275 lines
12 KiB
Text
275 lines
12 KiB
Text
/-
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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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Module: algebra.category.morphism
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Author: Floris van Doorn
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-/
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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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variables {ob : Type} [C : category ob] include C
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variables {a b c : ob} {g : b ⟶ c} {f : a ⟶ b} {h : b ⟶ a}
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inductive is_section [class] (f : a ⟶ b) : Type
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:= mk : ∀{g}, g ∘ f = id → is_section f
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inductive is_retraction [class] (f : a ⟶ b) : Type
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:= mk : ∀{g}, f ∘ g = id → is_retraction f
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inductive is_iso [class] (f : a ⟶ b) : Type
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:= mk : ∀{g}, g ∘ f = id → f ∘ g = id → is_iso f
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attribute is_iso [multiple-instances]
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definition retraction_of (f : 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 : 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 : 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 : 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 : 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 : 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 : 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 : 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 : 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 : 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 : (id_right g)⁻¹
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... = g ∘ f ∘ g' : by rewrite -Hr
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... = (g ∘ f) ∘ g' : by rewrite assoc
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... = id ∘ g' : by rewrite Hl
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... = g' : by rewrite id_left
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theorem retraction_eq_intro [H : is_section f] (H2 : f ∘ h = id) : retraction_of f = h :=
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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 : h ∘ f = id) : section_of f = h :=
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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 ∘ h = id) : f⁻¹ = h :=
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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 : h ∘ f = id) : f⁻¹ = h :=
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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 : by rewrite -assoc
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... = retraction_of f ∘ (retraction_of g ∘ g) ∘ f : by rewrite (assoc _ g f)
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... = retraction_of f ∘ id ∘ f : by rewrite retraction_compose
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... = retraction_of f ∘ f : by rewrite id_left
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... = id : by rewrite 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
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= g ∘ f ∘ section_of f ∘ section_of g : by rewrite -assoc
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... = g ∘ (f ∘ section_of f) ∘ section_of g : by rewrite -assoc
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... = g ∘ id ∘ section_of g : by rewrite compose_section
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... = g ∘ section_of g : by rewrite id_left
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... = id : by rewrite 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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structure isomorphic (a b : ob) :=
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(iso : a ⟶ b)
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[is_iso : is_iso iso]
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infix `≅`:50 := morphism.isomorphic
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namespace isomorphic
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open relation
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attribute is_iso [instance]
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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 refl symm trans
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end isomorphic
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inductive is_mono [class] (f : 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 : 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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match H with
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is_mono.mk H3 := H3 c g h H2
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end
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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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match H with
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is_epi.mk H3 := H3 c g h H2
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end
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theorem section_is_mono [instance] (f : 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, calc
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g = id ∘ g : by rewrite id_left
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... = (retraction_of f ∘ f) ∘ g : by rewrite -retraction_compose
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... = (retraction_of f ∘ f) ∘ h : by rewrite [-assoc, H, -assoc]
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... = id ∘ h : by rewrite retraction_compose
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... = h : by rewrite id_left)
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theorem retraction_is_epi [instance] (f : 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, calc
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g = g ∘ id : by rewrite id_right
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... = g ∘ f ∘ section_of f : by rewrite -compose_section
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... = h ∘ f ∘ section_of f : by rewrite [assoc, H, -assoc]
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... = h ∘ id : by rewrite compose_section
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... = h : by rewrite 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),
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begin
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rewrite *assoc, exact H
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end,
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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,
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begin
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rewrite -*assoc, exact H
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end,
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epi_elim (epi_elim H2))
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end morphism
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namespace morphism
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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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variables {ob : Type} [C : category ob] include C
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variables {a b c d : ob}
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variables (f : b ⟶ a) (r : c ⟶ d) (q : b ⟶ c) (p : a ⟶ b)
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variables (g : 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 : by rewrite assoc
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... = id ∘ p : by rewrite inverse_compose
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... = p : by rewrite id_left
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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 : by rewrite assoc
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... = id ∘ g : by rewrite compose_inverse
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... = g : by rewrite id_left
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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 : by rewrite assoc
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... = r ∘ id : by rewrite compose_inverse
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... = r : by rewrite id_right
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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 : by rewrite assoc
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... = f ∘ id : by rewrite inverse_compose
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... = f : by rewrite id_right
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theorem inv_pp [H' : is_iso p] : (q ∘ p)⁻¹ = p⁻¹ ∘ q⁻¹ :=
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inverse_eq_intro_left
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(show (p⁻¹ ∘ (q⁻¹)) ∘ q ∘ p = id, from
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by rewrite [-assoc, compose_V_pp, inverse_compose])
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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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variables {ob : Type} {C : category ob} include C
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variables {d c b a : ob}
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variables {i : b ⟶ c} {f : b ⟶ a}
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{r : c ⟶ d} {q : b ⟶ c} {p : a ⟶ b}
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{g : d ⟶ c} {h : c ⟶ b}
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{x : b ⟶ d} {z : a ⟶ c}
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{y : d ⟶ b} {w : c ⟶ a}
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variable [Hq : is_iso q] include Hq
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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
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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
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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
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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
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theorem moveL_1V (H : h ∘ q = id) : h = q⁻¹ := (inverse_eq_intro_left H)⁻¹
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theorem moveL_V1 (H : q ∘ h = id) : h = q⁻¹ := (inverse_eq_intro_right H)⁻¹
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theorem moveL_1M (H : i ∘ q⁻¹ = id) : i = q := moveL_1V H ⬝ inverse_involutive q
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theorem moveL_M1 (H : q⁻¹ ∘ i = id) : i = q := moveL_V1 H ⬝ inverse_involutive q
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theorem moveR_1M (H : id = i ∘ q⁻¹) : q = i := (moveL_1M (H⁻¹))⁻¹
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theorem moveR_M1 (H : id = q⁻¹ ∘ i) : q = i := (moveL_M1 (H⁻¹))⁻¹
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theorem moveR_1V (H : id = h ∘ q) : q⁻¹ = h := (moveL_1V (H⁻¹))⁻¹
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theorem moveR_V1 (H : id = q ∘ h) : q⁻¹ = h := (moveL_V1 (H⁻¹))⁻¹
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end
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end iso
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end morphism
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