feat(category): split category.lean in different files; add more constructions and theorems about isos
This commit is contained in:
Floris van Doorn
Leonardo de Moura
parent
ae3419f82f
commit
8d376b93cd
7 changed files with 827 additions and 554 deletions
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-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Floris van Doorn
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-- category
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import logic.eq logic.connectives
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import data.unit data.sigma data.prod
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import algebra.function
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import logic.axioms.funext
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open eq eq.ops
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inductive category [class] (ob : Type) : Type :=
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mk : Π (mor : ob → ob → Type) (comp : Π⦃A B C : ob⦄, mor B C → mor A B → mor A C)
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(id : Π {A : ob}, mor A A),
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(Π ⦃A B C D : ob⦄ {h : mor C D} {g : mor B C} {f : mor A B},
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comp h (comp g f) = comp (comp h g) f) →
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(Π ⦃A B : ob⦄ {f : mor A B}, comp id f = f) →
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(Π ⦃A B : ob⦄ {f : mor A B}, comp f id = f) →
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category ob
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inductive Category : Type :=
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mk : Π (A : Type), category A → Category
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namespace category
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section
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parameters {ob : Type} {Cat : category ob} {A B C D : ob}
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definition mor : ob → ob → Type := rec (λ mor compose id assoc idr idl, mor) Cat
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definition compose : Π {A B C : ob}, mor B C → mor A B → mor A C :=
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rec (λ mor compose id assoc idr idl, compose) Cat
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definition id : Π {A : ob}, mor A A :=
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rec (λ mor compose id assoc idr idl, id) Cat
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definition ID (A : ob) : mor A A := @id A
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precedence `∘` : 60
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infixr `∘` := compose
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infixl `=>`:25 := mor
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theorem assoc : Π {A B C D : ob} {h : mor C D} {g : mor B C} {f : mor A B},
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h ∘ (g ∘ f) = (h ∘ g) ∘ f :=
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rec (λ mor comp id assoc idr idl, assoc) Cat
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theorem id_left : Π {A B : ob} {f : mor A B}, id ∘ f = f :=
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rec (λ mor comp id assoc idl idr, idl) Cat
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theorem id_right : Π {A B : ob} {f : mor A B}, f ∘ id = f :=
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rec (λ mor comp id assoc idl idr, idr) Cat
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theorem id_compose {A : ob} : (ID A) ∘ id = id :=
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id_left
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theorem left_id_unique (i : mor A A) (H : Π{B} {f : mor B A}, i ∘ f = f) : i = id :=
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calc
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i = i ∘ id : symm id_right
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... = id : H
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theorem right_id_unique (i : mor A A) (H : Π{B} {f : mor A B}, f ∘ i = f) : i = id :=
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calc
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i = id ∘ i : eq.symm id_left
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... = id : H
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inductive is_section [class] {A B : ob} (f : mor A B) : Type :=
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mk : ∀{g}, g ∘ f = id → is_section f
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inductive is_retraction [class] {A B : ob} (f : mor A B) : Type :=
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mk : ∀{g}, f ∘ g = id → is_retraction f
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inductive is_iso [class] {A B : ob} (f : mor A B) : Type :=
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mk : ∀{g}, g ∘ f = id → f ∘ g = id → is_iso f
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definition retraction_of {A B : ob} (f : mor A B) {H : is_section f} : mor B A :=
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is_section.rec (λg h, g) H
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definition section_of {A B : ob} (f : mor A B) {H : is_retraction f} : mor B A :=
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is_retraction.rec (λg h, g) H
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definition inverse {A B : ob} (f : mor A B) {H : is_iso f} : mor B A :=
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is_iso.rec (λg h1 h2, g) H
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postfix `⁻¹` := inverse
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theorem id_is_iso [instance] : is_iso (ID A) :=
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is_iso.mk id_compose id_compose
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theorem inverse_compose {A B : ob} {f : mor A B} {H : is_iso f} : f⁻¹ ∘ f = id :=
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is_iso.rec (λg h1 h2, h1) H
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theorem compose_inverse {A B : ob} {f : mor A B} {H : is_iso f} : f ∘ f⁻¹ = id :=
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is_iso.rec (λg h1 h2, h2) H
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theorem iso_imp_retraction [instance] {A B : ob} (f : mor A B) {H : is_iso f} : is_section f :=
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is_section.mk inverse_compose
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theorem iso_imp_section [instance] {A B : ob} (f : mor A B) {H : is_iso f} : is_retraction f :=
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is_retraction.mk compose_inverse
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theorem retraction_compose {A B : ob} {f : mor A B} {H : is_section f} :
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retraction_of f ∘ f = id :=
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is_section.rec (λg h, h) H
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theorem compose_section {A B : ob} {f : mor A B} {H : is_retraction f} :
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f ∘ section_of f = id :=
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is_retraction.rec (λg h, h) H
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theorem left_inverse_eq_right_inverse {A B : ob} {f : mor A B} {g g' : mor B A}
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(Hl : g ∘ f = id) (Hr : f ∘ g' = id) : g = g' :=
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calc
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g = g ∘ id : symm id_right
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... = g ∘ f ∘ g' : {symm Hr}
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... = (g ∘ f) ∘ g' : assoc
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... = id ∘ g' : {Hl}
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... = g' : id_left
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theorem section_eq_retraction {A B : ob} {f : mor A B}
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(Hl : is_section f) (Hr : is_retraction f) : retraction_of f = section_of f :=
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left_inverse_eq_right_inverse retraction_compose compose_section
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theorem section_retraction_imp_iso {A B : ob} {f : mor A B}
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(Hl : is_section f) (Hr : is_retraction f) : is_iso f :=
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is_iso.mk (subst (section_eq_retraction Hl Hr) retraction_compose) compose_section
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theorem inverse_unique {A B : ob} {f : mor A B} (H H' : is_iso f)
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: @inverse _ _ f H = @inverse _ _ f H' :=
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left_inverse_eq_right_inverse inverse_compose compose_inverse
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theorem retraction_of_id {A : ob} : retraction_of (ID A) = id :=
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left_inverse_eq_right_inverse retraction_compose id_compose
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theorem section_of_id {A : ob} : section_of (ID A) = id :=
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symm (left_inverse_eq_right_inverse id_compose compose_section)
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theorem iso_of_id {A : ob} : ID A⁻¹ = id :=
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left_inverse_eq_right_inverse inverse_compose id_compose
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theorem composition_is_section [instance] {f : mor A B} {g : mor B C}
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(Hf : is_section f) (Hg : is_section g) : is_section (g ∘ f) :=
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is_section.mk
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(calc
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(retraction_of f ∘ retraction_of g) ∘ g ∘ f
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= retraction_of f ∘ retraction_of g ∘ g ∘ f : symm assoc
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... = retraction_of f ∘ (retraction_of g ∘ g) ∘ f : {assoc}
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... = retraction_of f ∘ id ∘ f : {retraction_compose}
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... = retraction_of f ∘ f : {id_left}
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... = id : retraction_compose)
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theorem composition_is_retraction [instance] {f : mor A B} {g : mor B C}
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(Hf : is_retraction f) (Hg : is_retraction g) : is_retraction (g ∘ f) :=
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is_retraction.mk
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(calc
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(g ∘ f) ∘ section_of f ∘ section_of g = g ∘ f ∘ section_of f ∘ section_of g : symm assoc
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... = g ∘ (f ∘ section_of f) ∘ section_of g : {assoc}
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... = g ∘ id ∘ section_of g : {compose_section}
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... = g ∘ section_of g : {id_left}
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... = id : compose_section)
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theorem composition_is_inverse [instance] {f : mor A B} {g : mor B C}
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(Hf : is_iso f) (Hg : is_iso g) : is_iso (g ∘ f) :=
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section_retraction_imp_iso _ _
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definition mono {A B : ob} (f : mor A B) : Prop :=
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∀⦃C⦄ {g h : mor C A}, f ∘ g = f ∘ h → g = h
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definition epi {A B : ob} (f : mor A B) : Prop :=
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∀⦃C⦄ {g h : mor B C}, g ∘ f = h ∘ f → g = h
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theorem section_is_mono {f : mor A B} (H : is_section f) : mono f :=
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λ C g h H,
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calc
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g = id ∘ g : symm id_left
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... = (retraction_of f ∘ f) ∘ g : {symm retraction_compose}
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... = retraction_of f ∘ f ∘ g : symm assoc
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... = retraction_of f ∘ f ∘ h : {H}
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... = (retraction_of f ∘ f) ∘ h : assoc
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... = id ∘ h : {retraction_compose}
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... = h : id_left
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theorem retraction_is_epi {f : mor A B} (H : is_retraction f) : epi f :=
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λ C g h H,
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calc
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g = g ∘ id : symm id_right
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... = g ∘ f ∘ section_of f : {symm compose_section}
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... = (g ∘ f) ∘ section_of f : assoc
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... = (h ∘ f) ∘ section_of f : {H}
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... = h ∘ f ∘ section_of f : symm assoc
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... = h ∘ id : {compose_section}
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... = h : id_right
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end
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section
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definition objects [coercion] (C : Category) : Type
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:= Category.rec (fun c s, c) C
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definition category_instance [instance] (C : Category) : category (objects C)
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:= Category.rec (fun c s, s) C
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end
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end category
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open category
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inductive functor {obC obD : Type} (C : category obC) (D : category obD) : Type :=
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mk : Π (obF : obC → obD) (morF : Π⦃A B : obC⦄, mor A B → mor (obF A) (obF B)),
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(Π ⦃A : obC⦄, morF (ID A) = ID (obF A)) →
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(Π ⦃A B C : obC⦄ {f : mor A B} {g : mor B C}, morF (g ∘ f) = morF g ∘ morF f) →
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functor C D
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inductive Functor (C D : Category) : Type :=
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mk : functor (category_instance C) (category_instance D) → Functor C D
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infixl `⇒`:25 := functor
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namespace functor
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section basic_functor
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parameters {obC obD : Type} {C : category obC} {D : category obD}
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definition object [coercion] (F : C ⇒ D) : obC → obD := rec (λ obF morF Hid Hcomp, obF) F
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definition morphism [coercion] (F : C ⇒ D) : Π{A B : obC}, mor A B → mor (F A) (F B) :=
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rec (λ obF morF Hid Hcomp, morF) F
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theorem respect_id (F : C ⇒ D) : Π {A : obC}, F (ID A) = ID (F A) :=
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rec (λ obF morF Hid Hcomp, Hid) F
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theorem respect_comp (F : C ⇒ D) : Π {a b c : obC} {f : mor a b} {g : mor b c},
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F (g ∘ f) = F g ∘ F f :=
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rec (λ obF morF Hid Hcomp, Hcomp) F
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end basic_functor
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section category_functor
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protected definition compose {obC obD obE : Type} {C : category obC} {D : category obD} {E : category obE}
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(G : D ⇒ E) (F : C ⇒ D) : C ⇒ E :=
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functor.mk
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(λx, G (F x))
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(λ a b f, G (F f))
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(λ a, calc
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G (F (ID a)) = G id : {respect_id F}
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... = id : respect_id G)
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(λ a b c f g, calc
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G (F (g ∘ f)) = G (F g ∘ F f) : {respect_comp F}
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... = G (F g) ∘ G (F f) : respect_comp G)
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precedence `∘∘` : 60
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infixr `∘∘` := compose
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protected theorem assoc {obA obB obC obD : Type} {A : category obA} {B : category obB}
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{C : category obC} {D : category obD} {H : C ⇒ D} {G : B ⇒ C} {F : A ⇒ B} :
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H ∘∘ (G ∘∘ F) = (H ∘∘ G) ∘∘ F :=
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rfl
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-- later check whether we want implicit or explicit arguments here. For the moment, define both
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protected definition id {ob : Type} {C : category ob} : functor C C :=
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mk (λa, a) (λ a b f, f) (λ a, rfl) (λ a b c f g, rfl)
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protected definition ID {ob : Type} (C : category ob) : functor C C := id
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protected definition Id {C : Category} : Functor C C := Functor.mk id
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protected definition iD (C : Category) : Functor C C := Functor.mk id
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protected theorem id_left {obC obB : Type} {B : category obB} {C : category obC} {F : B ⇒ C}
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: id ∘∘ F = F :=
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rec (λ obF morF idF compF, rfl) F
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protected theorem id_right {obC obB : Type} {B : category obB} {C : category obC} {F : B ⇒ C}
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: F ∘∘ id = F :=
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rec (λ obF morF idF compF, rfl) F
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end category_functor
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section Functor
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-- parameters {C D E : Category} (G : Functor D E) (F : Functor C D)
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definition Functor_functor {C D : Category} (F : Functor C D) : functor (category_instance C) (category_instance D) :=
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Functor.rec (λ x, x) F
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protected definition Compose {C D E : Category} (G : Functor D E) (F : Functor C D) : Functor C E :=
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Functor.mk (compose (Functor_functor G) (Functor_functor F))
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-- namespace Functor
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precedence `∘∘` : 60
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infixr `∘∘` := Compose
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-- end Functor
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protected definition Assoc {A B C D : Category} {H : Functor C D} {G : Functor B C} {F : Functor A B}
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: H ∘∘ (G ∘∘ F) = (H ∘∘ G) ∘∘ F :=
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rfl
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protected theorem Id_left {B : Category} {C : Category} {F : Functor B C}
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: Id ∘∘ F = F :=
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Functor.rec (λ f, subst id_left rfl) F
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protected theorem Id_right {B : Category} {C : Category} {F : Functor B C}
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: F ∘∘ Id = F :=
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Functor.rec (λ f, subst id_right rfl) F
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end Functor
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end functor
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open functor
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inductive natural_transformation {obC obD : Type} {C : category obC} {D : category obD}
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(F G : functor C D) : Type :=
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mk : Π (η : Π(a : obC), mor (object F a) (object G a)), (Π{a b : obC} (f : mor a b), morphism G f ∘ η a = η b ∘ morphism F f)
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→ natural_transformation F G
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-- inductive Natural_transformation {C D : Category} (F G : Functor C D) : Type :=
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-- mk : natural_transformation (Functor_functor F) (Functor_functor G) → Natural_transformation F G
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infixl `==>`:25 := natural_transformation
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namespace natural_transformation
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section
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parameters {obC obD : Type} {C : category obC} {D : category obD} {F G : C ⇒ D}
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definition natural_map [coercion] (η : F ==> G) :
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Π(a : obC), mor (object F a) (object G a) :=
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rec (λ x y, x) η
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definition naturality (η : F ==> G) :
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Π{a b : obC} (f : mor a b), morphism G f ∘ η a = η b ∘ morphism F f :=
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rec (λ x y, y) η
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end
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section
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parameters {obC obD : Type} {C : category obC} {D : category obD} {F G H : C ⇒ D}
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protected definition compose (η : G ==> H) (θ : F ==> G) : F ==> H :=
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natural_transformation.mk
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(λ a, η a ∘ θ a)
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(λ a b f,
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calc
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morphism H f ∘ (η a ∘ θ a) = (morphism H f ∘ η a) ∘ θ a : assoc
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... = (η b ∘ morphism G f) ∘ θ a : {naturality η f}
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... = η b ∘ (morphism G f ∘ θ a) : symm assoc
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... = η b ∘ (θ b ∘ morphism F f) : {naturality θ f}
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... = (η b ∘ θ b) ∘ morphism F f : assoc)
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end
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precedence `∘n` : 60
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infixr `∘n` := compose
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section
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protected theorem assoc {obC obD : Type} {C : category obC} {D : category obD} {F₄ F₃ F₂ F₁ : C ⇒ D} {η₃ : F₃ ==> F₄} {η₂ : F₂ ==> F₃} {η₁ : F₁ ==> F₂} : η₃ ∘n (η₂ ∘n η₁) = (η₃ ∘n η₂) ∘n η₁ :=
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congr_arg2_dep mk (funext (take x, assoc)) proof_irrel
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--TODO: check whether some of the below identities are superfluous
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protected definition id {obC obD : Type} {C : category obC} {D : category obD} {F : C ⇒ D}
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: natural_transformation F F :=
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mk (λa, id) (λa b f, id_right ⬝ symm id_left)
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protected definition ID {obC obD : Type} {C : category obC} {D : category obD} (F : C ⇒ D)
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: natural_transformation F F := id
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-- protected definition Id {C D : Category} {F : Functor C D} : Natural_transformation F F :=
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-- Natural_transformation.mk id
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-- protected definition iD {C D : Category} (F : Functor C D) : Natural_transformation F F :=
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-- Natural_transformation.mk id
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protected theorem id_left {obC obD : Type} {C : category obC} {D : category obD} {F G : C ⇒ D}
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{η : F ==> G} : natural_transformation.compose id η = η :=
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rec (λf H, congr_arg2_dep mk (funext (take x, id_left)) proof_irrel) η
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protected theorem id_right {obC obD : Type} {C : category obC} {D : category obD} {F G : C ⇒ D}
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{η : F ==> G} : natural_transformation.compose η id = η :=
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rec (λf H, congr_arg2_dep mk (funext (take x, id_right)) proof_irrel) η
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||||
end
|
||||
end natural_transformation
|
||||
|
||||
-- examples of categories / basic constructions (TODO: move to separate file)
|
||||
|
||||
open functor
|
||||
namespace category
|
||||
section
|
||||
open unit
|
||||
definition one [instance] : category unit :=
|
||||
category.mk (λa b, unit) (λ a b c f g, star) (λ a, star) (λ a b c d f g h, unit.equal _ _)
|
||||
(λ a b f, unit.equal _ _) (λ a b f, unit.equal _ _)
|
||||
end
|
||||
|
||||
section
|
||||
parameter {ob : Type}
|
||||
definition opposite (C : category ob) : category ob :=
|
||||
category.mk (λa b, mor b a) (λ a b c f g, g ∘ f) (λ a, id) (λ a b c d f g h, symm assoc)
|
||||
(λ a b f, id_right) (λ a b f, id_left)
|
||||
precedence `∘op` : 60
|
||||
infixr `∘op` := @compose _ (opposite _) _ _ _
|
||||
|
||||
parameters {C : category ob} {a b c : ob}
|
||||
|
||||
theorem compose_op {f : @mor ob C a b} {g : mor b c} : f ∘op g = g ∘ f :=
|
||||
rfl
|
||||
|
||||
theorem op_op {C : category ob} : opposite (opposite C) = C :=
|
||||
category.rec (λ mor comp id assoc idl idr, refl (mk _ _ _ _ _ _)) C
|
||||
end
|
||||
|
||||
definition Opposite (C : Category) : Category :=
|
||||
Category.mk (objects C) (opposite (category_instance C))
|
||||
|
||||
|
||||
section
|
||||
definition type_category : category Type :=
|
||||
mk (λA B, A → B) (λ a b c, function.compose) (λ a, function.id)
|
||||
(λ a b c d h g f, symm (function.compose_assoc h g f))
|
||||
(λ a b f, function.compose_id_left f) (λ a b f, function.compose_id_right f)
|
||||
end
|
||||
|
||||
section cat_Cat
|
||||
|
||||
definition Cat : category Category :=
|
||||
mk (λ a b, Functor a b) (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)
|
||||
|
||||
end cat_Cat
|
||||
|
||||
section functor_category
|
||||
parameters {obC obD : Type} (C : category obC) (D : category obD)
|
||||
definition functor_category : category (functor C D) :=
|
||||
mk (λa b, natural_transformation a b)
|
||||
(λ a b c g f, natural_transformation.compose g f)
|
||||
(λ a, natural_transformation.id)
|
||||
(λ a b c d h g f, natural_transformation.assoc)
|
||||
(λ a b f, natural_transformation.id_left)
|
||||
(λ a b f, natural_transformation.id_right)
|
||||
end functor_category
|
||||
|
||||
|
||||
section slice
|
||||
open sigma
|
||||
|
||||
definition slice {ob : Type} (C : category ob) (c : ob) : category (Σ(b : ob), mor b c) :=
|
||||
mk (λa b, Σ(g : mor (dpr1 a) (dpr1 b)), dpr2 b ∘ g = dpr2 a)
|
||||
(λ a b c g f, dpair (dpr1 g ∘ dpr1 f)
|
||||
(show dpr2 c ∘ (dpr1 g ∘ dpr1 f) = dpr2 a,
|
||||
proof
|
||||
calc
|
||||
dpr2 c ∘ (dpr1 g ∘ dpr1 f) = (dpr2 c ∘ dpr1 g) ∘ dpr1 f : assoc
|
||||
... = dpr2 b ∘ dpr1 f : {dpr2 g}
|
||||
... = dpr2 a : {dpr2 f}
|
||||
qed))
|
||||
(λ a, dpair id id_right)
|
||||
(λ a b c d h g f, dpair_eq assoc proof_irrel)
|
||||
(λ a b f, sigma.equal id_left proof_irrel)
|
||||
(λ a b f, sigma.equal id_right proof_irrel)
|
||||
-- We give proof_irrel instead of rfl, to give the unifier an easier time
|
||||
end slice
|
||||
|
||||
section coslice
|
||||
open sigma
|
||||
|
||||
definition coslice {ob : Type} (C : category ob) (c : ob) : category (Σ(b : ob), mor c b) :=
|
||||
mk (λa b, Σ(g : mor (dpr1 a) (dpr1 b)), g ∘ dpr2 a = dpr2 b)
|
||||
(λ a b c g f, dpair (dpr1 g ∘ dpr1 f)
|
||||
(show (dpr1 g ∘ dpr1 f) ∘ dpr2 a = dpr2 c,
|
||||
proof
|
||||
calc
|
||||
(dpr1 g ∘ dpr1 f) ∘ dpr2 a = dpr1 g ∘ (dpr1 f ∘ dpr2 a): symm assoc
|
||||
... = dpr1 g ∘ dpr2 b : {dpr2 f}
|
||||
... = dpr2 c : {dpr2 g}
|
||||
qed))
|
||||
(λ a, dpair id id_left)
|
||||
(λ a b c d h g f, dpair_eq assoc proof_irrel)
|
||||
(λ a b f, sigma.equal id_left proof_irrel)
|
||||
(λ a b f, sigma.equal id_right proof_irrel)
|
||||
|
||||
-- theorem slice_coslice_opp {ob : Type} (C : category ob) (c : ob) :
|
||||
-- coslice C c = opposite (slice (opposite C) c) :=
|
||||
-- sorry
|
||||
end coslice
|
||||
|
||||
section product
|
||||
open prod
|
||||
definition product {obC obD : Type} (C : category obC) (D : category obD)
|
||||
: category (obC × obD) :=
|
||||
mk (λa b, mor (pr1 a) (pr1 b) × mor (pr2 a) (pr2 b))
|
||||
(λ a b c g f, (pr1 g ∘ pr1 f , pr2 g ∘ pr2 f) )
|
||||
(λ a, (id,id))
|
||||
(λ a b c d h g f, pair_eq assoc assoc )
|
||||
(λ a b f, prod.equal id_left id_left )
|
||||
(λ a b f, prod.equal id_right id_right)
|
||||
|
||||
end product
|
||||
|
||||
section arrow
|
||||
open sigma eq.ops
|
||||
-- theorem concat_commutative_squares {ob : Type} {C : category ob} {a1 a2 a3 b1 b2 b3 : ob}
|
||||
-- {f1 : a1 => b1} {f2 : a2 => b2} {f3 : a3 => b3} {g2 : a2 => a3} {g1 : a1 => a2}
|
||||
-- {h2 : b2 => b3} {h1 : b1 => b2} (H1 : f2 ∘ g1 = h1 ∘ f1) (H2 : f3 ∘ g2 = h2 ∘ f2)
|
||||
-- : f3 ∘ (g2 ∘ g1) = (h2 ∘ h1) ∘ f1 :=
|
||||
-- calc
|
||||
-- f3 ∘ (g2 ∘ g1) = (f3 ∘ g2) ∘ g1 : assoc
|
||||
-- ... = (h2 ∘ f2) ∘ g1 : {H2}
|
||||
-- ... = h2 ∘ (f2 ∘ g1) : symm assoc
|
||||
-- ... = h2 ∘ (h1 ∘ f1) : {H1}
|
||||
-- ... = (h2 ∘ h1) ∘ f1 : assoc
|
||||
|
||||
-- definition arrow {ob : Type} (C : category ob) : category (Σ(a b : ob), mor a b) :=
|
||||
-- mk (λa b, Σ(g : mor (dpr1 a) (dpr1 b)) (h : mor (dpr2' a) (dpr2' b)),
|
||||
-- dpr3 b ∘ g = h ∘ dpr3 a)
|
||||
-- (λ a b c g f, dpair (dpr1 g ∘ dpr1 f) (dpair (dpr2' g ∘ dpr2' f) (concat_commutative_squares (dpr3 f) (dpr3 g))))
|
||||
-- (λ a, dpair id (dpair id (id_right ⬝ (symm id_left))))
|
||||
-- (λ a b c d h g f, dtrip_eq2 assoc assoc proof_irrel)
|
||||
-- (λ a b f, trip.equal2 id_left id_left proof_irrel)
|
||||
-- (λ a b f, trip.equal2 id_right id_right proof_irrel)
|
||||
|
||||
definition arrow_obs (ob : Type) (C : category ob) :=
|
||||
Σ(a b : ob), mor a b
|
||||
|
||||
definition src {ob : Type} {C : category ob} (a : arrow_obs ob C) : ob :=
|
||||
dpr1 a
|
||||
|
||||
definition dst {ob : Type} {C : category ob} (a : arrow_obs ob C) : ob :=
|
||||
dpr2' a
|
||||
|
||||
definition to_mor {ob : Type} {C : category ob} (a : arrow_obs ob C) : mor (src a) (dst a) :=
|
||||
dpr3 a
|
||||
|
||||
definition arrow_mor (ob : Type) (C : category ob) (a b : arrow_obs ob C) : Type :=
|
||||
Σ (g : mor (src a) (src b)) (h : mor (dst a) (dst b)), to_mor b ∘ g = h ∘ to_mor a
|
||||
|
||||
definition mor_src {ob : Type} {C : category ob} {a b : arrow_obs ob C} (m : arrow_mor ob C a b) : mor (src a) (src b) :=
|
||||
dpr1 m
|
||||
|
||||
definition mor_dst {ob : Type} {C : category ob} {a b : arrow_obs ob C} (m : arrow_mor ob C a b) : mor (dst a) (dst b) :=
|
||||
dpr2' m
|
||||
|
||||
definition commute {ob : Type} {C : category ob} {a b : arrow_obs ob C} (m : arrow_mor ob C a b) :
|
||||
to_mor b ∘ (mor_src m) = (mor_dst m) ∘ to_mor a :=
|
||||
dpr3 m
|
||||
|
||||
definition arrow (ob : Type) (C : category ob) : category (arrow_obs ob C) :=
|
||||
mk (λa b, arrow_mor ob C a b)
|
||||
(λ a b c g f, dpair (mor_src g ∘ mor_src f) (dpair (mor_dst g ∘ mor_dst f)
|
||||
(show to_mor c ∘ (mor_src g ∘ mor_src f) = (mor_dst g ∘ mor_dst f) ∘ to_mor a,
|
||||
proof
|
||||
calc
|
||||
to_mor c ∘ (mor_src g ∘ mor_src f) = (to_mor c ∘ mor_src g) ∘ mor_src f : assoc
|
||||
... = (mor_dst g ∘ to_mor b) ∘ mor_src f : {commute g}
|
||||
... = mor_dst g ∘ (to_mor b ∘ mor_src f) : symm assoc
|
||||
... = mor_dst g ∘ (mor_dst f ∘ to_mor a) : {commute f}
|
||||
... = (mor_dst g ∘ mor_dst f) ∘ to_mor a : assoc
|
||||
qed)
|
||||
))
|
||||
(λ a, dpair id (dpair id (id_right ⬝ (symm id_left))))
|
||||
(λ a b c d h g f, dtrip_eq_ndep assoc assoc proof_irrel)
|
||||
(λ a b f, trip.equal_ndep id_left id_left proof_irrel)
|
||||
(λ a b f, trip.equal_ndep id_right id_right proof_irrel)
|
||||
|
||||
end arrow
|
||||
|
||||
-- definition foo
|
||||
-- : category (sorry) :=
|
||||
-- mk (λa b, sorry)
|
||||
-- (λ a b c g f, sorry)
|
||||
-- (λ a, sorry)
|
||||
-- (λ a b c d h g f, sorry)
|
||||
-- (λ a b f, sorry)
|
||||
-- (λ a b f, sorry)
|
||||
|
||||
end category
|
||||
223
library/algebra/category/basic.lean
Normal file
223
library/algebra/category/basic.lean
Normal file
|
|
@ -0,0 +1,223 @@
|
|||
-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
|
||||
-- Released under Apache 2.0 license as described in the file LICENSE.
|
||||
-- Author: Floris van Doorn
|
||||
|
||||
import logic.axioms.funext
|
||||
|
||||
open eq eq.ops
|
||||
|
||||
inductive category [class] (ob : Type) : Type :=
|
||||
mk : Π (hom : ob → ob → Type)
|
||||
(comp : Π⦃a b c : ob⦄, hom b c → hom a b → hom a c)
|
||||
(id : Π {a : ob}, hom a a),
|
||||
(Π ⦃a b c d : ob⦄ {h : hom c d} {g : hom b c} {f : hom a b},
|
||||
comp h (comp g f) = comp (comp h g) f) →
|
||||
(Π ⦃a b : ob⦄ {f : hom a b}, comp id f = f) →
|
||||
(Π ⦃a b : ob⦄ {f : hom a b}, comp f id = f) →
|
||||
category ob
|
||||
|
||||
inductive Category : Type := mk : Π (ob : Type), category ob → Category
|
||||
|
||||
namespace category
|
||||
section
|
||||
parameters {ob : Type} {C : category ob}
|
||||
variables {a b c d : ob}
|
||||
|
||||
definition hom [reducible] : ob → ob → Type := rec (λ hom compose id assoc idr idl, hom) C
|
||||
-- note: needs to be reducible to typecheck composition in opposite category
|
||||
definition compose [reducible] : Π {a b c : ob}, hom b c → hom a b → hom a c :=
|
||||
rec (λ hom compose id assoc idr idl, compose) C
|
||||
|
||||
definition id [reducible] : Π {a : ob}, hom a a := rec (λ hom compose id assoc idr idl, id) C
|
||||
definition ID [reducible] : Π (a : ob), hom a a := @id
|
||||
|
||||
infixr `∘`:60 := compose
|
||||
infixl `⟶`:25 := hom -- input ⟶ using \-->
|
||||
|
||||
variables {h : hom c d} {g : hom b c} {f : hom a b} {i : hom a a}
|
||||
|
||||
theorem assoc : Π ⦃a b c d : ob⦄ (h : hom c d) (g : hom b c) (f : hom a b),
|
||||
h ∘ (g ∘ f) = (h ∘ g) ∘ f :=
|
||||
rec (λ hom comp id assoc idr idl, assoc) C
|
||||
|
||||
theorem id_left : Π ⦃a b : ob⦄ (f : hom a b), id ∘ f = f :=
|
||||
rec (λ hom comp id assoc idl idr, idl) C
|
||||
theorem id_right : Π ⦃a b : ob⦄ (f : hom a b), f ∘ id = f :=
|
||||
rec (λ hom comp id assoc idl idr, idr) C
|
||||
|
||||
theorem id_compose (a : ob) : (ID a) ∘ id = id := !id_left
|
||||
|
||||
theorem left_id_unique (H : Π{b} {f : hom b a}, i ∘ f = f) : i = id :=
|
||||
calc
|
||||
i = i ∘ id : symm !id_right
|
||||
... = id : H
|
||||
|
||||
theorem right_id_unique (H : Π{b} {f : hom a b}, f ∘ i = f) : i = id :=
|
||||
calc
|
||||
i = id ∘ i : eq.symm !id_left
|
||||
... = id : H
|
||||
end
|
||||
|
||||
section
|
||||
|
||||
definition objects [coercion] (C : Category) : Type
|
||||
:= Category.rec (fun c s, c) C
|
||||
|
||||
definition category_instance [instance] (C : Category) : category (objects C)
|
||||
:= Category.rec (fun c s, s) C
|
||||
|
||||
end
|
||||
|
||||
end category
|
||||
|
||||
open category
|
||||
|
||||
inductive functor {obC obD : Type} (C : category obC) (D : category obD) : Type :=
|
||||
mk : Π (obF : obC → obD) (homF : Π⦃a b : obC⦄, hom a b → hom (obF a) (obF b)),
|
||||
(Π ⦃a : obC⦄, homF (ID a) = ID (obF a)) →
|
||||
(Π ⦃a b c : obC⦄ {g : hom b c} {f : hom a b}, homF (g ∘ f) = homF g ∘ homF f) →
|
||||
functor C D
|
||||
|
||||
inductive Functor (C D : Category) : Type :=
|
||||
mk : functor (category_instance C) (category_instance D) → Functor C D
|
||||
|
||||
infixl `⇒`:25 := functor
|
||||
|
||||
namespace functor
|
||||
section basic_functor
|
||||
variables {obC obD obE : Type} {C : category obC} {D : category obD} {E : category obE}
|
||||
definition object [coercion] (F : C ⇒ D) : obC → obD := rec (λ obF homF Hid Hcomp, obF) F
|
||||
|
||||
definition morphism [coercion] (F : C ⇒ D) : Π{a b : obC}, hom a b → hom (F a) (F b) :=
|
||||
rec (λ obF homF Hid Hcomp, homF) F
|
||||
|
||||
theorem respect_id (F : C ⇒ D) : Π (a : obC), F (ID a) = id :=
|
||||
rec (λ obF homF Hid Hcomp, Hid) F
|
||||
|
||||
theorem respect_comp (F : C ⇒ D) : Π ⦃a b c : obC⦄ (g : hom b c) (f : hom a b),
|
||||
F (g ∘ f) = F g ∘ F f :=
|
||||
rec (λ obF homF Hid Hcomp, Hcomp) F
|
||||
|
||||
protected definition compose (G : D ⇒ E) (F : C ⇒ D) : C ⇒ E :=
|
||||
functor.mk
|
||||
(λx, G (F x))
|
||||
(λ a b f, G (F f))
|
||||
(λ a, calc
|
||||
G (F (ID a)) = G id : {respect_id F a}
|
||||
... = id : respect_id G (F a))
|
||||
(λ a b c g f, calc
|
||||
G (F (g ∘ f)) = G (F g ∘ F f) : {respect_comp F g f}
|
||||
... = G (F g) ∘ G (F f) : respect_comp G (F g) (F f))
|
||||
|
||||
infixr `∘∘`:60 := compose
|
||||
|
||||
protected theorem assoc {obA obB obC obD : Type} {A : category obA} {B : category obB}
|
||||
{C : category obC} {D : category obD} (H : C ⇒ D) (G : B ⇒ C) (F : A ⇒ B) :
|
||||
H ∘∘ (G ∘∘ F) = (H ∘∘ G) ∘∘ F :=
|
||||
rfl
|
||||
|
||||
-- later check whether we want implicit or explicit arguments here. For the moment, define both
|
||||
protected definition id {ob : Type} {C : category ob} : functor C C :=
|
||||
mk (λa, a) (λ a b f, f) (λ a, rfl) (λ a b c f g, rfl)
|
||||
protected definition ID {ob : Type} (C : category ob) : functor C C := id
|
||||
protected definition Id {C : Category} : Functor C C := Functor.mk id
|
||||
protected definition iD (C : Category) : Functor C C := Functor.mk id
|
||||
|
||||
protected theorem id_left (F : C ⇒ D) : id ∘∘ F = F := rec (λ obF homF idF compF, rfl) F
|
||||
protected theorem id_right (F : C ⇒ D) : F ∘∘ id = F := rec (λ obF homF idF compF, rfl) F
|
||||
|
||||
end basic_functor
|
||||
|
||||
section Functor
|
||||
variables {C₁ C₂ C₃ C₄: Category} --(G : Functor D E) (F : Functor C D)
|
||||
definition Functor_functor {C₁ C₂ : Category} (F : Functor C₁ C₂) : --remove params
|
||||
functor (category_instance C₁) (category_instance C₂) :=
|
||||
Functor.rec (λ x, x) F
|
||||
|
||||
protected definition Compose (G : Functor C₂ C₃) (F : Functor C₁ C₂) : Functor C₁ C₃ :=
|
||||
Functor.mk (compose (Functor_functor G) (Functor_functor F))
|
||||
|
||||
-- namespace Functor
|
||||
infixr `∘∘`:60 := Compose
|
||||
-- end Functor
|
||||
|
||||
protected definition Assoc (H : Functor C₃ C₄) (G : Functor C₂ C₃) (F : Functor C₁ C₂)
|
||||
: H ∘∘ (G ∘∘ F) = (H ∘∘ G) ∘∘ F :=
|
||||
rfl
|
||||
|
||||
protected theorem Id_left (F : Functor C₁ C₂) : Id ∘∘ F = F :=
|
||||
Functor.rec (λ f, subst !id_left rfl) F
|
||||
|
||||
protected theorem Id_right {F : Functor C₁ C₂} : F ∘∘ Id = F :=
|
||||
Functor.rec (λ f, subst !id_right rfl) F
|
||||
end Functor
|
||||
|
||||
end functor
|
||||
|
||||
open functor
|
||||
|
||||
inductive natural_transformation {obC obD : Type} {C : category obC} {D : category obD}
|
||||
(F G : functor C D) : Type :=
|
||||
mk : Π (η : Π(a : obC), hom (object F a) (object G a)), (Π{a b : obC} (f : hom a b), morphism G f ∘ η a = η b ∘ morphism F f)
|
||||
→ natural_transformation F G
|
||||
|
||||
-- inductive Natural_transformation {C D : Category} (F G : Functor C D) : Type :=
|
||||
-- mk : natural_transformation (Functor_functor F) (Functor_functor G) → Natural_transformation F G
|
||||
|
||||
infixl `⟹`:25 := natural_transformation
|
||||
|
||||
namespace natural_transformation
|
||||
section
|
||||
parameters {obC obD : Type} {C : category obC} {D : category obD} {F G : C ⇒ D}
|
||||
|
||||
definition natural_map [coercion] (η : F ⟹ G) :
|
||||
Π(a : obC), hom (object F a) (object G a) :=
|
||||
rec (λ x y, x) η
|
||||
|
||||
definition naturality (η : F ⟹ G) :
|
||||
Π{a b : obC} (f : hom a b), morphism G f ∘ η a = η b ∘ morphism F f :=
|
||||
rec (λ x y, y) η
|
||||
end
|
||||
|
||||
section
|
||||
parameters {obC obD : Type} {C : category obC} {D : category obD} {F G H : C ⇒ D}
|
||||
protected definition compose (η : G ⟹ H) (θ : F ⟹ G) : F ⟹ H :=
|
||||
natural_transformation.mk
|
||||
(λ a, η a ∘ θ a)
|
||||
(λ a b f,
|
||||
calc
|
||||
morphism H f ∘ (η a ∘ θ a) = (morphism H f ∘ η a) ∘ θ a : !assoc
|
||||
... = (η b ∘ morphism G f) ∘ θ a : {naturality η f}
|
||||
... = η b ∘ (morphism G f ∘ θ a) : symm !assoc
|
||||
... = η b ∘ (θ b ∘ morphism F f) : {naturality θ f}
|
||||
... = (η b ∘ θ b) ∘ morphism F f : !assoc)
|
||||
end
|
||||
precedence `∘n` : 60
|
||||
infixr `∘n` := compose
|
||||
section
|
||||
variables {obC obD : Type} {C : category obC} {D : category obD} {F₁ F₂ F₃ F₄ : C ⇒ D}
|
||||
protected theorem assoc (η₃ : F₃ ⟹ F₄) (η₂ : F₂ ⟹ F₃) (η₁ : F₁ ⟹ F₂) :
|
||||
η₃ ∘n (η₂ ∘n η₁) = (η₃ ∘n η₂) ∘n η₁ :=
|
||||
congr_arg2_dep mk (funext (take x, !assoc)) !proof_irrel
|
||||
|
||||
--TODO: check whether some of the below identities are superfluous
|
||||
protected definition id {obC obD : Type} {C : category obC} {D : category obD} {F : C ⇒ D}
|
||||
: natural_transformation F F :=
|
||||
mk (λa, id) (λa b f, !id_right ⬝ symm !id_left)
|
||||
protected definition ID {obC obD : Type} {C : category obC} {D : category obD} (F : C ⇒ D)
|
||||
: natural_transformation F F := id
|
||||
-- protected definition Id {C D : Category} {F : Functor C D} : Natural_transformation F F :=
|
||||
-- Natural_transformation.mk id
|
||||
-- protected definition iD {C D : Category} (F : Functor C D) : Natural_transformation F F :=
|
||||
-- Natural_transformation.mk id
|
||||
|
||||
protected theorem id_left (η : F₁ ⟹ F₂) : natural_transformation.compose id η = η :=
|
||||
rec (λf H, congr_arg2_dep mk (funext (take x, !id_left)) !proof_irrel) η
|
||||
|
||||
protected theorem id_right (η : F₁ ⟹ F₂) : natural_transformation.compose η id = η :=
|
||||
rec (λf H, congr_arg2_dep mk (funext (take x, !id_right)) !proof_irrel) η
|
||||
|
||||
end
|
||||
end natural_transformation
|
||||
|
||||
|
||||
270
library/algebra/category/constructions.lean
Normal file
270
library/algebra/category/constructions.lean
Normal file
|
|
@ -0,0 +1,270 @@
|
|||
-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
|
||||
-- Released under Apache 2.0 license as described in the file LICENSE.
|
||||
-- Author: Floris van Doorn
|
||||
|
||||
-- This file contains basic constructions on categories, including common categories
|
||||
|
||||
|
||||
import .basic
|
||||
import data.unit data.sigma data.prod data.empty data.bool
|
||||
|
||||
open eq eq.ops prod
|
||||
namespace category
|
||||
section
|
||||
open unit
|
||||
definition category_one : category unit :=
|
||||
mk (λa b, unit)
|
||||
(λ a b c f g, star)
|
||||
(λ a, star)
|
||||
(λ a b c d f g h, !unit.equal)
|
||||
(λ a b f, !unit.equal)
|
||||
(λ a b f, !unit.equal)
|
||||
end
|
||||
|
||||
section
|
||||
variables {ob : Type} {C : category ob} {a b c : ob}
|
||||
definition opposite (C : category ob) : category ob :=
|
||||
mk (λa b, hom b a)
|
||||
(λ a b c f g, g ∘ f)
|
||||
(λ a, id)
|
||||
(λ a b c d f g h, symm !assoc)
|
||||
(λ a b f, !id_right)
|
||||
(λ a b f, !id_left)
|
||||
--definition compose_opposite {C : category ob} {a b c : ob} {g : a => b} {f : b => c} : compose
|
||||
precedence `∘op` : 60
|
||||
infixr `∘op` := @compose _ (opposite _) _ _ _
|
||||
|
||||
theorem compose_op {f : @hom ob C a b} {g : hom b c} : f ∘op g = g ∘ f :=
|
||||
rfl
|
||||
|
||||
theorem op_op {C : category ob} : opposite (opposite C) = C :=
|
||||
category.rec (λ hom comp id assoc idl idr, refl (mk _ _ _ _ _ _)) C
|
||||
end
|
||||
|
||||
definition Opposite (C : Category) : Category :=
|
||||
Category.mk (objects C) (opposite (category_instance C))
|
||||
|
||||
section
|
||||
definition type_category : category Type :=
|
||||
mk (λa b, a → b)
|
||||
(λ a b c, function.compose)
|
||||
(λ a, function.id)
|
||||
(λ a b c d h g f, symm (function.compose_assoc h g f))
|
||||
(λ a b f, function.compose_id_left f)
|
||||
(λ a b f, function.compose_id_right f)
|
||||
end
|
||||
|
||||
section
|
||||
open decidable unit empty
|
||||
parameters (A : Type) {H : decidable_eq A}
|
||||
private definition set_hom (a b : A) := decidable.rec_on (H a b) (λh, unit) (λh, empty)
|
||||
private theorem set_hom_subsingleton [instance] (a b : A) : subsingleton (set_hom a b) := _
|
||||
private definition set_compose {a b c : A} (g : set_hom b c) (f : set_hom a b) : set_hom a c :=
|
||||
decidable.rec_on
|
||||
(H b c)
|
||||
(λ Hbc g, decidable.rec_on
|
||||
(H a b)
|
||||
(λ Hab f, rec_on_true (trans Hab Hbc) ⋆)
|
||||
(λh f, empty.rec _ f) f)
|
||||
(λh (g : empty), empty.rec _ g) g
|
||||
|
||||
definition set_category : category A :=
|
||||
mk (λa b, set_hom a b)
|
||||
(λ a b c g f, set_compose g f)
|
||||
(λ a, rec_on_true rfl ⋆)
|
||||
(λ a b c d h g f, subsingleton.elim _ _ _)
|
||||
(λ a b f, subsingleton.elim _ _ _)
|
||||
(λ a b f, subsingleton.elim _ _ _)
|
||||
end
|
||||
|
||||
section
|
||||
open bool
|
||||
definition category_two := set_category bool
|
||||
end
|
||||
|
||||
|
||||
section cat_of_cat
|
||||
definition category_of_categories : category Category :=
|
||||
mk (λ a b, Functor a b)
|
||||
(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)
|
||||
end cat_of_cat
|
||||
|
||||
section product
|
||||
open prod
|
||||
definition product_category {obC obD : Type} (C : category obC) (D : category obD)
|
||||
: category (obC × obD) :=
|
||||
mk (λa b, hom (pr1 a) (pr1 b) × hom (pr2 a) (pr2 b))
|
||||
(λ a b c g f, (pr1 g ∘ pr1 f , pr2 g ∘ pr2 f) )
|
||||
(λ a, (id,id))
|
||||
(λ a b c d h g f, pair_eq !assoc !assoc )
|
||||
(λ a b f, prod.equal !id_left !id_left )
|
||||
(λ a b f, prod.equal !id_right !id_right)
|
||||
|
||||
end product
|
||||
|
||||
namespace ops
|
||||
notation `Cat` := category_of_categories
|
||||
notation `type` := type_category
|
||||
notation 1 := category_one
|
||||
postfix `ᵒᵖ`:max := opposite
|
||||
infixr `×` := product_category
|
||||
instance category_of_categories type_category category_one product_category
|
||||
end ops
|
||||
|
||||
section functor_category
|
||||
parameters {obC obD : Type} (C : category obC) (D : category obD)
|
||||
definition functor_category : category (functor C D) :=
|
||||
mk (λa b, natural_transformation a b)
|
||||
(λ a b c g f, natural_transformation.compose g f)
|
||||
(λ a, natural_transformation.id)
|
||||
(λ a b c d h g f, !natural_transformation.assoc)
|
||||
(λ a b f, !natural_transformation.id_left)
|
||||
(λ a b f, !natural_transformation.id_right)
|
||||
end functor_category
|
||||
|
||||
section
|
||||
open sigma
|
||||
|
||||
definition slice_category [reducible] {ob : Type} (C : category ob) (c : ob) : category (Σ(b : ob), hom b c) :=
|
||||
mk (λa b, Σ(g : hom (dpr1 a) (dpr1 b)), dpr2 b ∘ g = dpr2 a)
|
||||
(λ a b c g f, dpair (dpr1 g ∘ dpr1 f)
|
||||
(show dpr2 c ∘ (dpr1 g ∘ dpr1 f) = dpr2 a,
|
||||
proof
|
||||
calc
|
||||
dpr2 c ∘ (dpr1 g ∘ dpr1 f) = (dpr2 c ∘ dpr1 g) ∘ dpr1 f : !assoc
|
||||
... = dpr2 b ∘ dpr1 f : {dpr2 g}
|
||||
... = dpr2 a : {dpr2 f}
|
||||
qed))
|
||||
(λ a, dpair id !id_right)
|
||||
(λ a b c d h g f, dpair_eq !assoc !proof_irrel)
|
||||
(λ a b f, sigma.equal !id_left !proof_irrel)
|
||||
(λ a b f, sigma.equal !id_right !proof_irrel)
|
||||
-- We use !proof_irrel instead of rfl, to give the unifier an easier time
|
||||
end --remove
|
||||
namespace slice
|
||||
section --remove
|
||||
open sigma category.ops --remove sigma
|
||||
instance slice_category
|
||||
parameters {ob : Type} (C : category ob)
|
||||
definition forgetful (x : ob) : (slice_category C x) ⇒ C :=
|
||||
functor.mk (λ a, dpr1 a)
|
||||
(λ a b f, dpr1 f)
|
||||
(λ a, rfl)
|
||||
(λ a b c g f, rfl)
|
||||
definition composition_functor {x y : ob} (h : x ⟶ y) : slice_category C x ⇒ slice_category C y :=
|
||||
functor.mk (λ a, dpair (dpr1 a) (h ∘ dpr2 a))
|
||||
(λ a b f, dpair (dpr1 f)
|
||||
(calc
|
||||
(h ∘ dpr2 b) ∘ dpr1 f = h ∘ (dpr2 b ∘ dpr1 f) : !assoc⁻¹
|
||||
... = h ∘ dpr2 a : {dpr2 f}))
|
||||
(λ a, rfl)
|
||||
(λ a b c g f, dpair_eq rfl !proof_irrel)
|
||||
-- the following definition becomes complicated
|
||||
-- definition slice_functor : C ⇒ category_of_categories :=
|
||||
-- functor.mk (λ a, Category.mk _ (slice_category C a))
|
||||
-- (λ a b f, Functor.mk (composition_functor f))
|
||||
-- (λ a, congr_arg Functor.mk
|
||||
-- (congr_arg4_dep functor.mk
|
||||
-- (funext (λx, sigma.equal rfl !id_left))
|
||||
-- sorry
|
||||
-- !proof_irrel
|
||||
-- !proof_irrel))
|
||||
-- (λ a b c g f, sorry)
|
||||
end
|
||||
end slice
|
||||
|
||||
section coslice
|
||||
open sigma
|
||||
|
||||
definition coslice {ob : Type} (C : category ob) (c : ob) : category (Σ(b : ob), hom c b) :=
|
||||
mk (λa b, Σ(g : hom (dpr1 a) (dpr1 b)), g ∘ dpr2 a = dpr2 b)
|
||||
(λ a b c g f, dpair (dpr1 g ∘ dpr1 f)
|
||||
(show (dpr1 g ∘ dpr1 f) ∘ dpr2 a = dpr2 c,
|
||||
proof
|
||||
calc
|
||||
(dpr1 g ∘ dpr1 f) ∘ dpr2 a = dpr1 g ∘ (dpr1 f ∘ dpr2 a): symm !assoc
|
||||
... = dpr1 g ∘ dpr2 b : {dpr2 f}
|
||||
... = dpr2 c : {dpr2 g}
|
||||
qed))
|
||||
(λ a, dpair id !id_left)
|
||||
(λ a b c d h g f, dpair_eq !assoc !proof_irrel)
|
||||
(λ a b f, sigma.equal !id_left !proof_irrel)
|
||||
(λ a b f, sigma.equal !id_right !proof_irrel)
|
||||
|
||||
-- theorem slice_coslice_opp {ob : Type} (C : category ob) (c : ob) :
|
||||
-- coslice C c = opposite (slice (opposite C) c) :=
|
||||
-- sorry
|
||||
end coslice
|
||||
|
||||
section arrow
|
||||
open sigma eq.ops
|
||||
-- theorem concat_commutative_squares {ob : Type} {C : category ob} {a1 a2 a3 b1 b2 b3 : ob}
|
||||
-- {f1 : a1 => b1} {f2 : a2 => b2} {f3 : a3 => b3} {g2 : a2 => a3} {g1 : a1 => a2}
|
||||
-- {h2 : b2 => b3} {h1 : b1 => b2} (H1 : f2 ∘ g1 = h1 ∘ f1) (H2 : f3 ∘ g2 = h2 ∘ f2)
|
||||
-- : f3 ∘ (g2 ∘ g1) = (h2 ∘ h1) ∘ f1 :=
|
||||
-- calc
|
||||
-- f3 ∘ (g2 ∘ g1) = (f3 ∘ g2) ∘ g1 : assoc
|
||||
-- ... = (h2 ∘ f2) ∘ g1 : {H2}
|
||||
-- ... = h2 ∘ (f2 ∘ g1) : symm assoc
|
||||
-- ... = h2 ∘ (h1 ∘ f1) : {H1}
|
||||
-- ... = (h2 ∘ h1) ∘ f1 : assoc
|
||||
|
||||
-- definition arrow {ob : Type} (C : category ob) : category (Σ(a b : ob), hom a b) :=
|
||||
-- mk (λa b, Σ(g : hom (dpr1 a) (dpr1 b)) (h : hom (dpr2' a) (dpr2' b)),
|
||||
-- dpr3 b ∘ g = h ∘ dpr3 a)
|
||||
-- (λ a b c g f, dpair (dpr1 g ∘ dpr1 f) (dpair (dpr2' g ∘ dpr2' f) (concat_commutative_squares (dpr3 f) (dpr3 g))))
|
||||
-- (λ a, dpair id (dpair id (id_right ⬝ (symm id_left))))
|
||||
-- (λ a b c d h g f, dtrip_eq2 assoc assoc !proof_irrel)
|
||||
-- (λ a b f, trip.equal2 id_left id_left !proof_irrel)
|
||||
-- (λ a b f, trip.equal2 id_right id_right !proof_irrel)
|
||||
|
||||
-- make these definitions private?
|
||||
variables {ob : Type} {C : category ob}
|
||||
protected definition arrow_obs (ob : Type) (C : category ob) := Σ(a b : ob), hom a b
|
||||
variables {a b : arrow_obs ob C}
|
||||
protected definition src (a : arrow_obs ob C) : ob := dpr1 a
|
||||
protected definition dst (a : arrow_obs ob C) : ob := dpr2' a
|
||||
protected definition to_hom (a : arrow_obs ob C) : hom (src a) (dst a) := dpr3 a
|
||||
|
||||
protected definition arrow_hom (a b : arrow_obs ob C) : Type :=
|
||||
Σ (g : hom (src a) (src b)) (h : hom (dst a) (dst b)), to_hom b ∘ g = h ∘ to_hom a
|
||||
|
||||
protected definition hom_src (m : arrow_hom a b) : hom (src a) (src b) := dpr1 m
|
||||
protected definition hom_dst (m : arrow_hom a b) : hom (dst a) (dst b) := dpr2' m
|
||||
protected definition commute (m : arrow_hom a b) : to_hom b ∘ (hom_src m) = (hom_dst m) ∘ to_hom a
|
||||
:= dpr3 m
|
||||
|
||||
definition arrow (ob : Type) (C : category ob) : category (arrow_obs ob C) :=
|
||||
mk (λa b, arrow_hom a b)
|
||||
(λ a b c g f, dpair (hom_src g ∘ hom_src f) (dpair (hom_dst g ∘ hom_dst f)
|
||||
(show to_hom c ∘ (hom_src g ∘ hom_src f) = (hom_dst g ∘ hom_dst f) ∘ to_hom a,
|
||||
proof
|
||||
calc
|
||||
to_hom c ∘ (hom_src g ∘ hom_src f) = (to_hom c ∘ hom_src g) ∘ hom_src f : !assoc
|
||||
... = (hom_dst g ∘ to_hom b) ∘ hom_src f : {commute g}
|
||||
... = hom_dst g ∘ (to_hom b ∘ hom_src f) : symm !assoc
|
||||
... = hom_dst g ∘ (hom_dst f ∘ to_hom a) : {commute f}
|
||||
... = (hom_dst g ∘ hom_dst f) ∘ to_hom a : !assoc
|
||||
qed)
|
||||
))
|
||||
(λ a, dpair id (dpair id (!id_right ⬝ (symm !id_left))))
|
||||
(λ a b c d h g f, dtrip_eq_ndep !assoc !assoc !proof_irrel)
|
||||
(λ a b f, trip.equal_ndep !id_left !id_left !proof_irrel)
|
||||
(λ a b f, trip.equal_ndep !id_right !id_right !proof_irrel)
|
||||
|
||||
end arrow
|
||||
|
||||
end category
|
||||
|
||||
-- definition foo
|
||||
-- : category (sorry) :=
|
||||
-- mk (λa b, sorry)
|
||||
-- (λ a b c g f, sorry)
|
||||
-- (λ a, sorry)
|
||||
-- (λ a b c d h g f, sorry)
|
||||
-- (λ a b f, sorry)
|
||||
-- (λ a b f, sorry)
|
||||
5
library/algebra/category/default.lean
Normal file
5
library/algebra/category/default.lean
Normal file
|
|
@ -0,0 +1,5 @@
|
|||
-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
|
||||
-- Released under Apache 2.0 license as described in the file LICENSE.
|
||||
-- Author: Floris van Doorn
|
||||
|
||||
import .basic .constructions
|
||||
35
library/algebra/category/limit.lean
Normal file
35
library/algebra/category/limit.lean
Normal file
|
|
@ -0,0 +1,35 @@
|
|||
-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
|
||||
-- Released under Apache 2.0 license as described in the file LICENSE.
|
||||
-- Author: Floris van Doorn
|
||||
|
||||
import .basic
|
||||
import data.sigma
|
||||
|
||||
open eq eq.ops category functor natural_transformation
|
||||
|
||||
namespace limits
|
||||
--representable functor
|
||||
section
|
||||
parameters {obI ob : Type} {I : category obI} {C : category ob} {D : I ⇒ C}
|
||||
|
||||
definition constant_diagram (a : ob) : I ⇒ C :=
|
||||
mk (λ i, a)
|
||||
(λ i j u, id)
|
||||
(λ i, rfl)
|
||||
(λ i j k v u, symm !id_compose)
|
||||
|
||||
definition cone := Σ(a : ob), constant_diagram a ⟹ D
|
||||
-- definition cone_category : category cone :=
|
||||
-- mk (λa b, sorry)
|
||||
-- (λ a b c g f, sorry)
|
||||
-- (λ a, sorry)
|
||||
-- (λ a b c d h g f, sorry)
|
||||
-- (λ a b f, sorry)
|
||||
-- (λ a b f, sorry)
|
||||
|
||||
end
|
||||
end limits
|
||||
-- functor.mk (λ a, sorry)
|
||||
-- (λ a b f, sorry)
|
||||
-- (λ a, sorry)
|
||||
-- (λ a b c g f, sorry)
|
||||
278
library/algebra/category/morphism.lean
Normal file
278
library/algebra/category/morphism.lean
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-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Floris van Doorn
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import .basic algebra.relation algebra.binary
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open eq eq.ops category
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namespace morphism
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section
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parameters {ob : Type} {C : category ob} include C
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variables {a b c d : ob} {h : @hom ob C c d} {g : @hom ob C b c} {f : @hom ob C a b} {i : @hom ob C b a}
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inductive is_section [class] (f : @hom ob C a b) : Type
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:= mk : ∀{g}, g ∘ f = id → is_section f
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inductive is_retraction [class] (f : @hom ob C a b) : Type
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:= mk : ∀{g}, f ∘ g = id → is_retraction f
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inductive is_iso [class] (f : @hom ob C a b) : Type
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:= mk : ∀{g}, g ∘ f = id → f ∘ g = id → is_iso f
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--remove implicit arguments in above lines
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definition retraction_of (f : hom a b) {H : is_section f} : hom b a :=
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is_section.rec (λg h, g) H
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definition section_of (f : hom a b) {H : is_retraction f} : hom b a :=
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is_retraction.rec (λg h, g) H
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definition inverse (f : hom a b) {H : is_iso f} : hom b a :=
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is_iso.rec (λg h1 h2, g) H
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postfix `⁻¹` := inverse
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theorem inverse_compose (f : hom a b) {H : is_iso f} : f⁻¹ ∘ f = id :=
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is_iso.rec (λg h1 h2, h1) H
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theorem compose_inverse (f : hom a b) {H : is_iso f} : f ∘ f⁻¹ = id :=
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is_iso.rec (λg h1 h2, h2) H
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theorem retraction_compose (f : hom a b) {H : is_section f} : retraction_of f ∘ f = id :=
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is_section.rec (λg h, h) H
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theorem compose_section (f : hom a b) {H : is_retraction f} : f ∘ section_of f = id :=
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is_retraction.rec (λg h, h) H
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theorem iso_imp_retraction [instance] (f : hom a b) {H : is_iso f} : is_section f :=
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is_section.mk !inverse_compose
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theorem iso_imp_section [instance] (f : hom a b) {H : is_iso f} : is_retraction f :=
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is_retraction.mk !compose_inverse
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theorem id_is_iso [instance] : is_iso (ID a) :=
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is_iso.mk !id_compose !id_compose
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theorem inverse_is_iso [instance] (f : a ⟶ b) {H : is_iso f} : is_iso (f⁻¹) :=
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is_iso.mk !compose_inverse !inverse_compose
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theorem left_inverse_eq_right_inverse {f : hom a b} {g g' : hom b a}
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(Hl : g ∘ f = id) (Hr : f ∘ g' = id) : g = g' :=
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calc
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g = g ∘ id : symm !id_right
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... = g ∘ f ∘ g' : {symm Hr}
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... = (g ∘ f) ∘ g' : !assoc
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... = id ∘ g' : {Hl}
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... = g' : !id_left
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theorem retraction_eq_intro {H : is_section f} (H2 : f ∘ i = id) : retraction_of f = i
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:= left_inverse_eq_right_inverse !retraction_compose H2
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theorem section_eq_intro {H : is_retraction f} (H2 : i ∘ f = id) : section_of f = i
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:= symm (left_inverse_eq_right_inverse H2 !compose_section)
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theorem inverse_eq_intro_right {H : is_iso f} (H2 : f ∘ i = id) : f⁻¹ = i
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:= left_inverse_eq_right_inverse !inverse_compose H2
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theorem inverse_eq_intro_left {H : is_iso f} (H2 : i ∘ f = id) : f⁻¹ = i
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:= symm (left_inverse_eq_right_inverse H2 !compose_inverse)
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theorem section_eq_retraction (f : a ⟶ b) {Hl : is_section f} {Hr : is_retraction f} :
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retraction_of f = section_of f :=
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retraction_eq_intro !compose_section
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theorem section_retraction_imp_iso (f : a ⟶ b) {Hl : is_section f} {Hr : is_retraction f}
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: is_iso f :=
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is_iso.mk (subst (section_eq_retraction f) (retraction_compose f)) (compose_section f)
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theorem inverse_unique (H H' : is_iso f) : @inverse _ _ f H = @inverse _ _ f H' :=
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inverse_eq_intro_left !inverse_compose
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theorem inverse_involutive (f : a ⟶ b) {H : is_iso f} : (f⁻¹)⁻¹ = f :=
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inverse_eq_intro_right !inverse_compose
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theorem retraction_of_id : retraction_of (ID a) = id :=
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retraction_eq_intro !id_compose
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theorem section_of_id : section_of (ID a) = id :=
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section_eq_intro !id_compose
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theorem iso_of_id : ID a⁻¹ = id :=
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inverse_eq_intro_left !id_compose
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theorem composition_is_section [instance] {Hf : is_section f} {Hg : is_section g}
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: is_section (g ∘ f) :=
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is_section.mk
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(calc
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(retraction_of f ∘ retraction_of g) ∘ g ∘ f
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= retraction_of f ∘ retraction_of g ∘ g ∘ f : symm !assoc
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... = retraction_of f ∘ (retraction_of g ∘ g) ∘ f : {assoc _ g f}
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... = retraction_of f ∘ id ∘ f : {!retraction_compose}
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... = retraction_of f ∘ f : {!id_left}
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... = id : !retraction_compose)
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theorem composition_is_retraction [instance] (Hf : is_retraction f) (Hg : is_retraction g)
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: is_retraction (g ∘ f) :=
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is_retraction.mk
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(calc
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(g ∘ f) ∘ section_of f ∘ section_of g = g ∘ f ∘ section_of f ∘ section_of g : symm !assoc
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... = g ∘ (f ∘ section_of f) ∘ section_of g : {assoc f _ _}
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... = g ∘ id ∘ section_of g : {!compose_section}
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... = g ∘ section_of g : {!id_left}
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... = id : !compose_section)
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theorem composition_is_inverse [instance] (Hf : is_iso f) (Hg : is_iso g) : is_iso (g ∘ f) :=
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!section_retraction_imp_iso
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inductive isomorphic (a b : ob) : Type := mk : ∀(g : a ⟶ b) {H : is_iso g}, isomorphic a b
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end -- remove
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namespace isomorphic
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section --remove
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parameters {ob : Type} {C : category ob} include C --remove
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variables {a b c d : ob} {h : @hom ob C c d} {g : @hom ob C b c} {f : @hom ob C a b} --remove
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open relation
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-- should these be coercions?
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definition iso [coercion] (H : isomorphic a b) : hom a b :=
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isomorphic.rec (λg h, g) H
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theorem is_iso [coercion][instance] (H : isomorphic a b) : is_iso (isomorphic.iso H) :=
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isomorphic.rec (λg h, h) H
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infix `≅`:50 := morphism.isomorphic -- why does "isomorphic" not work here?
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theorem refl (a : ob) : a ≅ a := mk id
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theorem symm ⦃a b : ob⦄ (H : a ≅ b) : b ≅ a := mk (inverse (iso H))
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theorem trans ⦃a b c : ob⦄ (H1 : a ≅ b) (H2 : b ≅ c) : a ≅ c := mk (iso H2 ∘ iso H1)
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theorem is_equivalence_eq [instance] (T : Type) : is_equivalence isomorphic :=
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is_equivalence.mk (is_reflexive.mk refl) (is_symmetric.mk symm) (is_transitive.mk trans)
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end -- remove
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end isomorphic
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section --remove
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parameters {ob : Type} {C : category ob} include C --remove
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variables {a b c d : ob} {h : @hom ob C c d} {g : @hom ob C b c} {f : @hom ob C a b} --remove
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--remove implicit arguments below
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inductive is_mono [class] (f : @hom ob C a b) : Prop :=
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mk : (∀c (g h : hom c a), f ∘ g = f ∘ h → g = h) → is_mono f
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inductive is_epi [class] (f : @hom ob C a b) : Prop :=
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mk : (∀c (g h : hom b c), g ∘ f = h ∘ f → g = h) → is_epi f
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theorem mono_elim {H : is_mono f} {g h : c ⟶ a} (H2 : f ∘ g = f ∘ h) : g = h
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:= is_mono.rec (λH3, H3 c g h H2) H
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theorem epi_elim {H : is_epi f} {g h : b ⟶ c} (H2 : g ∘ f = h ∘ f) : g = h
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:= is_epi.rec (λH3, H3 c g h H2) H
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theorem section_is_mono [instance] (f : hom a b) {H : is_section f} : is_mono f :=
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is_mono.mk
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(λ c g h H,
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calc
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g = id ∘ g : symm !id_left
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... = (retraction_of f ∘ f) ∘ g : {symm (retraction_compose f)}
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... = retraction_of f ∘ f ∘ g : symm !assoc
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... = retraction_of f ∘ f ∘ h : {H}
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... = (retraction_of f ∘ f) ∘ h : !assoc
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... = id ∘ h : {retraction_compose f}
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... = h : !id_left)
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theorem retraction_is_epi [instance] (f : hom a b) {H : is_retraction f} : is_epi f :=
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is_epi.mk
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(λ c g h H,
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calc
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g = g ∘ id : symm !id_right
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... = g ∘ f ∘ section_of f : {symm (compose_section f)}
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... = (g ∘ f) ∘ section_of f : !assoc
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... = (h ∘ f) ∘ section_of f : {H}
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... = h ∘ f ∘ section_of f : symm !assoc
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... = h ∘ id : {compose_section f}
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... = h : !id_right)
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--these theorems are now proven automatically using type classes
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--should they be instances?
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theorem id_is_mono : is_mono (ID a)
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theorem id_is_epi : is_epi (ID a)
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theorem composition_is_mono [instance] {Hf : is_mono f} {Hg : is_mono g} : is_mono (g ∘ f) :=
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is_mono.mk
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(λ d h₁ h₂ H,
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have H2 : g ∘ (f ∘ h₁) = g ∘ (f ∘ h₂), from symm (assoc g f h₁) ▸ symm (assoc g f h₂) ▸ H,
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mono_elim (mono_elim H2))
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theorem composition_is_epi [instance] {Hf : is_epi f} {Hg : is_epi g} : is_epi (g ∘ f) :=
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is_epi.mk
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(λ d h₁ h₂ H,
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have H2 : (h₁ ∘ g) ∘ f = (h₂ ∘ g) ∘ f, from assoc h₁ g f ▸ assoc h₂ g f ▸ H,
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epi_elim (epi_elim H2))
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end
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--rewrite lemmas for inverses, modified from
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--https://github.com/JasonGross/HoTT-categories/blob/master/theories/Categories/Category/Morphisms.v
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namespace iso
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section
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parameters {ob : Type} {C : category ob} include C
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variables {a b c d : ob} (f : @hom ob C b a)
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(r : @hom ob C c d) (q : @hom ob C b c) (p : @hom ob C a b)
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(g : @hom ob C d c)
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variable {Hq : is_iso q} include Hq
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theorem compose_pV : q ∘ q⁻¹ = id := !compose_inverse
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theorem compose_Vp : q⁻¹ ∘ q = id := !inverse_compose
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theorem compose_V_pp : q⁻¹ ∘ (q ∘ p) = p :=
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calc
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q⁻¹ ∘ (q ∘ p) = (q⁻¹ ∘ q) ∘ p : assoc (q⁻¹) q p
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... = id ∘ p : {inverse_compose q}
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... = p : id_left p
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theorem compose_p_Vp : q ∘ (q⁻¹ ∘ g) = g :=
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calc
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q ∘ (q⁻¹ ∘ g) = (q ∘ q⁻¹) ∘ g : assoc q (q⁻¹) g
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... = id ∘ g : {compose_inverse q}
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... = g : id_left g
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theorem compose_pp_V : (r ∘ q) ∘ q⁻¹ = r :=
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calc
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(r ∘ q) ∘ q⁻¹ = r ∘ q ∘ q⁻¹ : assoc r q (q⁻¹)⁻¹
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... = r ∘ id : {compose_inverse q}
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... = r : id_right r
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theorem compose_pV_p : (f ∘ q⁻¹) ∘ q = f :=
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calc
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(f ∘ q⁻¹) ∘ q = f ∘ q⁻¹ ∘ q : assoc f (q⁻¹) q⁻¹
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... = f ∘ id : {inverse_compose q}
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... = f : id_right f
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theorem inv_pp {H' : is_iso p} : (q ∘ p)⁻¹ = p⁻¹ ∘ q⁻¹ :=
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have H1 : (p⁻¹ ∘ q⁻¹) ∘ q ∘ p = p⁻¹ ∘ (q⁻¹ ∘ (q ∘ p)), from assoc (p⁻¹) (q⁻¹) (q ∘ p)⁻¹,
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have H2 : (p⁻¹) ∘ (q⁻¹ ∘ (q ∘ p)) = p⁻¹ ∘ p, from congr_arg _ (compose_V_pp q p),
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have H3 : p⁻¹ ∘ p = id, from inverse_compose p,
|
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inverse_eq_intro_left (H1 ⬝ H2 ⬝ H3)
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--the following calc proof is hard for the unifier (needs ~90k steps)
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-- inverse_eq_intro_left
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-- (calc
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-- (p⁻¹ ∘ (q⁻¹)) ∘ q ∘ p = p⁻¹ ∘ (q⁻¹ ∘ (q ∘ p)) : assoc (p⁻¹) (q⁻¹) (q ∘ p)⁻¹
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-- ... = (p⁻¹) ∘ p : congr_arg (λx, p⁻¹ ∘ x) (compose_V_pp q p)
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-- ... = id : inverse_compose p)
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theorem inv_Vp {H' : is_iso g} : (q⁻¹ ∘ g)⁻¹ = g⁻¹ ∘ q := inverse_involutive q ▸ inv_pp (q⁻¹) g
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theorem inv_pV {H' : is_iso f} : (q ∘ f⁻¹)⁻¹ = f ∘ q⁻¹ := inverse_involutive f ▸ inv_pp q (f⁻¹)
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theorem inv_VV {H' : is_iso r} : (q⁻¹ ∘ r⁻¹)⁻¹ = r ∘ q := inverse_involutive r ▸ inv_Vp q (r⁻¹)
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end
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section
|
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parameters {ob : Type} {C : category ob} include C
|
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variables {d c b a : ob}
|
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{i : @hom ob C b c} {f : @hom ob C b a}
|
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{r : @hom ob C c d} {q : @hom ob C b c} {p : @hom ob C a b}
|
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{g : @hom ob C d c} {h : @hom ob C c b}
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{x : @hom ob C b d} {z : @hom ob C a c}
|
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{y : @hom ob C d b} {w : @hom ob C c a}
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variable {Hq : is_iso q} include Hq
|
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|
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theorem moveR_Mp (H : y = q⁻¹ ∘ g) : q ∘ y = g := H⁻¹ ▸ compose_p_Vp q g
|
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theorem moveR_pM (H : w = f ∘ q⁻¹) : w ∘ q = f := H⁻¹ ▸ compose_pV_p f q
|
||||
theorem moveR_Vp (H : z = q ∘ p) : q⁻¹ ∘ z = p := H⁻¹ ▸ compose_V_pp q p
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theorem moveR_pV (H : x = r ∘ q) : x ∘ q⁻¹ = r := H⁻¹ ▸ compose_pp_V r q
|
||||
theorem moveL_Mp (H : q⁻¹ ∘ g = y) : g = q ∘ y := moveR_Mp (H⁻¹)⁻¹
|
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theorem moveL_pM (H : f ∘ q⁻¹ = w) : f = w ∘ q := moveR_pM (H⁻¹)⁻¹
|
||||
theorem moveL_Vp (H : q ∘ p = z) : p = q⁻¹ ∘ z := moveR_Vp (H⁻¹)⁻¹
|
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theorem moveL_pV (H : r ∘ q = x) : r = x ∘ q⁻¹ := moveR_pV (H⁻¹)⁻¹
|
||||
theorem moveL_1V (H : h ∘ q = id) : h = q⁻¹ := inverse_eq_intro_left H⁻¹
|
||||
theorem moveL_V1 (H : q ∘ h = id) : h = q⁻¹ := inverse_eq_intro_right H⁻¹
|
||||
theorem moveL_1M (H : i ∘ q⁻¹ = id) : i = q := moveL_1V H ⬝ inverse_involutive q
|
||||
theorem moveL_M1 (H : q⁻¹ ∘ i = id) : i = q := moveL_V1 H ⬝ inverse_involutive q
|
||||
theorem moveR_1M (H : id = i ∘ q⁻¹) : q = i := moveL_1M (H⁻¹)⁻¹
|
||||
theorem moveR_M1 (H : id = q⁻¹ ∘ i) : q = i := moveL_M1 (H⁻¹)⁻¹
|
||||
theorem moveR_1V (H : id = h ∘ q) : q⁻¹ = h := moveL_1V (H⁻¹)⁻¹
|
||||
theorem moveR_V1 (H : id = q ∘ h) : q⁻¹ = h := moveL_V1 (H⁻¹)⁻¹
|
||||
end
|
||||
end iso
|
||||
|
||||
end morphism
|
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16
library/algebra/category/yoneda.lean
Normal file
16
library/algebra/category/yoneda.lean
Normal file
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|
@ -0,0 +1,16 @@
|
|||
-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
|
||||
-- Released under Apache 2.0 license as described in the file LICENSE.
|
||||
-- Author: Floris van Doorn
|
||||
|
||||
import .basic .constructions
|
||||
|
||||
open eq eq.ops category functor category.ops
|
||||
|
||||
namespace yoneda
|
||||
--representable functor
|
||||
section
|
||||
parameters {ob : Type} {C : category ob}
|
||||
-- definition Hom : Cᵒᵖ × C ⇒ type :=
|
||||
-- sorry
|
||||
end
|
||||
end yoneda
|
||||
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Reference in a new issue