redefine maxm2 in strunc

This commit is contained in:
Floris van Doorn 2017-06-30 15:14:55 +01:00
parent 057980ca1f
commit dce2832ead
2 changed files with 35 additions and 26 deletions

View file

@ -5,19 +5,18 @@ open trunc_index nat
namespace int
definition maxm2 : → ℕ₋₂ :=
λ n, int.cases_on n trunc_index.of_nat
(λ m, nat.cases_on m -1 (λ a, -2))
attribute maxm2 [unfold 1]
/-
The function from integers to truncation indices which sends positive numbers to themselves, and negative
numbers to negative 2. In particular -1 is sent to -2, but since we only work with pointed types, that
doesn't matter for us -/
definition maxm2 [unfold 1] : → ℕ₋₂ :=
λ n, int.cases_on n trunc_index.of_nat (λk, -2)
definition maxm2_le_maxm0 (n : ) : maxm2 n ≤ max0 n :=
begin
induction n with n n,
{ exact le.tr_refl n },
{ cases n with n,
{ exact le.step (le.tr_refl -1) },
{ exact minus_two_le 0 } }
{ exact minus_two_le 0 }
end
definition max0_le_of_le {n : } {m : } (H : n ≤ of_nat m)
@ -43,14 +42,11 @@ definition loop_ptrunc_maxm2_pequiv (k : ) (X : Type*) :
begin
induction k with k k,
{ exact loop_ptrunc_pequiv k X },
{ cases k with k,
{ exact loop_ptrunc_pequiv -1 X },
{ cases k with k,
{ exact loop_ptrunc_pequiv -2 X },
{ exact loop_pequiv_punit_of_is_set (pType.mk (trunc -2 X) (tr pt))
⬝e* (pequiv_punit_of_is_contr
(pType.mk (trunc -2 (Point X = Point X)) (tr idp))
(is_trunc_trunc -2 (Point X = Point X)))⁻¹ᵉ* } } }
{ refine _ ⬝e* (pequiv_punit_of_is_contr _ !is_trunc_trunc)⁻¹ᵉ*,
apply @loop_pequiv_punit_of_is_set,
cases k with k,
{ change is_set (trunc 0 X), apply _ },
{ change is_set (trunc -2 X), apply _ }}
end
definition is_trunc_of_is_trunc_maxm2 (k : ) (X : Type)
@ -73,14 +69,12 @@ definition is_trunc_maxm2_loop (A : pType) (k : )
begin
intro H, induction k with k k,
{ apply is_trunc_loop, exact H },
{ cases k with k,
{ apply is_trunc_loop, exact H},
{ apply is_trunc_loop, cases k with k,
{ apply is_contr_loop, cases k with k,
{ exact H },
{ apply is_trunc_succ, exact H } } }
{ have H2 : is_contr A, from H, apply _ } }
end
definition is_strunc (k : ) (E : spectrum) : Type :=
definition is_strunc [reducible] (k : ) (E : spectrum) : Type :=
Π (n : ), is_trunc (maxm2 (k + n)) (E n)
definition is_strunc_change_int {k l : } (E : spectrum) (p : k = l)
@ -95,6 +89,10 @@ definition is_trunc_maxm2_change_int {k l : } (X : pType) (p : k = l)
: is_trunc (maxm2 k) X → is_trunc (maxm2 l) X :=
by induction p; exact id
definition strunc_functor [constructor] (k : ) {E F : spectrum} (f : E →ₛ F) :
strunc k E →ₛ strunc k F :=
smap.mk (λn, ptrunc_functor (maxm2 (k + n)) (f n)) sorry
definition is_strunc_EM_spectrum (G : AbGroup)
: is_strunc 0 (EM_spectrum G) :=
begin
@ -102,13 +100,19 @@ begin
{ -- case ≥ 0
apply is_trunc_maxm2_change_int (EM G n) (zero_add n)⁻¹,
apply is_trunc_EM },
{ induction n with n IH,
{ change is_contr (EM_spectrum G (-[1+n])),
induction n with n IH,
{ -- case = -1
apply is_trunc_loop, exact ab_group.is_set_carrier G },
apply is_contr_loop, exact is_trunc_EM G 0 },
{ -- case < -1
apply is_trunc_maxm2_loop, exact IH }}
apply is_trunc_loop, apply is_trunc_succ, exact IH }}
end
definition strunc_elim [constructor] {k : } {E F : spectrum} (f : E →ₛ F)
(H : is_strunc k F) : strunc k E →ₛ F :=
smap.mk (λn, ptrunc.elim (maxm2 (k + n)) (f n))
(λn, sorry)
definition trivial_shomotopy_group_of_is_strunc (E : spectrum)
{n k : } (K : is_strunc n E) (H : n < k)
: is_contr (πₛ[k] E) :=

View file

@ -192,8 +192,11 @@ namespace pointed
psquare (pequiv_ap B p) (pequiv_ap C p) (f a) (f a') :=
begin induction p, exact phrfl end
definition is_contr_loop (A : Type*) [is_set A] : is_contr (Ω A) :=
is_contr.mk idp (λa, !is_prop.elim)
definition loop_pequiv_punit_of_is_set (X : Type*) [is_set X] : Ω X ≃* punit :=
pequiv_punit_of_is_contr _ (is_contr_of_inhabited_prop pt)
pequiv_punit_of_is_contr _ (is_contr_loop X)
definition loop_punit : Ω punit ≃* punit :=
loop_pequiv_punit_of_is_set punit
@ -202,4 +205,6 @@ namespace pointed
f ~* g :=
phomotopy.mk (λa, !eq_of_is_contr) !eq_of_is_contr
end pointed