lean2/hott/algebra/category/iso.hlean
Floris van Doorn 17ccc283a9 feat(hott): move basic theorems from colimit development to library.
Most notable changes:
rename apo011 -> apd011 and apd011 -> apdt011
make an argument of pathover_of_eq explicit
2016-07-09 10:20:22 -07:00

417 lines
16 KiB
Text

/-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Floris van Doorn, Jakob von Raumer
-/
import .precategory types.sigma arity
open eq category prod equiv is_equiv sigma sigma.ops is_trunc
namespace iso
structure split_mono [class] {ob : Type} [C : precategory ob] {a b : ob} (f : a ⟶ b) :=
{retraction_of : b ⟶ a}
(retraction_comp : retraction_of ∘ f = id)
structure split_epi [class] {ob : Type} [C : precategory ob] {a b : ob} (f : a ⟶ b) :=
{section_of : b ⟶ a}
(comp_section : f ∘ section_of = id)
structure is_iso [class] {ob : Type} [C : precategory ob] {a b : ob} (f : a ⟶ b) :=
(inverse : b ⟶ a)
(left_inverse : inverse ∘ f = id)
(right_inverse : f ∘ inverse = id)
attribute is_iso.inverse [reducible]
open split_mono split_epi is_iso
abbreviation retraction_of [unfold 6] := @split_mono.retraction_of
abbreviation retraction_comp [unfold 6] := @split_mono.retraction_comp
abbreviation section_of [unfold 6] := @split_epi.section_of
abbreviation comp_section [unfold 6] := @split_epi.comp_section
abbreviation inverse [unfold 6] := @is_iso.inverse
abbreviation left_inverse [unfold 6] := @is_iso.left_inverse
abbreviation right_inverse [unfold 6] := @is_iso.right_inverse
postfix ⁻¹ := inverse
--a second notation for the inverse, which is not overloaded
postfix [parsing_only] `⁻¹ʰ`:std.prec.max_plus := inverse -- input using \-1h
variables {ob : Type} [C : precategory ob]
variables {a b c : ob} {g : b ⟶ c} {f : a ⟶ b} {h : b ⟶ a}
include C
definition split_mono_of_is_iso [constructor] [instance] [priority 300]
(f : a ⟶ b) [H : is_iso f] : split_mono f :=
split_mono.mk !left_inverse
definition split_epi_of_is_iso [constructor] [instance] [priority 300]
(f : a ⟶ b) [H : is_iso f] : split_epi f :=
split_epi.mk !right_inverse
definition is_iso_id [constructor] [instance] [priority 500] (a : ob) : is_iso (ID a) :=
is_iso.mk _ !id_id !id_id
definition is_iso_inverse [constructor] [instance] [priority 200] (f : a ⟶ b) {H : is_iso f}
: is_iso f⁻¹ :=
is_iso.mk _ !right_inverse !left_inverse
theorem left_inverse_eq_right_inverse {f : a ⟶ b} {g g' : hom b a}
(Hl : g ∘ f = id) (Hr : f ∘ g' = id) : g = g' :=
by rewrite [-(id_right g), -Hr, assoc, Hl, id_left]
theorem retraction_eq [H : split_mono f] (H2 : f ∘ h = id) : retraction_of f = h :=
left_inverse_eq_right_inverse !retraction_comp H2
theorem section_eq [H : split_epi f] (H2 : h ∘ f = id) : section_of f = h :=
(left_inverse_eq_right_inverse H2 !comp_section)⁻¹
theorem inverse_eq_right [H : is_iso f] (H2 : f ∘ h = id) : f⁻¹ = h :=
left_inverse_eq_right_inverse !left_inverse H2
theorem inverse_eq_left [H : is_iso f] (H2 : h ∘ f = id) : f⁻¹ = h :=
(left_inverse_eq_right_inverse H2 !right_inverse)⁻¹
theorem retraction_eq_section (f : a ⟶ b) [Hl : split_mono f] [Hr : split_epi f] :
retraction_of f = section_of f :=
retraction_eq !comp_section
definition is_iso_of_split_epi_of_split_mono [constructor] (f : a ⟶ b)
[Hl : split_mono f] [Hr : split_epi f] : is_iso f :=
is_iso.mk _ ((retraction_eq_section f) ▸ (retraction_comp f)) (comp_section f)
theorem inverse_unique (H H' : is_iso f) : @inverse _ _ _ _ f H = @inverse _ _ _ _ f H' :=
@inverse_eq_left _ _ _ _ _ _ H !left_inverse
theorem inverse_involutive (f : a ⟶ b) [H : is_iso f] [H : is_iso (f⁻¹)]
: (f⁻¹)⁻¹ = f :=
inverse_eq_right !left_inverse
theorem inverse_eq_inverse {f g : a ⟶ b} [H : is_iso f] [H : is_iso g] (p : f = g)
: f⁻¹ = g⁻¹ :=
by cases p;apply inverse_unique
theorem retraction_id (a : ob) : retraction_of (ID a) = id :=
retraction_eq !id_id
theorem section_id (a : ob) : section_of (ID a) = id :=
section_eq !id_id
theorem id_inverse (a : ob) [H : is_iso (ID a)] : (ID a)⁻¹ = id :=
inverse_eq_left !id_id
definition split_mono_comp [constructor] [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b)
[Hf : split_mono f] [Hg : split_mono g] : split_mono (g ∘ f) :=
split_mono.mk
(show (retraction_of f ∘ retraction_of g) ∘ g ∘ f = id,
by rewrite [-assoc, assoc _ g f, retraction_comp, id_left, retraction_comp])
definition split_epi_comp [constructor] [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b)
[Hf : split_epi f] [Hg : split_epi g] : split_epi (g ∘ f) :=
split_epi.mk
(show (g ∘ f) ∘ section_of f ∘ section_of g = id,
by rewrite [-assoc, {f ∘ _}assoc, comp_section, id_left, comp_section])
definition is_iso_comp [constructor] [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b)
[Hf : is_iso f] [Hg : is_iso g] : is_iso (g ∘ f) :=
!is_iso_of_split_epi_of_split_mono
theorem is_prop_is_iso [instance] (f : hom a b) : is_prop (is_iso f) :=
begin
apply is_prop.mk, intro H H',
cases H with g li ri, cases H' with g' li' ri',
fapply (apd0111 (@is_iso.mk ob C a b f)),
apply left_inverse_eq_right_inverse,
apply li,
apply ri',
apply is_prop.elimo,
apply is_prop.elimo,
end
end iso open iso
/- isomorphic objects -/
structure iso {ob : Type} [C : precategory ob] (a b : ob) :=
(to_hom : hom a b)
(struct : is_iso to_hom)
infix ` ≅ `:50 := iso
notation c ` ≅[`:50 C:0 `] `:0 c':50 := @iso C _ c c'
attribute iso.struct [instance] [priority 2000]
namespace iso
variables {ob : Type} [C : precategory ob]
variables {a b c : ob} {g : b ⟶ c} {f : a ⟶ b} {h : b ⟶ a}
include C
attribute to_hom [coercion]
protected definition MK [constructor] (f : a ⟶ b) (g : b ⟶ a)
(H1 : g ∘ f = id) (H2 : f ∘ g = id) :=
@(mk f) (is_iso.mk _ H1 H2)
variable {C}
definition to_inv [reducible] [unfold 5] (f : a ≅ b) : b ⟶ a := (to_hom f)⁻¹
definition to_left_inverse [unfold 5] (f : a ≅ b) : (to_hom f)⁻¹ ∘ (to_hom f) = id :=
left_inverse (to_hom f)
definition to_right_inverse [unfold 5] (f : a ≅ b) : (to_hom f) ∘ (to_hom f)⁻¹ = id :=
right_inverse (to_hom f)
variable [C]
protected definition refl [constructor] (a : ob) : a ≅ a :=
mk (ID a) _
protected definition symm [constructor] ⦃a b : ob⦄ (H : a ≅ b) : b ≅ a :=
mk (to_hom H)⁻¹ _
protected definition trans [constructor] ⦃a b c : ob⦄ (H1 : a ≅ b) (H2 : b ≅ c) : a ≅ c :=
mk (to_hom H2 ∘ to_hom H1) _
infixl ` ⬝i `:75 := iso.trans
postfix [parsing_only] `⁻¹ⁱ`:(max + 1) := iso.symm
definition change_hom [constructor] (H : a ≅ b) (f : a ⟶ b) (p : to_hom H = f) : a ≅ b :=
iso.MK f (to_inv H) (p ▸ to_left_inverse H) (p ▸ to_right_inverse H)
definition change_inv [constructor] (H : a ≅ b) (g : b ⟶ a) (p : to_inv H = g) : a ≅ b :=
iso.MK (to_hom H) g (p ▸ to_left_inverse H) (p ▸ to_right_inverse H)
definition iso_mk_eq {f f' : a ⟶ b} [H : is_iso f] [H' : is_iso f'] (p : f = f')
: iso.mk f _ = iso.mk f' _ :=
apd011 iso.mk p !is_prop.elimo
variable {C}
definition iso_eq {f f' : a ≅ b} (p : to_hom f = to_hom f') : f = f' :=
by (cases f; cases f'; apply (iso_mk_eq p))
variable [C]
-- The structure for isomorphism can be characterized up to equivalence by a sigma type.
protected definition sigma_char ⦃a b : ob⦄ : (Σ (f : hom a b), is_iso f) ≃ (a ≅ b) :=
begin
fapply (equiv.mk),
{intro S, apply iso.mk, apply (S.2)},
{fapply adjointify,
{intro p, cases p with f H, exact sigma.mk f H},
{intro p, cases p, apply idp},
{intro S, cases S, apply idp}},
end
-- The type of isomorphisms between two objects is a set
definition is_set_iso [instance] : is_set (a ≅ b) :=
begin
apply is_trunc_is_equiv_closed,
apply equiv.to_is_equiv (!iso.sigma_char),
end
definition iso_of_eq [unfold 5] (p : a = b) : a ≅ b :=
eq.rec_on p (iso.refl a)
definition hom_of_eq [reducible] [unfold 5] (p : a = b) : a ⟶ b :=
iso.to_hom (iso_of_eq p)
definition inv_of_eq [reducible] [unfold 5] (p : a = b) : b ⟶ a :=
iso.to_inv (iso_of_eq p)
definition iso_of_eq_inv (p : a = b) : iso_of_eq p⁻¹ = iso.symm (iso_of_eq p) :=
eq.rec_on p idp
theorem hom_of_eq_inv (p : a = b) : hom_of_eq p⁻¹ = inv_of_eq p :=
eq.rec_on p idp
theorem inv_of_eq_inv (p : a = b) : inv_of_eq p⁻¹ = hom_of_eq p :=
eq.rec_on p idp
definition iso_of_eq_con (p : a = b) (q : b = c)
: iso_of_eq (p ⬝ q) = iso.trans (iso_of_eq p) (iso_of_eq q) :=
eq.rec_on q (eq.rec_on p (iso_eq !id_id⁻¹))
section
open funext
variables {X : Type} {x y : X} {F G : X → ob}
definition transport_hom_of_eq (p : F = G) (f : hom (F x) (F y))
: p ▸ f = hom_of_eq (apd10 p y) ∘ f ∘ inv_of_eq (apd10 p x) :=
by induction p; exact !id_leftright⁻¹
definition transport_hom_of_eq_right (p : x = y) (f : hom c (F x))
: p ▸ f = hom_of_eq (ap F p) ∘ f :=
by induction p; exact !id_left⁻¹
definition transport_hom_of_eq_left (p : x = y) (f : hom (F x) c)
: p ▸ f = f ∘ inv_of_eq (ap F p) :=
by induction p; exact !id_right⁻¹
definition transport_hom (p : F ~ G) (f : hom (F x) (F y))
: eq_of_homotopy p ▸ f = hom_of_eq (p y) ∘ f ∘ inv_of_eq (p x) :=
calc
eq_of_homotopy p ▸ f =
hom_of_eq (apd10 (eq_of_homotopy p) y) ∘ f ∘ inv_of_eq (apd10 (eq_of_homotopy p) x)
: transport_hom_of_eq
... = hom_of_eq (p y) ∘ f ∘ inv_of_eq (p x) : {right_inv apd10 p}
end
structure mono [class] (f : a ⟶ b) :=
(elim : ∀c (g h : hom c a), f ∘ g = f ∘ h → g = h)
structure epi [class] (f : a ⟶ b) :=
(elim : ∀c (g h : hom b c), g ∘ f = h ∘ f → g = h)
definition mono_of_split_mono [instance] (f : a ⟶ b) [H : split_mono f] : mono f :=
mono.mk
(λ c g h H,
calc
g = id ∘ g : by rewrite id_left
... = (retraction_of f ∘ f) ∘ g : by rewrite retraction_comp
... = (retraction_of f ∘ f) ∘ h : by rewrite [-assoc, H, -assoc]
... = id ∘ h : by rewrite retraction_comp
... = h : by rewrite id_left)
definition epi_of_split_epi [instance] (f : a ⟶ b) [H : split_epi f] : epi f :=
epi.mk
(λ c g h H,
calc
g = g ∘ id : by rewrite id_right
... = g ∘ f ∘ section_of f : by rewrite -(comp_section f)
... = h ∘ f ∘ section_of f : by rewrite [assoc, H, -assoc]
... = h ∘ id : by rewrite comp_section
... = h : by rewrite id_right)
definition mono_comp [instance] (g : b ⟶ c) (f : a ⟶ b) [Hf : mono f] [Hg : mono g]
: mono (g ∘ f) :=
mono.mk
(λ d h₁ h₂ H,
have H2 : g ∘ (f ∘ h₁) = g ∘ (f ∘ h₂),
begin
rewrite *assoc, exact H
end,
!mono.elim (!mono.elim H2))
definition epi_comp [instance] (g : b ⟶ c) (f : a ⟶ b) [Hf : epi f] [Hg : epi g]
: epi (g ∘ f) :=
epi.mk
(λ d h₁ h₂ H,
have H2 : (h₁ ∘ g) ∘ f = (h₂ ∘ g) ∘ f,
begin
rewrite -*assoc, exact H
end,
!epi.elim (!epi.elim H2))
end iso
attribute iso.refl [refl]
attribute iso.symm [symm]
attribute iso.trans [trans]
namespace iso
/-
rewrite lemmas for inverses, modified from
https://github.com/JasonGross/HoTT-categories/blob/master/theories/Categories/Category/Morphisms.v
-/
section
variables {ob : Type} [C : precategory ob] include C
variables {a b c d : ob} (f : b ⟶ a)
(r : c ⟶ d) (q : b ⟶ c) (p : a ⟶ b)
(g : d ⟶ c)
variable [Hq : is_iso q] include Hq
theorem comp.right_inverse : q ∘ q⁻¹ = id := !right_inverse
theorem comp.left_inverse : q⁻¹ ∘ q = id := !left_inverse
theorem inverse_comp_cancel_left : q⁻¹ ∘ (q ∘ p) = p :=
by rewrite [assoc, left_inverse, id_left]
theorem comp_inverse_cancel_left : q ∘ (q⁻¹ ∘ g) = g :=
by rewrite [assoc, right_inverse, id_left]
theorem comp_inverse_cancel_right : (r ∘ q) ∘ q⁻¹ = r :=
by rewrite [-assoc, right_inverse, id_right]
theorem inverse_comp_cancel_right : (f ∘ q⁻¹) ∘ q = f :=
by rewrite [-assoc, left_inverse, id_right]
theorem comp_inverse [Hp : is_iso p] [Hpq : is_iso (q ∘ p)] : (q ∘ p)⁻¹ʰ = p⁻¹ʰ ∘ q⁻¹ʰ :=
inverse_eq_left
(show (p⁻¹ʰ ∘ q⁻¹ʰ) ∘ q ∘ p = id, from
by rewrite [-assoc, inverse_comp_cancel_left, left_inverse])
theorem inverse_comp_inverse_left [H' : is_iso g] : (q⁻¹ ∘ g)⁻¹ = g⁻¹ ∘ q :=
inverse_involutive q ▸ comp_inverse q⁻¹ g
theorem inverse_comp_inverse_right [H' : is_iso f] : (q ∘ f⁻¹)⁻¹ = f ∘ q⁻¹ :=
inverse_involutive f ▸ comp_inverse q f⁻¹
theorem inverse_comp_inverse_inverse [H' : is_iso r] : (q⁻¹ ∘ r⁻¹)⁻¹ = r ∘ q :=
inverse_involutive r ▸ inverse_comp_inverse_left q r⁻¹
end
section
variables {ob : Type} {C : precategory ob} include C
variables {d c b a : ob}
{r' : c ⟶ d} {i : b ⟶ c} {f : b ⟶ a}
{r : c ⟶ d} {q : b ⟶ c} {p : a ⟶ b}
{g : d ⟶ c} {h : c ⟶ b} {p' : a ⟶ b}
{x : b ⟶ d} {z : a ⟶ c}
{y : d ⟶ b} {w : c ⟶ a}
variable [Hq : is_iso q] include Hq
theorem comp_eq_of_eq_inverse_comp (H : y = q⁻¹ ∘ g) : q ∘ y = g :=
H⁻¹ ▸ comp_inverse_cancel_left q g
theorem comp_eq_of_eq_comp_inverse (H : w = f ∘ q⁻¹) : w ∘ q = f :=
H⁻¹ ▸ inverse_comp_cancel_right f q
theorem eq_comp_of_inverse_comp_eq (H : q⁻¹ ∘ g = y) : g = q ∘ y :=
(comp_eq_of_eq_inverse_comp H⁻¹)⁻¹
theorem eq_comp_of_comp_inverse_eq (H : f ∘ q⁻¹ = w) : f = w ∘ q :=
(comp_eq_of_eq_comp_inverse H⁻¹)⁻¹
variable {Hq}
theorem inverse_comp_eq_of_eq_comp (H : z = q ∘ p) : q⁻¹ ∘ z = p :=
H⁻¹ ▸ inverse_comp_cancel_left q p
theorem comp_inverse_eq_of_eq_comp (H : x = r ∘ q) : x ∘ q⁻¹ = r :=
H⁻¹ ▸ comp_inverse_cancel_right r q
theorem eq_inverse_comp_of_comp_eq (H : q ∘ p = z) : p = q⁻¹ ∘ z :=
(inverse_comp_eq_of_eq_comp H⁻¹)⁻¹
theorem eq_comp_inverse_of_comp_eq (H : r ∘ q = x) : r = x ∘ q⁻¹ :=
(comp_inverse_eq_of_eq_comp H⁻¹)⁻¹
theorem eq_inverse_of_comp_eq_id' (H : h ∘ q = id) : h = q⁻¹ := (inverse_eq_left H)⁻¹
theorem eq_inverse_of_comp_eq_id (H : q ∘ h = id) : h = q⁻¹ := (inverse_eq_right H)⁻¹
theorem inverse_eq_of_id_eq_comp (H : id = h ∘ q) : q⁻¹ = h :=
(eq_inverse_of_comp_eq_id' H⁻¹)⁻¹
theorem inverse_eq_of_id_eq_comp' (H : id = q ∘ h) : q⁻¹ = h :=
(eq_inverse_of_comp_eq_id H⁻¹)⁻¹
variable [Hq]
theorem eq_of_comp_inverse_eq_id (H : i ∘ q⁻¹ = id) : i = q :=
eq_inverse_of_comp_eq_id' H ⬝ inverse_involutive q
theorem eq_of_inverse_comp_eq_id (H : q⁻¹ ∘ i = id) : i = q :=
eq_inverse_of_comp_eq_id H ⬝ inverse_involutive q
theorem eq_of_id_eq_comp_inverse (H : id = i ∘ q⁻¹) : q = i := (eq_of_comp_inverse_eq_id H⁻¹)⁻¹
theorem eq_of_id_eq_inverse_comp (H : id = q⁻¹ ∘ i) : q = i := (eq_of_inverse_comp_eq_id H⁻¹)⁻¹
variables (q)
theorem comp.cancel_left (H : q ∘ p = q ∘ p') : p = p' :=
by rewrite [-inverse_comp_cancel_left q p, H, inverse_comp_cancel_left q]
theorem comp.cancel_right (H : r ∘ q = r' ∘ q) : r = r' :=
by rewrite [-comp_inverse_cancel_right r q, H, comp_inverse_cancel_right _ q]
end
end iso
namespace iso
/- precomposition and postcomposition by an iso is an equivalence -/
definition is_equiv_postcompose [constructor] {ob : Type} [precategory ob] {a b c : ob}
(g : b ⟶ c) [is_iso g] : is_equiv (λ(f : a ⟶ b), g ∘ f) :=
begin
fapply adjointify,
{ exact λf', g⁻¹ ∘ f'},
{ intro f', apply comp_inverse_cancel_left},
{ intro f, apply inverse_comp_cancel_left}
end
definition equiv_postcompose [constructor] {ob : Type} [precategory ob] {a b c : ob}
(g : b ⟶ c) [is_iso g] : (a ⟶ b) ≃ (a ⟶ c) :=
equiv.mk (λ(f : a ⟶ b), g ∘ f) (is_equiv_postcompose g)
definition is_equiv_precompose [constructor] {ob : Type} [precategory ob] {a b c : ob}
(f : a ⟶ b) [is_iso f] : is_equiv (λ(g : b ⟶ c), g ∘ f) :=
begin
fapply adjointify,
{ exact λg', g' ∘ f⁻¹},
{ intro g', apply comp_inverse_cancel_right},
{ intro g, apply inverse_comp_cancel_right}
end
definition equiv_precompose [constructor] {ob : Type} [precategory ob] {a b c : ob}
(f : a ⟶ b) [is_iso f] : (b ⟶ c) ≃ (a ⟶ c) :=
equiv.mk (λ(g : b ⟶ c), g ∘ f) (is_equiv_precompose f)
end iso