lean2/library/algebra/category/morphism.lean

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-- 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 algebra.relation algebra.binary
open eq eq.ops category
namespace morphism
section
variables {ob : Type} [C : category ob] include C
variables {a b c d : ob} {h : hom c d} {g : hom b c} {f : hom a b} {i : hom b a}
inductive is_section [class] (f : hom a b) : Type
:= mk : ∀{g}, g ∘ f = id → is_section f
inductive is_retraction [class] (f : hom a b) : Type
:= mk : ∀{g}, f ∘ g = id → is_retraction f
inductive is_iso [class] (f : hom a b) : Type
:= mk : ∀{g}, g ∘ f = id → f ∘ g = id → is_iso f
--remove implicit arguments in above lines
definition retraction_of (f : hom a b) [H : is_section f] : hom b a :=
is_section.rec (λg h, g) H
definition section_of (f : hom a b) [H : is_retraction f] : hom b a :=
is_retraction.rec (λg h, g) H
definition inverse (f : hom a b) [H : is_iso f] : hom b a :=
is_iso.rec (λg h1 h2, g) H
notation H ⁻¹ := inverse H
theorem inverse_compose (f : hom a b) [H : is_iso f] : f⁻¹ ∘ f = id :=
is_iso.rec (λg h1 h2, h1) H
theorem compose_inverse (f : hom a b) [H : is_iso f] : f ∘ f⁻¹ = id :=
is_iso.rec (λg h1 h2, h2) H
theorem retraction_compose (f : hom a b) [H : is_section f] : retraction_of f ∘ f = id :=
is_section.rec (λg h, h) H
theorem compose_section (f : hom a b) [H : is_retraction f] : f ∘ section_of f = id :=
is_retraction.rec (λg h, h) H
theorem iso_imp_retraction [instance] (f : hom a b) [H : is_iso f] : is_section f :=
is_section.mk !inverse_compose
theorem iso_imp_section [instance] (f : hom a b) [H : is_iso f] : is_retraction f :=
is_retraction.mk !compose_inverse
theorem id_is_iso [instance] : is_iso (ID a) :=
is_iso.mk !id_compose !id_compose
theorem inverse_is_iso [instance] (f : a ⟶ b) [H : is_iso f] : is_iso (f⁻¹) :=
is_iso.mk !compose_inverse !inverse_compose
theorem left_inverse_eq_right_inverse {f : hom a b} {g g' : hom b a}
(Hl : g ∘ f = id) (Hr : f ∘ g' = id) : g = g' :=
calc
g = g ∘ id : symm !id_right
... = g ∘ f ∘ g' : {symm Hr}
... = (g ∘ f) ∘ g' : !assoc
... = id ∘ g' : {Hl}
... = g' : !id_left
theorem retraction_eq_intro [H : is_section f] (H2 : f ∘ i = id) : retraction_of f = i
:= left_inverse_eq_right_inverse !retraction_compose H2
theorem section_eq_intro [H : is_retraction f] (H2 : i ∘ f = id) : section_of f = i
:= symm (left_inverse_eq_right_inverse H2 !compose_section)
theorem inverse_eq_intro_right [H : is_iso f] (H2 : f ∘ i = id) : f⁻¹ = i
:= left_inverse_eq_right_inverse !inverse_compose H2
theorem inverse_eq_intro_left [H : is_iso f] (H2 : i ∘ f = id) : f⁻¹ = i
:= symm (left_inverse_eq_right_inverse H2 !compose_inverse)
theorem section_eq_retraction (f : a ⟶ b) [Hl : is_section f] [Hr : is_retraction f] :
retraction_of f = section_of f :=
retraction_eq_intro !compose_section
theorem section_retraction_imp_iso (f : a ⟶ b) [Hl : is_section f] [Hr : is_retraction f]
: is_iso f :=
is_iso.mk (subst (section_eq_retraction f) (retraction_compose f)) (compose_section f)
theorem inverse_unique (H H' : is_iso f) : @inverse _ _ _ _ f H = @inverse _ _ _ _ f H' :=
inverse_eq_intro_left !inverse_compose
theorem inverse_involutive (f : a ⟶ b) [H : is_iso f] : (f⁻¹)⁻¹ = f :=
inverse_eq_intro_right !inverse_compose
theorem retraction_of_id : retraction_of (ID a) = id :=
retraction_eq_intro !id_compose
theorem section_of_id : section_of (ID a) = id :=
section_eq_intro !id_compose
theorem iso_of_id : ID a⁻¹ = id :=
inverse_eq_intro_left !id_compose
theorem composition_is_section [instance] [Hf : is_section f] [Hg : is_section g]
: is_section (g ∘ f) :=
is_section.mk
(calc
(retraction_of f ∘ retraction_of g) ∘ g ∘ f
= retraction_of f ∘ retraction_of g ∘ g ∘ f : symm (assoc _ _ (g ∘ f))
... = retraction_of f ∘ (retraction_of g ∘ g) ∘ f : {assoc _ g f}
... = retraction_of f ∘ id ∘ f : {retraction_compose g}
... = retraction_of f ∘ f : {id_left f}
... = id : !retraction_compose)
theorem composition_is_retraction [instance] (Hf : is_retraction f) (Hg : is_retraction g)
: is_retraction (g ∘ f) :=
is_retraction.mk
(calc
(g ∘ f) ∘ section_of f ∘ section_of g = g ∘ f ∘ section_of f ∘ section_of g : symm !assoc
... = g ∘ (f ∘ section_of f) ∘ section_of g : {assoc f _ _}
... = g ∘ id ∘ section_of g : {compose_section f}
... = g ∘ section_of g : {id_left (section_of g)}
... = id : !compose_section)
theorem composition_is_inverse [instance] (Hf : is_iso f) (Hg : is_iso g) : is_iso (g ∘ f) :=
!section_retraction_imp_iso
inductive isomorphic (a b : ob) : Type := mk : ∀(g : a ⟶ b) [H : is_iso g], isomorphic a b
end -- remove
namespace isomorphic
section --remove
variables {ob : Type} {C : category ob} include C --remove
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
open relation
-- should these be coercions?
definition iso [coercion] (H : isomorphic a b) : hom a b :=
isomorphic.rec (λg h, g) H
theorem is_iso [coercion][instance] (H : isomorphic a b) : is_iso (isomorphic.iso H) :=
isomorphic.rec (λg h, h) H
infix `≅`:50 := morphism.isomorphic -- why does "isomorphic" not work here?
theorem refl (a : ob) : a ≅ a := mk id
theorem symm ⦃a b : ob⦄ (H : a ≅ b) : b ≅ a := mk (inverse (iso H))
theorem trans ⦃a b c : ob⦄ (H1 : a ≅ b) (H2 : b ≅ c) : a ≅ c := mk (iso H2 ∘ iso H1)
theorem is_equivalence_eq [instance] (T : Type) : is_equivalence isomorphic :=
is_equivalence.mk (is_reflexive.mk refl) (is_symmetric.mk symm) (is_transitive.mk trans)
end -- remove
end isomorphic
section --remove
variables {ob : Type} {C : category ob} include C --remove
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
--remove implicit arguments below
inductive is_mono [class] (f : @hom ob C a b) : Prop :=
mk : (∀c (g h : hom c a), f ∘ g = f ∘ h → g = h) → is_mono f
inductive is_epi [class] (f : @hom ob C a b) : Prop :=
mk : (∀c (g h : hom b c), g ∘ f = h ∘ f → g = h) → is_epi f
theorem mono_elim [H : is_mono f] {g h : c ⟶ a} (H2 : f ∘ g = f ∘ h) : g = h
:= is_mono.rec (λH3, H3 c g h H2) H
theorem epi_elim [H : is_epi f] {g h : b ⟶ c} (H2 : g ∘ f = h ∘ f) : g = h
:= is_epi.rec (λH3, H3 c g h H2) H
theorem section_is_mono [instance] (f : hom a b) [H : is_section f] : is_mono f :=
is_mono.mk
(λ c g h H,
calc
g = id ∘ g : symm !id_left
... = (retraction_of f ∘ f) ∘ g : {symm (retraction_compose f)}
... = retraction_of f ∘ f ∘ g : symm !assoc
... = retraction_of f ∘ f ∘ h : {H}
... = (retraction_of f ∘ f) ∘ h : !assoc
... = id ∘ h : {retraction_compose f}
... = h : !id_left)
theorem retraction_is_epi [instance] (f : hom a b) [H : is_retraction f] : is_epi f :=
is_epi.mk
(λ c g h H,
calc
g = g ∘ id : symm !id_right
... = g ∘ f ∘ section_of f : {symm (compose_section f)}
... = (g ∘ f) ∘ section_of f : !assoc
... = (h ∘ f) ∘ section_of f : {H}
... = h ∘ f ∘ section_of f : symm !assoc
... = h ∘ id : {compose_section f}
... = h : !id_right)
--these theorems are now proven automatically using type classes
--should they be instances?
theorem id_is_mono : is_mono (ID a)
theorem id_is_epi : is_epi (ID a)
theorem composition_is_mono [instance] [Hf : is_mono f] [Hg : is_mono g] : is_mono (g ∘ f) :=
is_mono.mk
(λ d h₁ h₂ H,
have H2 : g ∘ (f ∘ h₁) = g ∘ (f ∘ h₂), from symm (assoc g f h₁) ▸ symm (assoc g f h₂) ▸ H,
mono_elim (mono_elim H2))
theorem composition_is_epi [instance] [Hf : is_epi f] [Hg : is_epi g] : is_epi (g ∘ f) :=
is_epi.mk
(λ d h₁ h₂ H,
have H2 : (h₁ ∘ g) ∘ f = (h₂ ∘ g) ∘ f, from assoc h₁ g f ▸ assoc h₂ g f ▸ H,
epi_elim (epi_elim H2))
end
--rewrite lemmas for inverses, modified from
--https://github.com/JasonGross/HoTT-categories/blob/master/theories/Categories/Category/Morphisms.v
namespace iso
section
variables {ob : Type} [C : category ob] include C
variables {a b c d : ob} (f : @hom ob C b a)
(r : @hom ob C c d) (q : @hom ob C b c) (p : @hom ob C a b)
(g : @hom ob C d c)
variable [Hq : is_iso q] include Hq
theorem compose_pV : q ∘ q⁻¹ = id := !compose_inverse
theorem compose_Vp : q⁻¹ ∘ q = id := !inverse_compose
theorem compose_V_pp : q⁻¹ ∘ (q ∘ p) = p :=
calc
q⁻¹ ∘ (q ∘ p) = (q⁻¹ ∘ q) ∘ p : assoc (q⁻¹) q p
... = id ∘ p : {inverse_compose q}
... = p : id_left p
theorem compose_p_Vp : q ∘ (q⁻¹ ∘ g) = g :=
calc
q ∘ (q⁻¹ ∘ g) = (q ∘ q⁻¹) ∘ g : assoc q (q⁻¹) g
... = id ∘ g : {compose_inverse q}
... = g : id_left g
theorem compose_pp_V : (r ∘ q) ∘ q⁻¹ = r :=
calc
(r ∘ q) ∘ q⁻¹ = r ∘ q ∘ q⁻¹ : assoc r q (q⁻¹)⁻¹
... = r ∘ id : {compose_inverse q}
... = r : id_right r
theorem compose_pV_p : (f ∘ q⁻¹) ∘ q = f :=
calc
(f ∘ q⁻¹) ∘ q = f ∘ q⁻¹ ∘ q : assoc f (q⁻¹) q⁻¹
... = f ∘ id : {inverse_compose q}
... = f : id_right f
theorem inv_pp [H' : is_iso p] : (q ∘ p)⁻¹ = p⁻¹ ∘ q⁻¹ :=
have H1 : (p⁻¹ ∘ q⁻¹) ∘ q ∘ p = p⁻¹ ∘ (q⁻¹ ∘ (q ∘ p)), from assoc (p⁻¹) (q⁻¹) (q ∘ p)⁻¹,
have H2 : (p⁻¹) ∘ (q⁻¹ ∘ (q ∘ p)) = p⁻¹ ∘ p, from congr_arg _ (compose_V_pp q p),
have H3 : p⁻¹ ∘ p = id, from inverse_compose p,
inverse_eq_intro_left (H1 ⬝ H2 ⬝ H3)
--the following calc proof is hard for the unifier (needs ~90k steps)
-- inverse_eq_intro_left
-- (calc
-- (p⁻¹ ∘ (q⁻¹)) ∘ q ∘ p = p⁻¹ ∘ (q⁻¹ ∘ (q ∘ p)) : assoc (p⁻¹) (q⁻¹) (q ∘ p)⁻¹
-- ... = (p⁻¹) ∘ p : congr_arg (λx, p⁻¹ ∘ x) (compose_V_pp q p)
-- ... = id : inverse_compose p)
theorem inv_Vp [H' : is_iso g] : (q⁻¹ ∘ g)⁻¹ = g⁻¹ ∘ q := inverse_involutive q ▸ inv_pp (q⁻¹) g
theorem inv_pV [H' : is_iso f] : (q ∘ f⁻¹)⁻¹ = f ∘ q⁻¹ := inverse_involutive f ▸ inv_pp q (f⁻¹)
theorem inv_VV [H' : is_iso r] : (q⁻¹ ∘ r⁻¹)⁻¹ = r ∘ q := inverse_involutive r ▸ inv_Vp q (r⁻¹)
end
section
variables {ob : Type} {C : category ob} include C
variables {d c b a : ob}
{i : @hom ob C b c} {f : @hom ob C b a}
{r : @hom ob C c d} {q : @hom ob C b c} {p : @hom ob C a b}
{g : @hom ob C d c} {h : @hom ob C c b}
{x : @hom ob C b d} {z : @hom ob C a c}
{y : @hom ob C d b} {w : @hom ob C c a}
variable [Hq : is_iso q] include Hq
theorem moveR_Mp (H : y = q⁻¹ ∘ g) : q ∘ y = g := H⁻¹ ▸ compose_p_Vp q g
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
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
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
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