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style(hott/algebra): rename theorems in the HoTT category libraries

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This commit is contained in:
Floris van Doorn 2015-02-27 19:50:06 -05:00
parent 23a248ab28
commit 326eaffafb
13 changed files with 213 additions and 203 deletions
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16
hott/algebra/category/basic.hlean
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@ -16,14 +16,14 @@ open morphism is_equiv eq is_trunc
namespace category
structure category [class] (ob : Type) extends parent : precategory ob :=
(iso_of_path_equiv : Π (a b : ob), is_equiv (@iso_of_path ob parent a b))
(iso_of_path_equiv : Π (a b : ob), is_equiv (@iso_of_eq ob parent a b))
attribute category [multiple-instances]
abbreviation iso_of_path_equiv := @category.iso_of_path_equiv
definition category.mk' [reducible] (ob : Type) (C : precategory ob)
(H : Π (a b : ob), is_equiv (@iso_of_path ob C a b)) : category ob :=
(H : Π (a b : ob), is_equiv (@iso_of_eq ob C a b)) : category ob :=
precategory.rec_on C category.mk H
section basic
@ -34,18 +34,18 @@ namespace category
-- TODO: Unsafe class instance?
attribute iso_of_path_equiv [instance]
definition path_of_iso (a b : ob) : a ≅ b → a = b :=
iso_of_path⁻¹
definition eq_of_iso (a b : ob) : a ≅ b → a = b :=
iso_of_eq⁻¹ᵉ
set_option apply.class_instance false -- disable class instance resolution in the apply tactic
definition ob_1_type : is_trunc 1 ob :=
definition is_trunc_1_ob : is_trunc 1 ob :=
begin
apply is_trunc_succ_intro, intros (a, b),
fapply is_trunc_is_equiv_closed,
exact (@path_of_iso _ _ a b),
exact (@eq_of_iso _ _ a b),
apply is_equiv_inv,
apply is_hset_iso,
apply is_hset_isomorphic,
end
end basic
@ -64,7 +64,7 @@ namespace category
definition category.Mk [reducible] := Category.mk
definition category.MK [reducible] (C : Precategory)
(H : Π (a b : C), is_equiv (@iso_of_path C C a b)) : Category :=
(H : Π (a b : C), is_equiv (@iso_of_eq C C a b)) : Category :=
Category.mk C (category.mk' C C H)
definition Category.eta (C : Category) : Category.mk C C = C :=

4
hott/algebra/category/constructions.hlean
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@ -13,11 +13,11 @@ open eq prod eq eq.ops equiv is_trunc funext pi category.ops morphism category
namespace category
section hset
definition is_category_hset (a b : Precategory_hset) : is_equiv (@iso_of_path _ _ a b) :=
definition is_univalent_hset (a b : Precategory_hset) : is_equiv (@iso_of_eq _ _ a b) :=
sorry
definition category_hset [reducible] [instance] : category hset :=
category.mk' hset precategory_hset is_category_hset
category.mk' hset precategory_hset is_univalent_hset
definition Category_hset [reducible] : Category :=
Category.mk hset category_hset

12
hott/algebra/groupoid.hlean
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@ -35,7 +35,7 @@ namespace category
(λ (a b : A) (p : a = b), con_idp p)
(λ (a b : A) (p : a = b), idp_con p)
(λ (a b : A) (p : a = b), @is_iso.mk A _ a b p p⁻¹
!con.left_inv !con.right_inv)
!con.right_inv !con.left_inv)
-- A groupoid with a contractible carrier is a group
definition group_of_is_contr_groupoid {ob : Type} [H : is_contr ob]
@ -49,10 +49,10 @@ namespace category
intro f, apply id_left,
intro f, apply id_right,
intro f, exact (morphism.inverse f),
intro f, exact (morphism.inverse_compose f),
intro f, exact (morphism.inverse_comp f),
end
definition group_of_unit_groupoid [G : groupoid unit] : group (hom ⋆ ⋆) :=
definition group_of_groupoid_unit [G : groupoid unit] : group (hom ⋆ ⋆) :=
begin
fapply group.mk,
intros (f, g), apply (comp f g),
@ -62,11 +62,11 @@ namespace category
intro f, apply id_left,
intro f, apply id_right,
intro f, exact (morphism.inverse f),
intro f, exact (morphism.inverse_compose f),
intro f, exact (morphism.inverse_comp f),
end
-- Conversely we can turn each group into a groupoid on the unit type
definition of_group.{l} (A : Type.{l}) [G : group A] : groupoid.{l l} unit :=
definition groupoid_of_group.{l} (A : Type.{l}) [G : group A] : groupoid.{l l} unit :=
begin
fapply groupoid.mk,
intros, exact A,
@ -92,7 +92,7 @@ namespace category
intro f, apply id_left,
intro f, apply id_right,
intro f, exact (morphism.inverse f),
intro f, exact (morphism.inverse_compose f),
intro f, exact (morphism.inverse_comp f),
end
-- Bundled version of categories

2
hott/algebra/precategory/basic.hlean
View file

@ -45,7 +45,7 @@ namespace category
definition id [reducible] := ID a
definition id_compose (a : ob) : ID a ∘ ID a = ID a := !id_left
definition id_comp (a : ob) : ID a ∘ ID a = ID a := !id_left
definition left_id_unique (H : Π{b} {f : hom b a}, i ∘ f = f) : i = id :=
calc i = i ∘ id : id_right

24
hott/algebra/precategory/constructions.hlean
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@ -39,7 +39,7 @@ namespace category
-- take the trick they use in Coq-HoTT, and introduce a further
-- axiom in the definition of precategories that provides thee
-- symmetric associativity proof.
definition op_op' {ob : Type} (C : precategory ob) : opposite (opposite C) = C :=
definition opposite_opposite' {ob : Type} (C : precategory ob) : opposite (opposite C) = C :=
begin
apply (precategory.rec_on C), intros (hom', homH', comp', ID', assoc', id_left', id_right'),
apply (ap (λassoc'', precategory.mk hom' @homH' comp' ID' assoc'' id_left' id_right')),
@ -48,8 +48,8 @@ namespace category
apply (@is_hset.elim), apply !homH',
end
definition op_op : Opposite (Opposite C) = C :=
(ap (Precategory.mk C) (op_op' C)) ⬝ !Precategory.eta
definition opposite_opposite : Opposite (Opposite C) = C :=
(ap (Precategory.mk C) (opposite_opposite' C)) ⬝ !Precategory.eta
end opposite
@ -91,7 +91,7 @@ namespace category
section
open prod is_trunc
definition prod_precategory [reducible] {obC obD : Type} (C : precategory obC) (D : precategory obD)
definition precategory_prod [reducible] {obC obD : Type} (C : precategory obC) (D : precategory obD)
: precategory (obC × obD) :=
precategory.mk (λ a b, hom (pr1 a) (pr1 b) × hom (pr2 a) (pr2 b))
(λ a b, !is_trunc_prod)
@ -101,8 +101,8 @@ namespace category
(λ a b f, prod_eq !id_left !id_left )
(λ a b f, prod_eq !id_right !id_right)
definition Prod_precategory [reducible] (C D : Precategory) : Precategory :=
precategory.Mk (prod_precategory C D)
definition Precategory_prod [reducible] (C D : Precategory) : Precategory :=
precategory.Mk (precategory_prod C D)
end
end product
@ -111,7 +111,7 @@ namespace category
--notation 1 := Category_one
--notation 2 := Category_two
postfix `ᵒᵖ`:max := opposite.Opposite
infixr `×c`:30 := product.Prod_precategory
infixr `×c`:30 := product.Precategory_prod
--instance [persistent] type_category category_one
-- category_two product.prod_category
end ops
@ -169,7 +169,7 @@ namespace category
-- definition Precategory_functor_rev [reducible] (C D : Precategory) : Precategory :=
-- Precategory_functor D C
/- we prove that if a natural transformation is pointwise an iso, then it is an iso -/
/- we prove that if a natural transformation is pointwise an to_fun, then it is an to_fun -/
variables {C D : Precategory} {F G : C ⇒ D} (η : F ⟹ G) [iso : Π(a : C), is_iso (η a)]
include iso
definition nat_trans_inverse : G ⟹ F :=
@ -177,16 +177,16 @@ namespace category
(λc, (η c)⁻¹)
(λc d f,
begin
apply iso.con_inv_eq_of_eq_con,
apply to_fun.comp_inverse_eq_of_eq_comp,
apply concat, rotate_left 1, apply assoc,
apply iso.eq_inv_con_of_con_eq,
apply to_fun.eq_inverse_comp_of_comp_eq,
apply inverse,
apply naturality,
end)
definition nat_trans_left_inverse : nat_trans_inverse η ∘n η = nat_trans.id :=
begin
fapply (apD011 nat_trans.mk),
apply eq_of_homotopy, intro c, apply inverse_compose,
apply eq_of_homotopy, intro c, apply inverse_comp,
apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros,
apply is_hset.elim
end
@ -194,7 +194,7 @@ namespace category
definition nat_trans_right_inverse : η ∘n nat_trans_inverse η = nat_trans.id :=
begin
fapply (apD011 nat_trans.mk),
apply eq_of_homotopy, intro c, apply compose_inverse,
apply eq_of_homotopy, intro c, apply comp_inverse,
apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros,
apply is_hset.elim
end

32
hott/algebra/precategory/functor.hlean
View file

@ -12,19 +12,19 @@ open function category eq prod equiv is_equiv sigma sigma.ops is_trunc funext
open pi
structure functor (C D : Precategory) : Type :=
(obF : C → D)
(homF : Π ⦃a b : C⦄, hom a b → hom (obF a) (obF b))
(respect_id : Π (a : C), homF (ID a) = ID (obF a))
(to_fun_ob : C → D)
(to_fun_hom : Π ⦃a b : C⦄, hom a b → hom (to_fun_ob a) (to_fun_ob b))
(respect_id : Π (a : C), to_fun_hom (ID a) = ID (to_fun_ob a))
(respect_comp : Π {a b c : C} (g : hom b c) (f : hom a b),
homF (g ∘ f) = homF g ∘ homF f)
to_fun_hom (g ∘ f) = to_fun_hom g ∘ to_fun_hom f)
namespace functor
infixl `⇒`:25 := functor
variables {C D E : Precategory}
attribute obF [coercion]
attribute homF [coercion]
attribute to_fun_ob [coercion]
attribute to_fun_hom [coercion]
-- The following lemmas will later be used to prove that the type of
-- precategories forms a precategory itself
@ -78,7 +78,7 @@ namespace functor
functor_eq_mk'' id₁ id₂ comp₁ comp₂ idp
(eq_of_homotopy (λc, eq_of_homotopy (λc', eq_of_homotopy (pH c c'))))
definition functor_eq_mk {F₁ F₂ : C ⇒ D} : Π(p : obF F₁ ∼ obF F₂),
definition functor_eq_mk {F₁ F₂ : C ⇒ D} : Π(p : to_fun_ob F₁ ∼ to_fun_ob F₂),
(Π(a b : C) (f : hom a b), transport (λF, hom (F a) (F b)) (eq_of_homotopy p) (F₁ f) = F₂ f)
→ F₁ = F₂ :=
functor.rec_on F₁ (λO₁ H₁ id₁ comp₁, functor.rec_on F₂ (λO₂ H₂ id₂ comp₂ p, !functor_eq_mk'))
@ -97,11 +97,11 @@ namespace functor
-- "functor C D" is equivalent to a certain sigma type
set_option unifier.max_steps 38500
protected definition sigma_char :
(Σ (obF : C → D)
(homF : Π ⦃a b : C⦄, hom a b → hom (obF a) (obF b)),
(Π (a : C), homF (ID a) = ID (obF a)) ×
(Σ (to_fun_ob : C → D)
(to_fun_hom : Π ⦃a b : C⦄, hom a b → hom (to_fun_ob a) (to_fun_ob b)),
(Π (a : C), to_fun_hom (ID a) = ID (to_fun_ob a)) ×
(Π {a b c : C} (g : hom b c) (f : hom a b),
homF (g ∘ f) = homF g ∘ homF f)) ≃ (functor C D) :=
to_fun_hom (g ∘ f) = to_fun_hom g ∘ to_fun_hom f)) ≃ (functor C D) :=
begin
fapply equiv.MK,
{intro S, fapply functor.mk,
@ -120,7 +120,7 @@ namespace functor
apply idp},
end
protected definition strict_cat_has_functor_hset
protected definition is_hset_functor
[HD : is_hset D] : is_hset (functor C D) :=
begin
apply is_trunc_is_equiv_closed, apply equiv.to_is_equiv,
@ -143,21 +143,21 @@ namespace category
open functor
--TODO: make this a structure
definition precat_of_strict_precats : precategory (Σ (C : Precategory), is_hset C) :=
definition precat_strict_precat : precategory (Σ (C : Precategory), is_hset C) :=
precategory.mk (λ a b, functor a.1 b.1)
(λ a b, @functor.strict_cat_has_functor_hset a.1 b.1 b.2)
(λ a b, @functor.is_hset_functor a.1 b.1 b.2)
(λ a b c g f, functor.compose g f)
(λ a, functor.id)
(λ a b c d h g f, !functor.assoc)
(λ a b f, !functor.id_left)
(λ a b f, !functor.id_right)
definition Precat_of_strict_cats := precategory.Mk precat_of_strict_precats
definition Precat_of_strict_cats := precategory.Mk precat_strict_precat
namespace ops
abbreviation SPreCat := Precat_of_strict_cats
--attribute precat_of_strict_precats [instance]
--attribute precat_strict_precat [instance]
end ops

14
hott/algebra/precategory/iso.hlean
View file

@ -2,7 +2,7 @@
Copyright (c) 2014 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: algebra.precategory.iso
Module: algebra.precategory.to_fun
Authors: Floris van Doorn, Jakob von Raumer
-/
@ -32,7 +32,7 @@ namespace morphism
-- The structure for isomorphism can be characterized up to equivalence
-- by a sigma type.
definition sigma_is_iso_equiv ⦃a b : ob⦄ : (Σ (f : hom a b), is_iso f) ≃ (a ≅ b) :=
definition isomorphic.sigma_char ⦃a b : ob⦄ : (Σ (f : hom a b), is_iso f) ≃ (a ≅ b) :=
begin
fapply (equiv.mk),
{intro S, apply isomorphic.mk, apply (S.2)},
@ -45,7 +45,7 @@ namespace morphism
set_option apply.class_instance false -- disable class instance resolution in the apply tactic
-- The statement "f is an isomorphism" is a mere proposition
definition is_hprop_of_is_iso : is_hset (is_iso f) :=
definition is_hset_iso : is_hset (is_iso f) :=
begin
apply is_trunc_is_equiv_closed,
apply (equiv.to_is_equiv (!sigma_char)),
@ -56,17 +56,17 @@ namespace morphism
end
-- The type of isomorphisms between two objects is a set
definition is_hset_iso : is_hset (a ≅ b) :=
definition is_hset_isomorphic : is_hset (a ≅ b) :=
begin
apply is_trunc_is_equiv_closed,
apply (equiv.to_is_equiv (!sigma_is_iso_equiv)),
apply (equiv.to_is_equiv (!isomorphic.sigma_char)),
apply is_trunc_sigma,
apply homH,
{intro f, apply is_hprop_of_is_iso},
{intro f, apply is_hset_iso},
end
-- In a precategory, equal objects are isomorphic
definition iso_of_path (p : a = b) : a ≅ b :=
definition iso_of_eq (p : a = b) : a ≅ b :=
eq.rec_on p (isomorphic.mk id)
end morphism

230
hott/algebra/precategory/morphism.hlean
View file

@ -11,26 +11,26 @@ import algebra.precategory.basic
open eq category sigma sigma.ops equiv is_equiv is_trunc
namespace morphism
structure is_section [class] {ob : Type} [C : precategory ob] {a b : ob} (f : a ⟶ b) :=
structure split_mono [class] {ob : Type} [C : precategory ob] {a b : ob} (f : a ⟶ b) :=
{retraction_of : b ⟶ a}
(retraction_compose : retraction_of ∘ f = id)
structure is_retraction [class] {ob : Type} [C : precategory ob] {a b : ob} (f : a ⟶ b) :=
(retraction_comp : retraction_of ∘ f = id)
structure split_epi [class] {ob : Type} [C : precategory ob] {a b : ob} (f : a ⟶ b) :=
{section_of : b ⟶ a}
(compose_section : f ∘ section_of = id)
(comp_section : f ∘ section_of = id)
structure is_iso [class] {ob : Type} [C : precategory ob] {a b : ob} (f : a ⟶ b) :=
{inverse : b ⟶ a}
(inverse_compose : inverse ∘ f = id)
(compose_inverse : f ∘ inverse = id)
(inverse_comp : inverse ∘ f = id)
(comp_inverse : f ∘ inverse = id)
attribute is_iso [multiple-instances]
open is_section is_retraction is_iso
definition retraction_of [reducible] := @is_section.retraction_of
definition retraction_compose [reducible] := @is_section.retraction_compose
definition section_of [reducible] := @is_retraction.section_of
definition compose_section [reducible] := @is_retraction.compose_section
open split_mono split_epi is_iso
definition retraction_of [reducible] := @split_mono.retraction_of
definition retraction_comp [reducible] := @split_mono.retraction_comp
definition section_of [reducible] := @split_epi.section_of
definition comp_section [reducible] := @split_epi.comp_section
definition inverse [reducible] := @is_iso.inverse
definition inverse_compose [reducible] := @is_iso.inverse_compose
definition compose_inverse [reducible] := @is_iso.compose_inverse
definition inverse_comp [reducible] := @is_iso.inverse_comp
definition comp_inverse [reducible] := @is_iso.comp_inverse
postfix `⁻¹` := inverse
--a second notation for the inverse, which is not overloaded
postfix [parsing-only] `⁻¹ʰ`:std.prec.max_plus := inverse
@ -40,77 +40,77 @@ namespace morphism
include C
definition iso_imp_retraction [instance] [priority 300] [reducible]
(f : a ⟶ b) [H : is_iso f] : is_section f :=
is_section.mk !inverse_compose
(f : a ⟶ b) [H : is_iso f] : split_mono f :=
split_mono.mk !inverse_comp
definition iso_imp_section [instance] [priority 300] [reducible]
(f : a ⟶ b) [H : is_iso f] : is_retraction f :=
is_retraction.mk !compose_inverse
(f : a ⟶ b) [H : is_iso f] : split_epi f :=
split_epi.mk !comp_inverse
definition is_iso_id [instance] [priority 500] (a : ob) : is_iso (ID a) :=
is_iso.mk !id_compose !id_compose
is_iso.mk !id_comp !id_comp
definition is_iso_inverse [instance] [priority 200] (f : a ⟶ b) [H : is_iso f] : is_iso f⁻¹ :=
is_iso.mk !compose_inverse !inverse_compose
is_iso.mk !comp_inverse !inverse_comp
definition left_inverse_eq_right_inverse {f : a ⟶ b} {g g' : hom b a}
(Hl : g ∘ f = id) (Hr : f ∘ g' = id) : g = g' :=
by rewrite [-(id_right g), -Hr, assoc, Hl, id_left]
definition retraction_eq_intro [H : is_section f] (H2 : f ∘ h = id) : retraction_of f = h :=
left_inverse_eq_right_inverse !retraction_compose H2
definition retraction_eq [H : split_mono f] (H2 : f ∘ h = id) : retraction_of f = h :=
left_inverse_eq_right_inverse !retraction_comp H2
definition section_eq_intro [H : is_retraction f] (H2 : h ∘ f = id) : section_of f = h :=
(left_inverse_eq_right_inverse H2 !compose_section)⁻¹
definition section_eq [H : split_epi f] (H2 : h ∘ f = id) : section_of f = h :=
(left_inverse_eq_right_inverse H2 !comp_section)⁻¹
definition inverse_eq_intro_right [H : is_iso f] (H2 : f ∘ h = id) : f⁻¹ = h :=
left_inverse_eq_right_inverse !inverse_compose H2
definition inverse_eq_right [H : is_iso f] (H2 : f ∘ h = id) : f⁻¹ = h :=
left_inverse_eq_right_inverse !inverse_comp H2
definition inverse_eq_intro_left [H : is_iso f] (H2 : h ∘ f = id) : f⁻¹ = h :=
(left_inverse_eq_right_inverse H2 !compose_inverse)⁻¹
definition inverse_eq_left [H : is_iso f] (H2 : h ∘ f = id) : f⁻¹ = h :=
(left_inverse_eq_right_inverse H2 !comp_inverse)⁻¹
definition section_eq_retraction (f : a ⟶ b) [Hl : is_section f] [Hr : is_retraction f] :
definition retraction_eq_section (f : a ⟶ b) [Hl : split_mono f] [Hr : split_epi f] :
retraction_of f = section_of f :=
retraction_eq_intro !compose_section
retraction_eq !comp_section
definition section_retraction_imp_iso (f : a ⟶ b) [Hl : is_section f] [Hr : is_retraction f]
definition iso_of_split_epi_of_split_mono (f : a ⟶ b) [Hl : split_mono f] [Hr : split_epi f]
: is_iso f :=
is_iso.mk ((section_eq_retraction f) ▹ (retraction_compose f)) (compose_section f)
is_iso.mk ((retraction_eq_section f) ▹ (retraction_comp f)) (comp_section f)
definition inverse_unique (H H' : is_iso f) : @inverse _ _ _ _ f H = @inverse _ _ _ _ f H' :=
inverse_eq_intro_left !inverse_compose
inverse_eq_left !inverse_comp
definition inverse_involutive (f : a ⟶ b) [H : is_iso f] : (f⁻¹)⁻¹ = f :=
inverse_eq_intro_right !inverse_compose
inverse_eq_right !inverse_comp
definition retraction_of_id (a : ob) : retraction_of (ID a) = id :=
retraction_eq_intro !id_compose
definition retraction_id (a : ob) : retraction_of (ID a) = id :=
retraction_eq !id_comp
definition section_of_id (a : ob) : section_of (ID a) = id :=
section_eq_intro !id_compose
definition section_id (a : ob) : section_of (ID a) = id :=
section_eq !id_comp
definition iso_of_id (a : ob) [H : is_iso (ID a)] : (ID a)⁻¹ = id :=
inverse_eq_intro_left !id_compose
definition id_inverse (a : ob) [H : is_iso (ID a)] : (ID a)⁻¹ = id :=
inverse_eq_left !id_comp
definition composition_is_section [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b)
[Hf : is_section f] [Hg : is_section g] : is_section (g ∘ f) :=
is_section.mk
definition split_mono_comp [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b)
[Hf : split_mono f] [Hg : split_mono g] : split_mono (g ∘ f) :=
split_mono.mk
(show (retraction_of f ∘ retraction_of g) ∘ g ∘ f = id,
by rewrite [-assoc, assoc _ g f, retraction_compose, id_left, retraction_compose])
by rewrite [-assoc, assoc _ g f, retraction_comp, id_left, retraction_comp])
definition composition_is_retraction [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b)
[Hf : is_retraction f] [Hg : is_retraction g] : is_retraction (g ∘ f) :=
is_retraction.mk
definition split_epi_comp [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b)
[Hf : split_epi f] [Hg : split_epi g] : split_epi (g ∘ f) :=
split_epi.mk
(show (g ∘ f) ∘ section_of f ∘ section_of g = id,
by rewrite [-assoc, {f ∘ _}assoc, compose_section, id_left, compose_section])
by rewrite [-assoc, {f ∘ _}assoc, comp_section, id_left, comp_section])
definition composition_is_inverse [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b)
definition iso_comp [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b)
[Hf : is_iso f] [Hg : is_iso g] : is_iso (g ∘ f) :=
!section_retraction_imp_iso
!iso_of_split_epi_of_split_mono
structure isomorphic (a b : ob) :=
(iso : hom a b)
[struct : is_iso iso]
(to_fun : hom a b)
[struct : is_iso to_fun]
infix `≅`:50 := morphism.isomorphic
attribute isomorphic.struct [instance] [priority 400]
@ -122,57 +122,57 @@ namespace morphism
mk (ID a)
protected definition symm ⦃a b : ob⦄ (H : a ≅ b) : b ≅ a :=
mk (iso H)⁻¹
mk (to_fun H)⁻¹
protected definition trans ⦃a b c : ob⦄ (H1 : a ≅ b) (H2 : b ≅ c) : a ≅ c :=
mk (iso H2 ∘ iso H1)
mk (to_fun H2 ∘ to_fun H1)
end isomorphic
structure is_mono [class] (f : a ⟶ b) :=
structure mono [class] (f : a ⟶ b) :=
(elim : ∀c (g h : hom c a), f ∘ g = f ∘ h → g = h)
structure is_epi [class] (f : a ⟶ b) :=
structure epi [class] (f : a ⟶ b) :=
(elim : ∀c (g h : hom b c), g ∘ f = h ∘ f → g = h)
definition is_mono_of_is_section [instance] (f : a ⟶ b) [H : is_section f] : is_mono f :=
is_mono.mk
definition mono_of_split_mono [instance] (f : a ⟶ b) [H : split_mono f] : mono f :=
mono.mk
(λ c g h H,
calc
g = id ∘ g : by rewrite id_left
... = (retraction_of f ∘ f) ∘ g : by rewrite -retraction_compose
... = (retraction_of f ∘ f) ∘ g : by rewrite -retraction_comp
... = (retraction_of f ∘ f) ∘ h : by rewrite [-assoc, H, -assoc]
... = id ∘ h : by rewrite retraction_compose
... = id ∘ h : by rewrite retraction_comp
... = h : by rewrite id_left)
definition is_epi_of_is_retraction [instance] (f : a ⟶ b) [H : is_retraction f] : is_epi f :=
is_epi.mk
definition epi_of_split_epi [instance] (f : a ⟶ b) [H : split_epi f] : epi f :=
epi.mk
(λ c g h H,
calc
g = g ∘ id : by rewrite id_right
... = g ∘ f ∘ section_of f : by rewrite -compose_section
... = g ∘ f ∘ section_of f : by rewrite -comp_section
... = h ∘ f ∘ section_of f : by rewrite [assoc, H, -assoc]
... = h ∘ id : by rewrite compose_section
... = h ∘ id : by rewrite comp_section
... = h : by rewrite id_right)
definition is_mono_comp [instance] (g : b ⟶ c) (f : a ⟶ b) [Hf : is_mono f] [Hg : is_mono g]
: is_mono (g ∘ f) :=
is_mono.mk
definition mono_comp [instance] (g : b ⟶ c) (f : a ⟶ b) [Hf : mono f] [Hg : mono g]
: mono (g ∘ f) :=
mono.mk
(λ d h₁ h₂ H,
have H2 : g ∘ (f ∘ h₁) = g ∘ (f ∘ h₂),
begin
rewrite *assoc, exact H
end,
!is_mono.elim (!is_mono.elim H2))
!mono.elim (!mono.elim H2))
definition is_epi_comp [instance] (g : b ⟶ c) (f : a ⟶ b) [Hf : is_epi f] [Hg : is_epi g]
: is_epi (g ∘ f) :=
is_epi.mk
definition epi_comp [instance] (g : b ⟶ c) (f : a ⟶ b) [Hf : epi f] [Hg : epi g]
: epi (g ∘ f) :=
epi.mk
(λ d h₁ h₂ H,
have H2 : (h₁ ∘ g) ∘ f = (h₂ ∘ g) ∘ f,
begin
rewrite -*assoc, exact H
end,
!is_epi.elim (!is_epi.elim H2))
!epi.elim (!epi.elim H2))
end morphism
@ -181,37 +181,37 @@ namespace morphism
rewrite lemmas for inverses, modified from
https://github.com/JasonGross/HoTT-categories/blob/master/theories/Categories/Category/Morphisms.v
-/
namespace iso
namespace to_fun
section
variables {ob : Type} [C : precategory ob] include C
variables {a b c d : ob} (f : b ⟶ a)
(r : c ⟶ d) (q : b ⟶ c) (p : a ⟶ b)
(g : d ⟶ c)
variable [Hq : is_iso q] include Hq
definition compose_pV : q ∘ q⁻¹ = id := !compose_inverse
definition compose_Vp : q⁻¹ ∘ q = id := !inverse_compose
definition compose_V_pp : q⁻¹ ∘ (q ∘ p) = p :=
by rewrite [assoc, inverse_compose, id_left]
definition compose_p_Vp : q ∘ (q⁻¹ ∘ g) = g :=
by rewrite [assoc, compose_inverse, id_left]
definition compose_pp_V : (r ∘ q) ∘ q⁻¹ = r :=
by rewrite [-assoc, compose_inverse, id_right]
definition compose_pV_p : (f ∘ q⁻¹) ∘ q = f :=
by rewrite [-assoc, inverse_compose, id_right]
definition comp.right_inverse : q ∘ q⁻¹ = id := !comp_inverse
definition comp.left_inverse : q⁻¹ ∘ q = id := !inverse_comp
definition inverse_comp_cancel_left : q⁻¹ ∘ (q ∘ p) = p :=
by rewrite [assoc, inverse_comp, id_left]
definition comp_inverse_cancel_left : q ∘ (q⁻¹ ∘ g) = g :=
by rewrite [assoc, comp_inverse, id_left]
definition comp_inverse_cancel_right : (r ∘ q) ∘ q⁻¹ = r :=
by rewrite [-assoc, comp_inverse, id_right]
definition inverse_comp_cancel_right : (f ∘ q⁻¹) ∘ q = f :=
by rewrite [-assoc, inverse_comp, id_right]
definition con_inv [Hp : is_iso p] [Hpq : is_iso (q ∘ p)] : (q ∘ p)⁻¹ʰ = p⁻¹ʰ ∘ q⁻¹ʰ :=
inverse_eq_intro_left
definition comp_inverse [Hp : is_iso p] [Hpq : is_iso (q ∘ p)] : (q ∘ p)⁻¹ʰ = p⁻¹ʰ ∘ q⁻¹ʰ :=
inverse_eq_left
(show (p⁻¹ʰ ∘ q⁻¹ʰ) ∘ q ∘ p = id, from
by rewrite [-assoc, compose_V_pp, inverse_compose])
by rewrite [-assoc, inverse_comp_cancel_left, inverse_comp])
definition inv_con_inv_left [H' : is_iso g] : (q⁻¹ ∘ g)⁻¹ = g⁻¹ ∘ q :=
inverse_involutive q ▹ con_inv q⁻¹ g
definition inverse_comp_inverse_left [H' : is_iso g] : (q⁻¹ ∘ g)⁻¹ = g⁻¹ ∘ q :=
inverse_involutive q ▹ comp_inverse q⁻¹ g
definition inv_con_inv_right [H' : is_iso f] : (q ∘ f⁻¹)⁻¹ = f ∘ q⁻¹ :=
inverse_involutive f ▹ con_inv q f⁻¹
definition inverse_comp_inverse_right [H' : is_iso f] : (q ∘ f⁻¹)⁻¹ = f ∘ q⁻¹ :=
inverse_involutive f ▹ comp_inverse q f⁻¹
definition inv_con_inv_inv [H' : is_iso r] : (q⁻¹ ∘ r⁻¹)⁻¹ = r ∘ q :=
inverse_involutive r ▹ inv_con_inv_left q r⁻¹
definition inverse_comp_inverse_inverse [H' : is_iso r] : (q⁻¹ ∘ r⁻¹)⁻¹ = r ∘ q :=
inverse_involutive r ▹ inverse_comp_inverse_left q r⁻¹
end
section
@ -224,25 +224,35 @@ namespace morphism
{y : d ⟶ b} {w : c ⟶ a}
variable [Hq : is_iso q] include Hq
definition con_eq_of_eq_inv_con (H : y = q⁻¹ ∘ g) : q ∘ y = g := H⁻¹ ▹ compose_p_Vp q g
definition con_eq_of_eq_con_inv (H : w = f ∘ q⁻¹) : w ∘ q = f := H⁻¹ ▹ compose_pV_p f q
definition inv_con_eq_of_eq_con (H : z = q ∘ p) : q⁻¹ ∘ z = p := H⁻¹ ▹ compose_V_pp q p
definition con_inv_eq_of_eq_con (H : x = r ∘ q) : x ∘ q⁻¹ = r := H⁻¹ ▹ compose_pp_V r q
definition eq_con_of_inv_con_eq (H : q⁻¹ ∘ g = y) : g = q ∘ y := (con_eq_of_eq_inv_con H⁻¹)⁻¹
definition eq_con_of_con_inv_eq (H : f ∘ q⁻¹ = w) : f = w ∘ q := (con_eq_of_eq_con_inv H⁻¹)⁻¹
definition eq_inv_con_of_con_eq (H : q ∘ p = z) : p = q⁻¹ ∘ z := (inv_con_eq_of_eq_con H⁻¹)⁻¹
definition eq_con_inv_of_con_eq (H : r ∘ q = x) : r = x ∘ q⁻¹ := (con_inv_eq_of_eq_con H⁻¹)⁻¹
definition eq_inv_of_con_eq_idp' (H : h ∘ q = id) : h = q⁻¹ := (inverse_eq_intro_left H)⁻¹
definition eq_inv_of_con_eq_idp (H : q ∘ h = id) : h = q⁻¹ := (inverse_eq_intro_right H)⁻¹
definition eq_of_con_inv_eq_idp (H : i ∘ q⁻¹ = id) : i = q :=
eq_inv_of_con_eq_idp' H ⬝ inverse_involutive q
definition eq_of_inv_con_eq_idp (H : q⁻¹ ∘ i = id) : i = q :=
eq_inv_of_con_eq_idp H ⬝ inverse_involutive q
definition eq_of_idp_eq_con_inv (H : id = i ∘ q⁻¹) : q = i := (eq_of_con_inv_eq_idp H⁻¹)⁻¹
definition eq_of_idp_eq_inv_con (H : id = q⁻¹ ∘ i) : q = i := (eq_of_inv_con_eq_idp H⁻¹)⁻¹
definition inv_eq_of_idp_eq_con (H : id = h ∘ q) : q⁻¹ = h := (eq_inv_of_con_eq_idp' H⁻¹)⁻¹
definition inv_eq_of_idp_eq_con' (H : id = q ∘ h) : q⁻¹ = h := (eq_inv_of_con_eq_idp H⁻¹)⁻¹
definition comp_eq_of_eq_inverse_comp (H : y = q⁻¹ ∘ g) : q ∘ y = g :=
H⁻¹ ▹ comp_inverse_cancel_left q g
definition comp_eq_of_eq_comp_inverse (H : w = f ∘ q⁻¹) : w ∘ q = f :=
H⁻¹ ▹ inverse_comp_cancel_right f q
definition inverse_comp_eq_of_eq_comp (H : z = q ∘ p) : q⁻¹ ∘ z = p :=
H⁻¹ ▹ inverse_comp_cancel_left q p
definition comp_inverse_eq_of_eq_comp (H : x = r ∘ q) : x ∘ q⁻¹ = r :=
H⁻¹ ▹ comp_inverse_cancel_right r q
definition eq_comp_of_inverse_comp_eq (H : q⁻¹ ∘ g = y) : g = q ∘ y :=
(comp_eq_of_eq_inverse_comp H⁻¹)⁻¹
definition eq_comp_of_comp_inverse_eq (H : f ∘ q⁻¹ = w) : f = w ∘ q :=
(comp_eq_of_eq_comp_inverse H⁻¹)⁻¹
definition eq_inverse_comp_of_comp_eq (H : q ∘ p = z) : p = q⁻¹ ∘ z :=
(inverse_comp_eq_of_eq_comp H⁻¹)⁻¹
definition eq_comp_inverse_of_comp_eq (H : r ∘ q = x) : r = x ∘ q⁻¹ :=
(comp_inverse_eq_of_eq_comp H⁻¹)⁻¹
definition eq_inverse_of_comp_eq_id' (H : h ∘ q = id) : h = q⁻¹ := (inverse_eq_left H)⁻¹
definition eq_inverse_of_comp_eq_id (H : q ∘ h = id) : h = q⁻¹ := (inverse_eq_right H)⁻¹
definition eq_of_comp_inverse_eq_id (H : i ∘ q⁻¹ = id) : i = q :=
eq_inverse_of_comp_eq_id' H ⬝ inverse_involutive q
definition eq_of_inverse_comp_eq_id (H : q⁻¹ ∘ i = id) : i = q :=
eq_inverse_of_comp_eq_id H ⬝ inverse_involutive q
definition eq_of_id_eq_comp_inverse (H : id = i ∘ q⁻¹) : q = i := (eq_of_comp_inverse_eq_id H⁻¹)⁻¹
definition eq_of_id_eq_inverse_comp (H : id = q⁻¹ ∘ i) : q = i := (eq_of_inverse_comp_eq_id H⁻¹)⁻¹
definition inverse_eq_of_id_eq_comp (H : id = h ∘ q) : q⁻¹ = h :=
(eq_inverse_of_comp_eq_id' H⁻¹)⁻¹
definition inverse_eq_of_id_eq_comp' (H : id = q ∘ h) : q⁻¹ = h :=
(eq_inverse_of_comp_eq_id H⁻¹)⁻¹
end
end iso
end to_fun
end morphism

64
hott/algebra/precategory/yoneda.hlean
View file

@ -26,7 +26,7 @@ namespace yoneda
... = _ : assoc
--disturbing behaviour: giving the type of f "(x ⟶ y)" explicitly makes the unifier loop
definition representable_functor (C : Precategory) : Cᵒᵖ ×c C ⇒ set :=
definition hom_functor (C : Precategory) : Cᵒᵖ ×c C ⇒ set :=
functor.mk (λ(x : Cᵒᵖ ×c C), homset x.1 x.2)
(λ(x y : Cᵒᵖ ×c C) (f : _) (h : homset x.1 x.2), f.2 ∘⁅ C ⁆ (h ∘⁅ C ⁆ f.1))
proof (λ(x : Cᵒᵖ ×c C), eq_of_homotopy (λ(h : homset x.1 x.2), !id_left ⬝ !id_right)) qed
@ -48,7 +48,7 @@ namespace functor
(λd d' g, F (id, g))
(λd, !respect_id)
(λd₁ d₂ d₃ g' g, proof calc
F (id, g' ∘ g) = F (id ∘ id, g' ∘ g) : {(id_compose c)⁻¹}
F (id, g' ∘ g) = F (id ∘ id, g' ∘ g) : {(id_comp c)⁻¹}
... = F ((id,g') ∘ (id, g)) : idp
... = F (id,g') ∘ F (id, g) : respect_comp F qed)
local abbreviation Fob := @functor_curry_ob
@ -66,7 +66,7 @@ namespace functor
local abbreviation Fhom := @functor_curry_hom
definition functor_curry_hom_def ⦃c c' : C⦄ (f : c ⟶ c') (d : D) :
(Fhom F f) d = homF F (f, id) := idp
(Fhom F f) d = to_fun_hom F (f, id) := idp
definition functor_curry_id (c : C) : Fhom F (ID c) = nat_trans.id :=
nat_trans_eq_mk (λd, respect_id F _)
@ -75,7 +75,7 @@ namespace functor
: Fhom F (f' ∘ f) = Fhom F f' ∘n Fhom F f :=
nat_trans_eq_mk (λd, calc
natural_map (Fhom F (f' ∘ f)) d = F (f' ∘ f, id) : functor_curry_hom_def
... = F (f' ∘ f, id ∘ id) : {(id_compose d)⁻¹}
... = F (f' ∘ f, id ∘ id) : {(id_comp d)⁻¹}
... = F ((f',id) ∘ (f, id)) : idp
... = F (f',id) ∘ F (f, id) : respect_comp F
... = natural_map ((Fhom F f') ∘ (Fhom F f)) d : idp)
@ -87,35 +87,35 @@ namespace functor
(functor_curry_comp F)
definition functor_uncurry_ob [reducible] (p : C ×c D) : E :=
obF (G p.1) p.2
to_fun_ob (G p.1) p.2
local abbreviation Gob := @functor_uncurry_ob
definition functor_uncurry_hom ⦃p p' : C ×c D⦄ (f : hom p p') : Gob G p ⟶ Gob G p' :=
homF (obF G p'.1) f.2 ∘ natural_map (homF G f.1) p.2
to_fun_hom (to_fun_ob G p'.1) f.2 ∘ natural_map (to_fun_hom G f.1) p.2
local abbreviation Ghom := @functor_uncurry_hom
definition functor_uncurry_id (p : C ×c D) : Ghom G (ID p) = id :=
calc
Ghom G (ID p) = homF (obF G p.1) id ∘ natural_map (homF G id) p.2 : idp
... = id ∘ natural_map (homF G id) p.2 : ap (λx, x ∘ _) (respect_id (obF G p.1) p.2)
Ghom G (ID p) = to_fun_hom (to_fun_ob G p.1) id ∘ natural_map (to_fun_hom G id) p.2 : idp
... = id ∘ natural_map (to_fun_hom G id) p.2 : ap (λx, x ∘ _) (respect_id (to_fun_ob G p.1) p.2)
... = id ∘ natural_map nat_trans.id p.2 : {respect_id G p.1}
... = id : id_compose
... = id : id_comp
definition functor_uncurry_comp ⦃p p' p'' : C ×c D⦄ (f' : p' ⟶ p'') (f : p ⟶ p')
: Ghom G (f' ∘ f) = Ghom G f' ∘ Ghom G f :=
calc
Ghom G (f' ∘ f)
= homF (obF G p''.1) (f'.2 ∘ f.2) ∘ natural_map (homF G (f'.1 ∘ f.1)) p.2 : idp
... = (homF (obF G p''.1) f'.2 ∘ homF (obF G p''.1) f.2)
∘ natural_map (homF G (f'.1 ∘ f.1)) p.2 : {respect_comp (obF G p''.1) f'.2 f.2}
... = (homF (obF G p''.1) f'.2 ∘ homF (obF G p''.1) f.2)
∘ natural_map (homF G f'.1 ∘ homF G f.1) p.2 : {respect_comp G f'.1 f.1}
... = (homF (obF G p''.1) f'.2 ∘ homF (obF G p''.1) f.2)
∘ (natural_map (homF G f'.1) p.2 ∘ natural_map (homF G f.1) p.2) : idp
... = (homF (obF G p''.1) f'.2 ∘ homF (obF G p''.1) f.2)
∘ (natural_map (homF G f'.1) p.2 ∘ natural_map (homF G f.1) p.2) : idp
... = (homF (obF G p''.1) f'.2 ∘ natural_map (homF G f'.1) p'.2)
∘ (homF (obF G p'.1) f.2 ∘ natural_map (homF G f.1) p.2) :
= to_fun_hom (to_fun_ob G p''.1) (f'.2 ∘ f.2) ∘ natural_map (to_fun_hom G (f'.1 ∘ f.1)) p.2 : idp
... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
∘ natural_map (to_fun_hom G (f'.1 ∘ f.1)) p.2 : {respect_comp (to_fun_ob G p''.1) f'.2 f.2}
... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
∘ natural_map (to_fun_hom G f'.1 ∘ to_fun_hom G f.1) p.2 : {respect_comp G f'.1 f.1}
... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
∘ (natural_map (to_fun_hom G f'.1) p.2 ∘ natural_map (to_fun_hom G f.1) p.2) : idp
... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ to_fun_hom (to_fun_ob G p''.1) f.2)
∘ (natural_map (to_fun_hom G f'.1) p.2 ∘ natural_map (to_fun_hom G f.1) p.2) : idp
... = (to_fun_hom (to_fun_ob G p''.1) f'.2 ∘ natural_map (to_fun_hom G f'.1) p'.2)
∘ (to_fun_hom (to_fun_ob G p'.1) f.2 ∘ natural_map (to_fun_hom G f.1) p.2) :
square_prepostcompose (!naturality⁻¹ᵖ) _ _
... = Ghom G f' ∘ Ghom G f : idp
@ -144,20 +144,20 @@ namespace functor
-- apply idp
-- end))))
-- definition functor_eq_mk1 {F₁ F₂ : C ⇒ D} : Π(p : obF F₁ = obF F₂),
-- definition functor_eq_mk1 {F₁ F₂ : C ⇒ D} : Π(p : to_fun_ob F₁ = to_fun_ob F₂),
-- (Π(a b : C) (f : hom a b), transport (λF, hom (F a) (F b)) p (F₁ f) = F₂ f)
-- → F₁ = F₂ :=
-- functor.rec_on F₁ (λO₁ H₁ id₁ comp₁, functor.rec_on F₂ (λO₂ H₂ id₂ comp₂ p, !functor_eq_mk'1))
--set_option pp.notation false
definition functor_uncurry_functor_curry : functor_uncurry (functor_curry F) = F :=
functor_eq_mk (λp, ap (obF F) !prod.eta)
functor_eq_mk (λp, ap (to_fun_ob F) !prod.eta)
begin
intros (cd, cd', fg),
cases cd with (c,d), cases cd' with (c',d'), cases fg with (f,g),
have H : (functor_uncurry (functor_curry F)) (f, g) = F (f,g),
from calc
(functor_uncurry (functor_curry F)) (f, g) = homF F (id, g) ∘ homF F (f, id) : idp
(functor_uncurry (functor_curry F)) (f, g) = to_fun_hom F (id, g) ∘ to_fun_hom F (f, id) : idp
... = F (id ∘ f, g ∘ id) : respect_comp F (id,g) (f,id)
... = F (f, g ∘ id) : {id_left f}
... = F (f,g) : {id_right g},
@ -172,13 +172,13 @@ namespace functor
fapply functor_eq_mk,
{intro d, apply idp},
{intros (d, d', g),
have H : homF (functor_curry (functor_uncurry G) c) g = homF (G c) g,
have H : to_fun_hom (functor_curry (functor_uncurry G) c) g = to_fun_hom (G c) g,
from calc
homF (functor_curry (functor_uncurry G) c) g
= homF (G c) g ∘ natural_map (homF G (ID c)) d : idp
... = homF (G c) g ∘ natural_map (ID (G c)) d
: ap (λx, homF (G c) g ∘ natural_map x d) (respect_id G c)
... = homF (G c) g : id_right,
to_fun_hom (functor_curry (functor_uncurry G) c) g
= to_fun_hom (G c) g ∘ natural_map (to_fun_hom G (ID c)) d : idp
... = to_fun_hom (G c) g ∘ natural_map (ID (G c)) d
: ap (λx, to_fun_hom (G c) g ∘ natural_map x d) (respect_id G c)
... = to_fun_hom (G c) g : id_right,
rewrite H,
-- esimp {idp},
apply sorry
@ -199,7 +199,7 @@ namespace functor
definition contravariant_yoneda_embedding : Cᵒᵖ ⇒ set ^c C :=
functor_curry !yoneda.representable_functor
functor_curry !yoneda.hom_functor
end functor
@ -209,8 +209,8 @@ end functor
-- --universe levels are given explicitly because Lean uses 6 variables otherwise
-- structure adjoint.{u v} [C D : Precategory.{u v}] (F : C ⇒ D) (G : D ⇒ C) : Type.{max u v} :=
-- (nat_iso : (representable_functor D) ∘f (prod_functor (opposite_functor F) (functor.ID D)) ⟹
-- (representable_functor C) ∘f (prod_functor (functor.ID (Cᵒᵖ)) G))
-- (nat_iso : (hom_functor D) ∘f (prod_functor (opposite_functor F) (functor.ID D)) ⟹
-- (hom_functor C) ∘f (prod_functor (functor.ID (Cᵒᵖ)) G))
-- (is_iso_nat_iso : is_iso nat_iso)
-- infix `⊣`:55 := adjoint

6
hott/init/equiv.hlean
View file

@ -109,7 +109,7 @@ namespace is_equiv
have eq1 : ap f (sec a) = _,
from calc ap f (sec a)
= idp ⬝ ap f (sec a) : !idp_con⁻¹
... = (ret (f a) ⬝ (ret (f a))⁻¹) ⬝ ap f (sec a) : {!con.left_inv⁻¹}
... = (ret (f a) ⬝ (ret (f a))⁻¹) ⬝ ap f (sec a) : {!con.right_inv⁻¹}
... = ((ret fgfa)⁻¹ ⬝ ap (f ∘ g) (ret (f a))) ⬝ ap f (sec a) : {!con_ap_eq_con⁻¹}
... = ((ret fgfa)⁻¹ ⬝ fgretrfa) ⬝ ap f (sec a) : {ap_compose g f _}
... = (ret fgfa)⁻¹ ⬝ (fgretrfa ⬝ ap f (sec a)) : !con.assoc,
@ -180,11 +180,11 @@ namespace is_equiv
◾ (inverse (ap_compose f⁻¹ f _))
◾ (adj f _)⁻¹)
⬝ con_ap_con_eq_con_con (retr f) _ _
⬝ whisker_right !con.right_inv _
⬝ whisker_right !con.left_inv _
⬝ !idp_con)
(λp, whisker_right (whisker_left _ (ap_compose f f⁻¹ _)⁻¹) _
⬝ con_ap_con_eq_con_con (sect f) _ _
⬝ whisker_right !con.right_inv _
⬝ whisker_right !con.left_inv _
⬝ !idp_con)
-- The function equiv_rect says that given an equivalence f : A → B,

8
hott/init/path.hlean
View file

@ -66,11 +66,11 @@ namespace eq
eq.rec_on r (eq.rec_on q idp)
-- The left inverse law.
definition con.left_inv (p : x = y) : p ⬝ p⁻¹ = idp :=
definition con.right_inv (p : x = y) : p ⬝ p⁻¹ = idp :=
eq.rec_on p idp
-- The right inverse law.
definition con.right_inv (p : x = y) : p⁻¹ ⬝ p = idp :=
definition con.left_inv (p : x = y) : p⁻¹ ⬝ p = idp :=
eq.rec_on p idp
/- Several auxiliary theorems about canceling inverses across associativity. These are somewhat
@ -408,11 +408,11 @@ namespace eq
definition tr_inv_tr (P : A → Type) {x y : A} (p : x = y) (z : P y) :
p ▹ p⁻¹ ▹ z = z :=
(tr_con P p⁻¹ p z)⁻¹ ⬝ ap (λr, transport P r z) (con.right_inv p)
(tr_con P p⁻¹ p z)⁻¹ ⬝ ap (λr, transport P r z) (con.left_inv p)
definition inv_tr_tr (P : A → Type) {x y : A} (p : x = y) (z : P x) :
p⁻¹ ▹ p ▹ z = z :=
(tr_con P p p⁻¹ z)⁻¹ ⬝ ap (λr, transport P r z) (con.left_inv p)
(tr_con P p p⁻¹ z)⁻¹ ⬝ ap (λr, transport P r z) (con.right_inv p)
definition tr_con_lemma (P : A → Type)
{x y z w : A} (p : x = y) (q : y = z) (r : z = w) (u : P x) :

2
hott/init/trunc.hlean
View file

@ -105,7 +105,7 @@ namespace is_trunc
(contr x)⁻¹ ⬝ (contr y)
definition hprop_eq {A : Type} [H : is_contr A] {x y : A} (p q : x = y) : p = q :=
have K : ∀ (r : x = y), center_eq x y = r, from (λ r, eq.rec_on r !con.right_inv),
have K : ∀ (r : x = y), center_eq x y = r, from (λ r, eq.rec_on r !con.left_inv),
(K p)⁻¹ ⬝ K q
definition is_contr_eq [instance] {A : Type} [H : is_contr A] (x y : A) : is_contr (x = y)

2
script/rename.pl
View file

@ -65,7 +65,7 @@ sub rename_in_file {
foreach my $key (keys %renamings) {
# replace instances of key, not preceeded by a letter, and not
# followed by a letter, number, ', or .
s/(?<![a-zA-z])$key(?![w'.])/$renamings{$key}/g;
s/(?<![a-zA-z])$key(?![\w'])/$renamings{$key}/g;
}
print;
}