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feat(hott/algebra/bundled): add a parameter to Group to specify whether it's an additive or multiplicative group

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This commit is contained in:
Floris van Doorn 2016-09-17 18:37:49 -04:00
parent 467001c0a9
commit d70334d100
12 changed files with 236 additions and 124 deletions
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92
hott/algebra/bundled.hlean
View file

@ -5,7 +5,7 @@ Authors: Jeremy Avigad
Bundled structures
-/
import algebra.group
import algebra.group homotopy.interval
open algebra
namespace algebra
@ -33,51 +33,85 @@ structure CommMonoid :=
attribute CommMonoid.carrier [coercion]
attribute CommMonoid.struct [instance]
structure Group :=
abbreviation signature := interval
structure Group (i : signature) :=
(carrier : Type) (struct : group carrier)
definition MulGroup [reducible] : Type := Group interval.zero
definition AddGroup [reducible] : Type := Group interval.one
attribute Group.carrier [coercion]
attribute Group.struct [instance]
structure CommGroup :=
definition MulGroup.mk [constructor] (G : Type) (H : group G) : MulGroup := Group.mk _ G _
definition AddGroup.mk [constructor] (G : Type) (H : add_group G) : AddGroup :=
Group.mk _ G add_group.to_group
section
local attribute group.to_add_group Group.struct [instance]
definition MulGroup.struct (G : MulGroup) : group G := _
definition AddGroup.struct (G : AddGroup) : add_group G := _
end
attribute MulGroup.struct AddGroup.struct [instance] [priority 2000]
attribute Group.struct [instance] [priority 800]
structure CommGroup (i : signature) :=
(carrier : Type) (struct : comm_group carrier)
definition MulCommGroup [reducible] : Type := CommGroup interval.zero
definition AddCommGroup [reducible] : Type := CommGroup interval.one
attribute CommGroup.carrier [coercion]
attribute CommGroup.struct [instance]
structure AddSemigroup :=
(carrier : Type) (struct : add_semigroup carrier)
definition MulCommGroup.mk [constructor] (G : Type) (H : comm_group G) : MulCommGroup :=
CommGroup.mk _ G _
definition AddCommGroup.mk [constructor] (G : Type) (H : add_comm_group G) : AddCommGroup :=
CommGroup.mk _ G add_comm_group.to_comm_group
attribute AddSemigroup.carrier [coercion]
attribute AddSemigroup.struct [instance]
section
local attribute comm_group.to_add_comm_group CommGroup.struct [instance]
structure AddCommSemigroup :=
(carrier : Type) (struct : add_comm_semigroup carrier)
definition MulCommGroup.struct (G : MulCommGroup) : comm_group G := _
definition AddCommGroup.struct (G : AddCommGroup) : add_comm_group G := _
end
attribute AddCommSemigroup.carrier [coercion]
attribute AddCommSemigroup.struct [instance]
attribute MulCommGroup.struct AddCommGroup.struct [instance] [priority 2000]
attribute CommGroup.struct [instance] [priority 800]
structure AddMonoid :=
(carrier : Type) (struct : add_monoid carrier)
attribute AddMonoid.carrier [coercion]
attribute AddMonoid.struct [instance]
structure AddCommMonoid :=
(carrier : Type) (struct : add_comm_monoid carrier)
-- structure AddSemigroup :=
-- (carrier : Type) (struct : add_semigroup carrier)
attribute AddCommMonoid.carrier [coercion]
attribute AddCommMonoid.struct [instance]
-- attribute AddSemigroup.carrier [coercion]
-- attribute AddSemigroup.struct [instance]
structure AddGroup :=
(carrier : Type) (struct : add_group carrier)
-- structure AddCommSemigroup :=
-- (carrier : Type) (struct : add_comm_semigroup carrier)
attribute AddGroup.carrier [coercion]
attribute AddGroup.struct [instance]
-- attribute AddCommSemigroup.carrier [coercion]
-- attribute AddCommSemigroup.struct [instance]
structure AddCommGroup :=
(carrier : Type) (struct : add_comm_group carrier)
-- structure AddMonoid :=
-- (carrier : Type) (struct : add_monoid carrier)
attribute AddCommGroup.carrier [coercion]
attribute AddCommGroup.struct [instance]
-- attribute AddMonoid.carrier [coercion]
-- attribute AddMonoid.struct [instance]
-- structure AddCommMonoid :=
-- (carrier : Type) (struct : add_comm_monoid carrier)
-- attribute AddCommMonoid.carrier [coercion]
-- attribute AddCommMonoid.struct [instance]
-- structure AddGroup :=
-- (carrier : Type) (struct : add_group carrier)
-- attribute AddGroup.carrier [coercion]
-- attribute AddGroup.struct [instance]
-- structure AddCommGroup :=
-- (carrier : Type) (struct : add_comm_group carrier)
-- attribute AddCommGroup.carrier [coercion]
-- attribute AddCommGroup.struct [instance]
end algebra

3
hott/algebra/category/functor/attributes.hlean
View file

@ -21,6 +21,9 @@ namespace category
definition split_essentially_surjective [class] (F : C ⇒ D) := Π(d : D), Σ(c : C), F c ≅ d
definition essentially_surjective [class] (F : C ⇒ D) := Π(d : D), ∃(c : C), F c ≅ d
definition is_weak_equivalence [class] (F : C ⇒ D) :=
fully_faithful F × essentially_surjective F
definition is_equiv_of_fully_faithful [instance] (F : C ⇒ D)
[H : fully_faithful F] (c c' : C) : is_equiv (@(to_fun_hom F) c c') :=
!H

3
hott/algebra/category/functor/equivalence.hlean
View file

@ -18,9 +18,6 @@ namespace category
(is_iso_unit : is_iso η)
(is_iso_counit : is_iso ε)
definition is_weak_equivalence [class] (F : C ⇒ D) :=
fully_faithful F × essentially_surjective F
abbreviation inverse := @is_equivalence.G
postfix ⁻¹ := inverse
--a second notation for the inverse, which is not overloaded (there is no unicode superscript F)

2
hott/algebra/category/groupoid.hlean
View file

@ -79,7 +79,7 @@ namespace category
definition Groupoid.eta [unfold 1] (C : Groupoid) : Groupoid.mk C C = C :=
Groupoid.rec (λob c, idp) C
definition Groupoid_of_Group [constructor] (G : Group) : Groupoid :=
definition Groupoid_of_Group [constructor] {i : signature} (G : Group i) : Groupoid :=
Groupoid.mk unit (groupoid_of_group G)
end category

27
hott/algebra/group.hlean
View file

@ -137,6 +137,22 @@ definition add_comm_monoid.to_comm_monoid {A : Type} [s : add_comm_monoid A] : c
mul_comm := add_comm_monoid.add_comm
⦄
definition monoid.to_add_monoid {A : Type} [s : monoid A] : add_monoid A :=
⦃ add_monoid,
add := monoid.mul,
add_assoc := monoid.mul_assoc,
zero := monoid.one A,
add_zero := monoid.mul_one,
zero_add := monoid.one_mul,
is_set_carrier := _
⦄
definition comm_monoid.to_add_comm_monoid {A : Type} [s : comm_monoid A] : add_comm_monoid A :=
⦃ add_comm_monoid,
monoid.to_add_monoid,
add_comm := comm_monoid.mul_comm
⦄
section add_comm_monoid
variables [s : add_comm_monoid A]
include s
@ -337,6 +353,9 @@ definition add_group.to_group {A : Type} [s : add_group A] : group A :=
⦃ group, add_monoid.to_monoid,
mul_left_inv := add_group.add_left_inv ⦄
definition group.to_add_group {A : Type} [s : group A] : add_group A :=
⦃ add_group, monoid.to_add_monoid,
add_left_inv := group.mul_left_inv ⦄
section add_group
@ -528,6 +547,14 @@ end add_group
structure add_comm_group [class] (A : Type) extends add_group A, add_comm_monoid A
definition add_comm_group.to_comm_group {A : Type} [s : add_comm_group A] : comm_group A :=
⦃ comm_group, add_group.to_group,
mul_comm := add_comm_group.add_comm ⦄
definition comm_group.to_add_comm_group {A : Type} [s : comm_group A] : add_comm_group A :=
⦃ add_comm_group, group.to_add_group,
add_comm := comm_group.mul_comm ⦄
section add_comm_group
variable [s : add_comm_group A]
include s

63
hott/algebra/group_theory.hlean
View file

@ -15,19 +15,22 @@ open eq algebra pointed function is_trunc pi category equiv is_equiv
set_option class.force_new true
namespace group
definition pointed_Group [instance] [constructor] {i : signature} (G : Group i) : pointed G :=
pointed.mk 1
definition pType_of_Group [constructor] [reducible] {i : signature} (G : Group i) : Type* :=
pointed.MK G 1
definition Set_of_Group [constructor] {i : signature} (G : Group i) : Set := trunctype.mk G _
definition pointed_Group [instance] [constructor] (G : Group) : pointed G := pointed.mk 1
definition pType_of_Group [constructor] [reducible] (G : Group) : Type* := pointed.MK G 1
definition Set_of_Group [constructor] (G : Group) : Set := trunctype.mk G _
definition Group_of_CommGroup [coercion] [constructor] {i : signature} (G : CommGroup i) :
Group i :=
Group.mk i G _
definition Group_of_CommGroup [coercion] [constructor] (G : CommGroup) : Group :=
Group.mk G _
definition comm_group_Group_of_CommGroup [instance] [constructor] (G : CommGroup)
definition comm_group_Group_of_CommGroup [instance] [constructor] {i : signature} (G : CommGroup i)
: comm_group (Group_of_CommGroup G) :=
begin esimp, exact _ end
definition group_pType_of_Group [instance] (G : Group) : group (pType_of_Group G) :=
definition group_pType_of_Group [instance] {i : signature} (G : Group i) :
group (pType_of_Group G) :=
Group.struct G
/- group homomorphisms -/
@ -74,18 +77,20 @@ namespace group
end
structure homomorphism (G₁ G₂ : Group) : Type :=
structure homomorphism {i j : signature} (G₁ : Group i) (G₂ : Group j) : Type :=
(φ : G₁ → G₂)
(p : is_homomorphism φ)
infix ` →g `:55 := homomorphism
definition group_fun [unfold 3] [coercion] := @homomorphism.φ
definition homomorphism.struct [instance] [priority 2000] {G₁ G₂ : Group} (φ : G₁ →g G₂)
definition group_fun [unfold 5] [coercion] := @homomorphism.φ
definition homomorphism.struct [instance] [priority 2000] {i j : signature}
{G₁ : Group i} {G₂ : Group j} (φ : G₁ →g G₂)
: is_homomorphism φ :=
homomorphism.p φ
variables {G G₁ G₂ G₃ : Group} {g h : G₁} {ψ : G₂ →g G₃} {φ₁ φ₂ : G₁ →g G₂} (φ : G₁ →g G₂)
variables {i j k l : signature} {G : Group i} {G₁ : Group j} {G₂ : Group k} {G₃ : Group l}
{g h : G₁} {ψ : G₂ →g G₃} {φ₁ φ₂ : G₁ →g G₂} (φ : G₁ →g G₂)
definition to_respect_mul /- φ -/ (g h : G₁) : φ (g * h) = φ g * φ h :=
respect_mul φ g h
@ -99,7 +104,8 @@ namespace group
definition to_is_embedding_homomorphism /- φ -/ (H : Π{g}, φ g = 1 → g = 1) : is_embedding φ :=
is_embedding_homomorphism φ @H
definition is_set_homomorphism [instance] (G₁ G₂ : Group) : is_set (G₁ →g G₂) :=
variables (G₁ G₂)
definition is_set_homomorphism [instance] : is_set (G₁ →g G₂) :=
begin
have H : G₁ →g G₂ ≃ Σ(f : G₁ → G₂), Π(g₁ g₂ : G₁), f (g₁ * g₂) = f g₁ * f g₂,
begin
@ -112,6 +118,7 @@ namespace group
apply is_trunc_equiv_closed_rev, exact H
end
variables {G₁ G₂}
definition pmap_of_homomorphism [constructor] /- φ -/ : pType_of_Group G₁ →* pType_of_Group G₂ :=
pmap.mk φ begin esimp, exact respect_one φ end
@ -126,13 +133,15 @@ namespace group
definition homomorphism_compose [constructor] [trans] (ψ : G₂ →g G₃) (φ : G₁ →g G₂) : G₁ →g G₃ :=
homomorphism.mk (ψ ∘ φ) (is_homomorphism_compose _ _)
definition homomorphism_id [constructor] [refl] (G : Group) : G →g G :=
variable (G)
definition homomorphism_id [constructor] [refl] : G →g G :=
homomorphism.mk (@id G) (is_homomorphism_id G)
variable {G}
infixr ` ∘g `:75 := homomorphism_compose
notation 1 := homomorphism_id _
structure isomorphism (A B : Group) :=
structure isomorphism {i j : signature} (A : Group i) (B : Group j) :=
(to_hom : A →g B)
(is_equiv_to_hom : is_equiv to_hom)
@ -151,10 +160,10 @@ namespace group
(p : Πg₁ g₂, φ (g₁ * g₂) = φ g₁ * φ g₂) : G₁ ≃g G₂ :=
isomorphism.mk (homomorphism.mk φ p) !to_is_equiv
definition eq_of_isomorphism {G₁ G₂ : Group} (φ : G₁ ≃g G₂) : G₁ = G₂ :=
definition eq_of_isomorphism {G₁ G₂ : Group i} (φ : G₁ ≃g G₂) : G₁ = G₂ :=
Group_eq (equiv_of_isomorphism φ) (respect_mul φ)
definition isomorphism_of_eq {G₁ G₂ : Group} (φ : G₁ = G₂) : G₁ ≃g G₂ :=
definition isomorphism_of_eq {G₁ G₂ : Group i} (φ : G₁ = G₂) : G₁ ≃g G₂ :=
isomorphism_of_equiv (equiv_of_eq (ap Group.carrier φ))
begin intros, induction φ, reflexivity end
@ -165,8 +174,10 @@ namespace group
rewrite [respect_mul φ, +right_inv φ]
end end
definition isomorphism.refl [refl] [constructor] (G : Group) : G ≃g G :=
variable (G)
definition isomorphism.refl [refl] [constructor] : G ≃g G :=
isomorphism.mk 1 !is_equiv_id
variable {G}
definition isomorphism.symm [symm] [constructor] (φ : G₁ ≃g G₂) : G₂ ≃g G₁ :=
isomorphism.mk (to_ginv φ) !is_equiv_inv
@ -175,11 +186,11 @@ namespace group
isomorphism.mk (ψ ∘g φ) !is_equiv_compose
definition isomorphism.eq_trans [trans] [constructor]
{G₁ G₂ G₃ : Group} (φ : G₁ = G₂) (ψ : G₂ ≃g G₃) : G₁ ≃g G₃ :=
{G₁ G₂ : Group i} {G₃ : Group j} (φ : G₁ = G₂) (ψ : G₂ ≃g G₃) : G₁ ≃g G₃ :=
proof isomorphism.trans (isomorphism_of_eq φ) ψ qed
definition isomorphism.trans_eq [trans] [constructor]
{G₁ G₂ G₃ : Group} (φ : G₁ ≃g G₂) (ψ : G₂ = G₃) : G₁ ≃g G₃ :=
{G₁ : Group i} {G₂ G₃ : Group j} (φ : G₁ ≃g G₂) (ψ : G₂ = G₃) : G₁ ≃g G₃ :=
isomorphism.trans φ (isomorphism_of_eq ψ)
postfix `⁻¹ᵍ`:(max + 1) := isomorphism.symm
@ -199,10 +210,10 @@ namespace group
/- category of groups -/
definition precategory_group [constructor] : precategory Group :=
definition precategory_group [constructor] (i : signature) : precategory (Group i) :=
precategory.mk homomorphism
@homomorphism_compose
@homomorphism_id
(@homomorphism_compose _ _ _)
(@homomorphism_id _)
(λG₁ G₂ G₃ G₄ φ₃ φ₂ φ₁, homomorphism_eq (λg, idp))
(λG₁ G₂ φ, homomorphism_eq (λg, idp))
(λG₁ G₂ φ, homomorphism_eq (λg, idp))
@ -260,15 +271,17 @@ namespace group
end
definition trivial_group_of_is_contr (G : Group) [H : is_contr G] : G ≃g G0 :=
variable (G)
definition trivial_group_of_is_contr [H : is_contr G] : G ≃g G0 :=
begin
fapply isomorphism_of_equiv,
{ apply equiv_unit_of_is_contr},
{ intros, reflexivity}
end
definition trivial_group_of_is_contr' (G : Group) [H : is_contr G] : G = G0 :=
definition trivial_group_of_is_contr' (G : MulGroup) [H : is_contr G] : G = G0 :=
eq_of_isomorphism (trivial_group_of_is_contr G)
variable {G}
/-
A group where the point in the pointed type corresponds with 1 in the group.

10
hott/algebra/homotopy_group.hlean
View file

@ -36,13 +36,13 @@ namespace eq
local attribute comm_group_homotopy_group [instance]
definition ghomotopy_group [constructor] (n : ℕ) (A : Type*) : Group :=
Group.mk (π[succ n] A) _
definition ghomotopy_group [constructor] (n : ℕ) (A : Type*) : MulGroup :=
MulGroup.mk (π[succ n] A) _
definition cghomotopy_group [constructor] (n : ℕ) (A : Type*) : CommGroup :=
CommGroup.mk (π[succ (succ n)] A) _
definition cghomotopy_group [constructor] (n : ℕ) (A : Type*) : MulCommGroup :=
MulCommGroup.mk (π[succ (succ n)] A) _
definition fundamental_group [constructor] (A : Type*) : Group :=
definition fundamental_group [constructor] (A : Type*) : MulGroup :=
ghomotopy_group zero A
notation `πg[`:95 n:0 ` +1] `:0 A:95 := ghomotopy_group n A

10
hott/algebra/hott.hlean
View file

@ -15,8 +15,8 @@ namespace algebra
definition trivial_group [constructor] : group unit :=
group.mk (λx y, star) _ (λx y z, idp) star (unit.rec idp) (unit.rec idp) (λx, star) (λx, idp)
definition Trivial_group [constructor] : Group :=
Group.mk _ trivial_group
definition Trivial_group [constructor] : MulGroup :=
MulGroup.mk _ trivial_group
abbreviation G0 := Trivial_group
@ -65,15 +65,15 @@ namespace algebra
apply pathover_idp_of_eq, exact group_eq (resp_mul)
end
definition Group_eq_of_eq {G H : Group} (p : carrier G = carrier H)
definition Group_eq_of_eq {i : signature} {G H : Group i} (p : carrier G = carrier H)
(resp_mul : Π(g h : G), cast p (g * h) = cast p g * cast p h) : G = H :=
begin
cases G with Gc G, cases H with Hc H,
apply (apd011 mk p),
apply (apd011 (Group.mk i) p),
exact group_pathover resp_mul
end
definition Group_eq {G H : Group} (f : carrier G ≃ carrier H)
definition Group_eq {i : signature} {G H : Group i} (f : carrier G ≃ carrier H)
(resp_mul : Π(g h : G), f (g * h) = f g * f h) : G = H :=
Group_eq_of_eq (ua f) (λg h, !cast_ua ⬝ resp_mul g h ⬝ ap011 mul !cast_ua⁻¹ !cast_ua⁻¹)

102
hott/homotopy/EM.hlean
View file

@ -14,10 +14,10 @@ open algebra pointed nat eq category group algebra is_trunc iso pointed unit tru
namespace EM
open groupoid_quotient
variable {G : Group}
definition EM1 (G : Group) : Type :=
variables {i : signature} {G : Group i}
definition EM1 {i : signature} (G : Group i) : Type :=
groupoid_quotient (Groupoid_of_Group G)
definition pEM1 [constructor] (G : Group) : Type* :=
definition pEM1 [constructor] {i : signature} (G : Group i) : Type* :=
pointed.MK (EM1 G) (elt star)
definition base : EM1 G := elt star
@ -95,18 +95,19 @@ namespace EM
end EM
attribute EM.base [constructor]
attribute EM.rec EM.elim [unfold 7] [recursor 7]
attribute EM.rec_on EM.elim_on [unfold 4]
attribute EM.set_rec EM.set_elim [unfold 6]
attribute EM.prop_rec EM.prop_elim EM.elim_set [unfold 5]
attribute EM.rec EM.elim [unfold 8] [recursor 8]
attribute EM.rec_on EM.elim_on [unfold 5]
attribute EM.set_rec EM.set_elim [unfold 7]
attribute EM.prop_rec EM.prop_elim EM.elim_set [unfold 6]
namespace EM
open groupoid_quotient
definition base_eq_base_equiv [constructor] (G : Group) : (base = base :> pEM1 G) ≃ G :=
variables {i : signature} (G : Group i)
definition base_eq_base_equiv [constructor] : (base = base :> pEM1 G) ≃ G :=
!elt_eq_elt_equiv
definition fundamental_group_pEM1 (G : Group) : π₁ (pEM1 G) ≃g G :=
definition fundamental_group_pEM1 : π₁ (pEM1 G) ≃g G :=
begin
fapply isomorphism_of_equiv,
{ exact trunc_equiv_trunc 0 !base_eq_base_equiv ⬝e trunc_equiv 0 G},
@ -114,19 +115,20 @@ namespace EM
exact encode_con p q}
end
proposition is_trunc_pEM1 [instance] (G : Group) : is_trunc 1 (pEM1 G) :=
proposition is_trunc_pEM1 [instance] : is_trunc 1 (pEM1 G) :=
!is_trunc_groupoid_quotient
proposition is_trunc_EM1 [instance] (G : Group) : is_trunc 1 (EM1 G) :=
proposition is_trunc_EM1 [instance] : is_trunc 1 (EM1 G) :=
!is_trunc_groupoid_quotient
proposition is_conn_EM1 [instance] (G : Group) : is_conn 0 (EM1 G) :=
proposition is_conn_EM1 [instance] : is_conn 0 (EM1 G) :=
by apply @is_conn_groupoid_quotient; esimp; exact _
proposition is_conn_pEM1 [instance] (G : Group) : is_conn 0 (pEM1 G) :=
proposition is_conn_pEM1 [instance] : is_conn 0 (pEM1 G) :=
is_conn_EM1 G
variable {G}
definition EM1_map [unfold 7] {G : Group} {X : Type*} (e : Ω X ≃ G)
definition EM1_map [unfold 7] {X : Type*} (e : Ω X ≃ G)
(r : Πp q, e (p ⬝ q) = e p * e q) [is_conn 0 X] [is_trunc 1 X] : EM1 G → X :=
begin
intro x, induction x using EM.elim,
@ -140,9 +142,9 @@ end EM
open hopf susp
namespace EM
-- The K(G,n+1):
variables (G : CommGroup) (n : ℕ)
variables {i : signature} {G : CommGroup i} (n : ℕ)
definition EM1_mul [unfold 2 3] {G : CommGroup} (x x' : EM1 G) : EM1 G :=
definition EM1_mul [unfold 2 3] (x x' : EM1 G) : EM1 G :=
begin
induction x,
{ exact x'},
@ -154,14 +156,15 @@ namespace EM
{ refine EM.prop_rec _ x', apply resp_mul}
end
definition EM1_mul_one (G : CommGroup) (x : EM1 G) : EM1_mul x base = x :=
variable (G)
definition EM1_mul_one (x : EM1 G) : EM1_mul x base = x :=
begin
induction x using EM.set_rec,
{ reflexivity},
{ apply eq_pathover_id_right, apply hdeg_square, refine EM.elim_pth _ g}
end
definition h_space_EM1 [constructor] [instance] (G : CommGroup) : h_space (pEM1 G) :=
definition h_space_EM1 [constructor] [instance] : h_space (pEM1 G) :=
begin
fapply h_space.mk,
{ exact EM1_mul},
@ -171,22 +174,22 @@ namespace EM
end
/- K(G, n+1) -/
definition EMadd1 (G : CommGroup) (n : ℕ) : Type* :=
definition EMadd1 (n : ℕ) : Type* :=
ptrunc (n+1) (iterate_psusp n (pEM1 G))
definition loop_EM2 (G : CommGroup) : Ω[1] (EMadd1 G 1) ≃* pEM1 G :=
definition loop_EM2 : Ω[1] (EMadd1 G 1) ≃* pEM1 G :=
begin
apply hopf.delooping, reflexivity
end
definition homotopy_group_EM2 (G : CommGroup) : πg[1+1] (EMadd1 G 1) ≃g G :=
definition homotopy_group_EM2 : πg[1+1] (EMadd1 G 1) ≃g G :=
begin
refine ghomotopy_group_succ_in _ 0 ⬝g _,
refine homotopy_group_isomorphism_of_pequiv 0 (loop_EM2 G) ⬝g _,
apply fundamental_group_pEM1
end
definition homotopy_group_EMadd1 (G : CommGroup) (n : ℕ) : πg[n+1] (EMadd1 G n) ≃g G :=
definition homotopy_group_EMadd1 (n : ℕ) : πg[n+1] (EMadd1 G n) ≃g G :=
begin
cases n with n,
{ refine homotopy_group_isomorphism_of_pequiv 0 _ ⬝g fundamental_group_pEM1 G,
@ -201,42 +204,43 @@ namespace EM
section
local attribute EMadd1 [reducible]
definition is_conn_EMadd1 [instance] (G : CommGroup) (n : ℕ) : is_conn n (EMadd1 G n) := _
definition is_conn_EMadd1 [instance] (n : ℕ) : is_conn n (EMadd1 G n) := _
definition is_trunc_EMadd1 [instance] (G : CommGroup) (n : ℕ) : is_trunc (n+1) (EMadd1 G n) := _
definition is_trunc_EMadd1 [instance] (n : ℕ) : is_trunc (n+1) (EMadd1 G n) :=
_
end
/- K(G, n) -/
definition EM (G : CommGroup) : ℕ → Type*
definition EM {i : signature} (G : CommGroup i) : ℕ → Type*
| 0 := pType_of_Group G
| (k+1) := EMadd1 G k
namespace ops
abbreviation K := EM
abbreviation K := @EM
end ops
open ops
definition phomotopy_group_EM (G : CommGroup) (n : ℕ) : π*[n] (EM G n) ≃* pType_of_Group G :=
definition phomotopy_group_EM (n : ℕ) : π*[n] (EM G n) ≃* pType_of_Group G :=
begin
cases n with n,
{ rexact ptrunc_pequiv 0 (pType_of_Group G) _},
{ apply pequiv_of_isomorphism (homotopy_group_EMadd1 G n)}
end
definition ghomotopy_group_EM (G : CommGroup) (n : ℕ) : πg[n+1] (EM G (n+1)) ≃g G :=
definition ghomotopy_group_EM (n : ℕ) : πg[n+1] (EM G (n+1)) ≃g G :=
homotopy_group_EMadd1 G n
definition is_conn_EM [instance] (G : CommGroup) (n : ℕ) : is_conn (n.-1) (EM G n) :=
definition is_conn_EM [instance] (n : ℕ) : is_conn (n.-1) (EM G n) :=
begin
cases n with n,
{ apply is_conn_minus_one, apply tr, unfold [EM], exact 1},
{ apply is_conn_EMadd1}
end
definition is_conn_EM_succ [instance] (G : CommGroup) (n : ℕ) : is_conn n (EM G (succ n)) :=
definition is_conn_EM_succ [instance] (n : ℕ) : is_conn n (EM G (succ n)) :=
is_conn_EM G (succ n)
definition is_trunc_EM [instance] (G : CommGroup) (n : ℕ) : is_trunc n (EM G n) :=
definition is_trunc_EM [instance] (n : ℕ) : is_trunc n (EM G n) :=
begin
cases n with n,
{ unfold [EM], apply semigroup.is_set_carrier},
@ -244,26 +248,29 @@ namespace EM
end
/- Uniqueness of K(G, 1) -/
definition pEM1_pmap [constructor] {G : Group} {X : Type*} (e : Ω X ≃ G)
(r : Πp q, e (p ⬝ q) = e p * e q) [is_conn 0 X] [is_trunc 1 X] : pEM1 G →* X :=
variable {H : Group i}
definition pEM1_pmap [constructor] {X : Type*} (e : Ω X ≃ H)
(r : Πp q, e (p ⬝ q) = e p * e q) [is_conn 0 X] [is_trunc 1 X] : pEM1 H →* X :=
begin
apply pmap.mk (EM1_map e r),
reflexivity,
end
definition loop_pEM1 [constructor] (G : Group) : Ω (pEM1 G) ≃* pType_of_Group G :=
pequiv_of_equiv (base_eq_base_equiv G) idp
variable (H)
definition loop_pEM1 [constructor] : Ω (pEM1 H) ≃* pType_of_Group H :=
pequiv_of_equiv (base_eq_base_equiv H) idp
definition loop_pEM1_pmap {G : Group} {X : Type*} (e : Ω X ≃ G)
variable {H}
definition loop_pEM1_pmap {X : Type*} (e : Ω X ≃ H)
(r : Πp q, e (p ⬝ q) = e p * e q) [is_conn 0 X] [is_trunc 1 X] :
Ω→(pEM1_pmap e r) ~ e⁻¹ᵉ ∘ base_eq_base_equiv G :=
Ω→(pEM1_pmap e r) ~ e⁻¹ᵉ ∘ base_eq_base_equiv H :=
begin
apply homotopy_of_inv_homotopy_pre (base_eq_base_equiv G),
apply homotopy_of_inv_homotopy_pre (base_eq_base_equiv H),
intro g, exact !idp_con ⬝ !elim_pth
end
open trunc_index
definition pEM1_pequiv'.{u} {G : Group.{u}} {X : pType.{u}} (e : Ω X ≃ G)
definition pEM1_pequiv'.{u} {i : signature} {G : Group.{u} i} {X : pType.{u}} (e : Ω X ≃ G)
(r : Πp q, e (p ⬝ q) = e p * e q) [is_conn 0 X] [is_trunc 1 X] : pEM1 G ≃* X :=
begin
apply pequiv_of_pmap (pEM1_pmap e r),
@ -280,7 +287,7 @@ namespace EM
do 2 exact trivial_homotopy_group_of_is_trunc _ (succ_lt_succ !zero_lt_succ)}}
end
definition pEM1_pequiv.{u} {G : Group.{u}} {X : pType.{u}} (e : π₁ X ≃g G)
definition pEM1_pequiv.{u} {i : signature} {G : Group.{u} i} {X : pType.{u}} (e : π₁ X ≃g G)
[is_conn 0 X] [is_trunc 1 X] : pEM1 G ≃* X :=
begin
apply pEM1_pequiv' (!trunc_equiv⁻¹ᵉ ⬝e equiv_of_isomorphism e),
@ -290,7 +297,7 @@ namespace EM
definition pEM1_pequiv_type {X : Type*} [is_conn 0 X] [is_trunc 1 X] : pEM1 (π₁ X) ≃* X :=
pEM1_pequiv !isomorphism.refl
definition EM_pequiv_1.{u} {G : CommGroup.{u}} {X : pType.{u}} (e : π₁ X ≃g G)
definition EM_pequiv_1.{u} {i : signature} {G : CommGroup.{u} i} {X : pType.{u}} (e : π₁ X ≃g G)
[is_conn 0 X] [is_trunc 1 X] : EM G 1 ≃* X :=
begin
refine _ ⬝e* pEM1_pequiv e,
@ -298,11 +305,12 @@ namespace EM
apply is_trunc_pEM1
end
definition EMadd1_pequiv_pEM1 (G : CommGroup) : EMadd1 G 0 ≃* pEM1 G :=
variable (G)
definition EMadd1_pequiv_pEM1 : EMadd1 G 0 ≃* pEM1 G :=
begin apply ptrunc_pequiv, apply is_trunc_pEM1 end
definition EM1add1_pequiv_0.{u} {G : CommGroup.{u}} {X : pType.{u}} (e : π₁ X ≃g G)
[is_conn 0 X] [is_trunc 1 X] : EMadd1 G 0 ≃* X :=
definition EM1add1_pequiv_0.{u} {i : signature} {G : CommGroup.{u} i} {X : pType.{u}}
(e : π₁ X ≃g G) [is_conn 0 X] [is_trunc 1 X] : EMadd1 G 0 ≃* X :=
EMadd1_pequiv_pEM1 G ⬝e* pEM1_pequiv e
definition KG1_pequiv.{u} {X Y : pType.{u}} (e : π₁ X ≃g π₁ Y)
@ -314,7 +322,8 @@ namespace EM
!EMadd1_pequiv_pEM1 ⬝e* pEM1_pequiv fundamental_group_of_circle
/- loops of EM-spaces -/
definition loop_EMadd1 {G : CommGroup} (n : ℕ) : Ω (EMadd1 G (succ n)) ≃* EMadd1 G n :=
variable {G}
definition loop_EMadd1 (n : ℕ) : Ω (EMadd1 G (succ n)) ≃* EMadd1 G n :=
begin
cases n with n,
{ symmetry, apply EM1add1_pequiv_0, rexact homotopy_group_EMadd1 G 1,
@ -326,7 +335,8 @@ namespace EM
symmetry, refine freudenthal_pequiv _ this, }
end
definition loop_EM (G : CommGroup) (n : ℕ) : Ω (K G (succ n)) ≃* K G n :=
variable (G)
definition loop_EM (n : ℕ) : Ω (K G (succ n)) ≃* K G n :=
begin
cases n with n,
{ refine _ ⬝e* pequiv_of_isomorphism (fundamental_group_pEM1 G),

8
hott/homotopy/LES_of_homotopy_groups.hlean
View file

@ -729,12 +729,12 @@ namespace chain_complex
| (fin.mk 2 H) := proof comm_group_homotopy_group n (pfiber f) qed
| (fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
definition Group_LES_of_homotopy_groups (x : +3ℕ) : Group.{u} :=
Group.mk (LES_of_homotopy_groups (nat.succ (pr1 x), pr2 x))
definition Group_LES_of_homotopy_groups (x : +3ℕ) : MulGroup.{u} :=
MulGroup.mk (LES_of_homotopy_groups (nat.succ (pr1 x), pr2 x))
(group_LES_of_homotopy_groups (pr1 x) (pr2 x))
definition CommGroup_LES_of_homotopy_groups (n : +3ℕ) : CommGroup.{u} :=
CommGroup.mk (LES_of_homotopy_groups (pr1 n + 2, pr2 n))
definition CommGroup_LES_of_homotopy_groups (n : +3ℕ) : MulCommGroup.{u} :=
MulCommGroup.mk (LES_of_homotopy_groups (pr1 n + 2, pr2 n))
(comm_group_LES_of_homotopy_groups (pr1 n) (pr2 n))
definition homomorphism_LES_of_homotopy_groups_fun : Π(k : +3ℕ),

9
hott/types/int/hott.hlean
View file

@ -14,13 +14,14 @@ namespace int
section
open algebra
definition group_integers [constructor] : Group :=
Group.mk ℤ (group_of_add_group _)
definition group_integers [constructor] : AddGroup :=
AddGroup.mk ℤ _
notation `gℤ` := group_integers
definition CommGroup_int [constructor] : CommGroup :=
CommGroup.mk ℤ ⦃comm_group, group_of_add_group ℤ, mul_comm := add.comm⦄
definition CommGroup_int [constructor] : AddCommGroup :=
AddCommGroup.mk ℤ _
notation `agℤ` := CommGroup_int
end

27
library/algebra/group.lean
View file

@ -127,6 +127,21 @@ definition add_comm_monoid.to_comm_monoid {A : Type} [add_comm_monoid A] : comm_
mul_comm := add_comm_monoid.add_comm
⦄
definition monoid.to_add_monoid {A : Type} [s : monoid A] : add_monoid A :=
⦃ add_monoid,
add := monoid.mul,
add_assoc := monoid.mul_assoc,
zero := monoid.one A,
add_zero := monoid.mul_one,
zero_add := monoid.one_mul
⦄
definition comm_monoid.to_add_comm_monoid {A : Type} [s : comm_monoid A] : add_comm_monoid A :=
⦃ add_comm_monoid,
monoid.to_add_monoid,
add_comm := comm_monoid.mul_comm
⦄
section add_comm_monoid
variables [add_comm_monoid A]
@ -314,6 +329,9 @@ definition add_group.to_group {A : Type} [add_group A] : group A :=
⦃ group, add_monoid.to_monoid,
mul_left_inv := add_group.add_left_inv ⦄
definition group.to_add_group {A : Type} [s : group A] : add_group A :=
⦃ add_group, monoid.to_add_monoid,
add_left_inv := group.mul_left_inv ⦄
section add_group
variables [s : add_group A]
@ -541,6 +559,15 @@ end add_group
structure add_comm_group [class] (A : Type) extends add_group A, add_comm_monoid A
definition add_comm_group.to_comm_group (A : Type) [s : add_comm_group A] : comm_group A :=
⦃ comm_group, add_group.to_group,
mul_comm := add_comm_group.add_comm ⦄
definition comm_group.to_add_comm_group (A : Type) [s : comm_group A] : add_comm_group A :=
⦃ add_comm_group, group.to_add_group,
add_comm := comm_group.mul_comm ⦄
section add_comm_group
variable [s : add_comm_group A]
include s