤ m): n ≤ m.+1 :=
protected definition le_succ {n m : ℕ₋₂} (H1 : n ≤ m) : n ≤ m.+1 :=
le.step H1
protected definition self_le_succ (n : ℕ₋₂) : n ≤ n.+1 :=
le.step (trunc_index.le.tr_refl n)
-- the order is total
protected theorem le_sum_le (n m : ℕ₋₂) : n ≤ m ⊎ m ≤ n :=
begin
induction m with m IH,
{ exact inr !minus_two_le},
{ cases IH with H H,
{ exact inl (trunc_index.le_succ H)},
{ cases H with n' H,
{ exact inl !trunc_index.self_le_succ},
{ exact inr (succ_le_succ H)}}}
end
end trunc_index open trunc_index
definition weak_order_trunc_index [trans_instance] [reducible] : weak_order trunc_index :=
weak_order.mk le trunc_index.le.tr_refl @trunc_index.le_trans @trunc_index.le_antisymm
definition linear_weak_order_trunc_index [trans_instance] [reducible] :
linear_weak_order trunc_index :=
linear_weak_order.mk le trunc_index.le.tr_refl @trunc_index.le_trans @trunc_index.le_antisymm
trunc_index.le_sum_le
namespace trunc_index
@ -198,6 +215,15 @@ namespace trunc_index
exact le_of_succ_le_succ (le_of_succ_le_succ H)
end
protected theorem succ_le_of_not_le {n m : ℕ₋₂} (H : ¬ n ≤ m) : m.+1 ≤ n :=
begin
cases (le.total n m) with H2 H2,
{ exfalso, exact H H2},
{ cases H2 with n' H2',
{ exfalso, exact H !le.refl},
{ exact succ_le_succ H2'}}
end
end trunc_index open trunc_index
namespace is_trunc
@ -310,7 +336,7 @@ namespace is_trunc
is_set_of_double_neg_elim (λa b, by_contradiction)
end
theorem is_trunc_of_axiom_K_of_le {A : Type} (n : ℕ₋₂) (H : -1 ≤ n)
theorem is_trunc_of_axiom_K_of_le {A : Type} {n : ℕ₋₂} (H : -1 ≤ n)
(K : Π(a : A), is_trunc n (a = a)) : is_trunc (n.+1) A :=
@is_trunc_succ_intro _ _ (λa b, is_trunc_of_imp_is_trunc_of_le H (λp, eq.rec_on p !K))
@ -472,7 +498,7 @@ namespace trunc
end
/- equivalences between truncated types (see also hit.trunc) -/
definition trunc_trunc_equiv_left [constructor] (A : Type) (n m : ℕ₋₂) (H : n ≤ m)
definition trunc_trunc_equiv_left [constructor] (A : Type) {n m : ℕ₋₂} (H : n ≤ m)
: trunc n (trunc m A) ≃ trunc n A :=
begin
note H2 := is_trunc_of_le (trunc n A) H,
@ -483,7 +509,7 @@ namespace trunc
{ intro x, induction x with x, induction x with x, reflexivity}
end
definition trunc_trunc_equiv_right [constructor] (A : Type) (n m : ℕ₋₂) (H : n ≤ m)
definition trunc_trunc_equiv_right [constructor] (A : Type) {n m : ℕ₋₂} (H : n ≤ m)
: trunc m (trunc n A) ≃ trunc n A :=
begin
apply trunc_equiv,
@ -492,7 +518,7 @@ namespace trunc
definition trunc_equiv_trunc_of_le {n m : ℕ₋₂} {A B : Type} (H : n ≤ m)
(f : trunc m A ≃ trunc m B) : trunc n A ≃ trunc n B :=
(trunc_trunc_equiv_left A _ _ H)⁻¹ᵉ ⬝e trunc_equiv_trunc n f ⬝e trunc_trunc_equiv_left B _ _ H
(trunc_trunc_equiv_left A H)⁻¹ᵉ ⬝e trunc_equiv_trunc n f ⬝e trunc_trunc_equiv_left B H
definition trunc_trunc_equiv_trunc_trunc [constructor] (n m : ℕ₋₂) (A : Type)
: trunc n (trunc m A) ≃ trunc m (trunc n A) :=
@ -504,6 +530,27 @@ namespace trunc
{ reflexivity}
end
theorem is_trunc_trunc_of_le (A : Type)
(n : ℕ₋₂) {m k : ℕ₋₂} (H : m ≤ k) [is_trunc n (trunc k A)] : is_trunc n (trunc m A) :=
begin
apply is_trunc_equiv_closed,
{ apply trunc_trunc_equiv_left, exact H},
end
definition trunc_functor_homotopy_of_le {n k : ℕ₋₂} {A B : Type} (f : A → B) (H : n ≤ k) :
to_fun (trunc_trunc_equiv_left B H) ∘
trunc_functor n (trunc_functor k f) ∘
to_fun (trunc_trunc_equiv_left A H)⁻¹ᵉ ~
trunc_functor n f :=
begin
intro x, induction x with x, reflexivity
end
definition is_equiv_trunc_functor_of_le {n k : ℕ₋₂} {A B : Type} (f : A → B) (H : n ≤ k)
[is_equiv (trunc_functor k f)] : is_equiv (trunc_functor n f) :=
is_equiv_of_equiv_of_homotopy (trunc_equiv_trunc_of_le H (equiv.mk (trunc_functor k f) _))
(trunc_functor_homotopy_of_le f H)
/- trunc_functor preserves surjectivity -/
definition is_surjective_trunc_functor {A B : Type} (n : ℕ₋₂) (f : A → B) [H : is_surjective f]
@ -539,19 +586,19 @@ namespace trunc
: ptrunc n X ≃* X :=
pequiv_of_equiv (trunc_equiv n X) idp
definition ptrunc_ptrunc_pequiv_left [constructor] (A : Type*) (n m : ℕ₋₂) (H : n ≤ m)
definition ptrunc_ptrunc_pequiv_left [constructor] (A : Type*) {n m : ℕ₋₂} (H : n ≤ m)
: ptrunc n (ptrunc m A) ≃* ptrunc n A :=
pequiv_of_equiv (trunc_trunc_equiv_left A n m H) idp
pequiv_of_equiv (trunc_trunc_equiv_left A H) idp
definition ptrunc_ptrunc_pequiv_right [constructor] (A : Type*) (n m : ℕ₋₂) (H : n ≤ m)
definition ptrunc_ptrunc_pequiv_right [constructor] (A : Type*) {n m : ℕ₋₂} (H : n ≤ m)
: ptrunc m (ptrunc n A) ≃* ptrunc n A :=
pequiv_of_equiv (trunc_trunc_equiv_right A n m H) idp
pequiv_of_equiv (trunc_trunc_equiv_right A H) idp
definition ptrunc_pequiv_ptrunc_of_le {n m : ℕ₋₂} {A B : Type*} (H : n ≤ m)
(f : ptrunc m A ≃* ptrunc m B) : ptrunc n A ≃* ptrunc n B :=
(ptrunc_ptrunc_pequiv_left A _ _ H)⁻¹ᵉ* ⬝e*
(ptrunc_ptrunc_pequiv_left A H)⁻¹ᵉ* ⬝e*
ptrunc_pequiv_ptrunc n f ⬝e*
ptrunc_ptrunc_pequiv_left B _ _ H
ptrunc_ptrunc_pequiv_left B H
definition ptrunc_ptrunc_pequiv_ptrunc_ptrunc [constructor] (n m : ℕ₋₂) (A : Type*)
: ptrunc n (ptrunc m A) ≃ ptrunc m (ptrunc n A) :=

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@ -43,6 +43,13 @@ section
theorem le.total (a b : A) : a ≤ b b ≤ a := !linear_weak_order.le_total
theorem le_of_not_ge {a b : A} (H : ¬ a ≥ b) : a ≤ b := or.resolve_left !le.total H
theorem le_by_cases (a b : A) {P : Prop} (H1 : a ≤ b → P) (H2 : b ≤ a → P) : P :=
begin
cases (le.total a b) with H H,
{ exact H1 H},
{ exact H2 H}
end
end
/- strict orders -/