/- Copyright (c) 2015 Floris van Doorn. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Module: algebra.precategory.iso Author: Floris van Doorn, Jakob von Raumer -/ import algebra.precategory.basic 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 [quasireducible] attribute is_iso [multiple-instances] open split_mono split_epi is_iso definition retraction_of [reducible] := @split_mono.retraction_of definition retraction_comp [reducible] := @split_mono.retraction_comp definition section_of [reducible] := @split_epi.section_of definition comp_section [reducible] := @split_epi.comp_section definition inverse [reducible] := @is_iso.inverse definition left_inverse [reducible] := @is_iso.left_inverse definition right_inverse [reducible] := @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 [instance] [priority 300] [reducible] (f : a ⟶ b) [H : is_iso f] : split_mono f := split_mono.mk !left_inverse definition split_epi_of_is_iso [instance] [priority 300] [reducible] (f : a ⟶ b) [H : is_iso f] : split_epi f := split_epi.mk !right_inverse definition is_iso_id [instance] [priority 500] (a : ob) : is_iso (ID a) := is_iso.mk !id_comp !id_comp definition is_iso_inverse [instance] [priority 200] (f : a ⟶ b) [H : is_iso f] : is_iso f⁻¹ := is_iso.mk !right_inverse !left_inverse definition left_inverse_eq_right_inverse {f : a ⟶ b} {g g' : hom b a} (Hl : g ∘ f = id) (Hr : f ∘ g' = id) : g = g' := by rewrite [-(id_right g), -Hr, assoc, Hl, id_left] definition retraction_eq [H : split_mono f] (H2 : f ∘ h = id) : retraction_of f = h := left_inverse_eq_right_inverse !retraction_comp H2 definition section_eq [H : split_epi f] (H2 : h ∘ f = id) : section_of f = h := (left_inverse_eq_right_inverse H2 !comp_section)⁻¹ definition inverse_eq_right [H : is_iso f] (H2 : f ∘ h = id) : f⁻¹ = h := left_inverse_eq_right_inverse !left_inverse H2 definition inverse_eq_left [H : is_iso f] (H2 : h ∘ f = id) : f⁻¹ = h := (left_inverse_eq_right_inverse H2 !right_inverse)⁻¹ definition 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 (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) definition inverse_unique (H H' : is_iso f) : @inverse _ _ _ _ f H = @inverse _ _ _ _ f H' := inverse_eq_left !left_inverse definition inverse_involutive (f : a ⟶ b) [H : is_iso f] [H : is_iso (f⁻¹)] : (f⁻¹)⁻¹ = f := inverse_eq_right !left_inverse definition retraction_id (a : ob) : retraction_of (ID a) = id := retraction_eq !id_comp definition section_id (a : ob) : section_of (ID a) = id := section_eq !id_comp definition id_inverse (a : ob) [H : is_iso (ID a)] : (ID a)⁻¹ = id := inverse_eq_left !id_comp definition split_mono_comp [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b) [Hf : split_mono f] [Hg : split_mono g] : split_mono (g ∘ f) := split_mono.mk (show (retraction_of f ∘ retraction_of g) ∘ g ∘ f = id, by rewrite [-assoc, assoc _ g f, retraction_comp, id_left, retraction_comp]) definition split_epi_comp [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b) [Hf : split_epi f] [Hg : split_epi g] : split_epi (g ∘ f) := split_epi.mk (show (g ∘ f) ∘ section_of f ∘ section_of g = id, by rewrite [-assoc, {f ∘ _}assoc, comp_section, id_left, comp_section]) definition is_iso_comp [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 -- "is_iso f" is equivalent to a certain sigma type -- definition is_iso.sigma_char (f : hom a b) : -- (Σ (g : hom b a), (g ∘ f = id) × (f ∘ g = id)) ≃ is_iso f := -- begin -- fapply equiv.MK, -- {intro S, apply is_iso.mk, -- exact (pr₁ S.2), -- exact (pr₂ S.2)}, -- {intro H, cases H with (g, η, ε), -- exact (sigma.mk g (pair η ε))}, -- {intro H, cases H, apply idp}, -- {intro S, cases S with (g, ηε), cases ηε, apply idp}, -- end definition is_hprop_is_iso [instance] (f : hom a b) : is_hprop (is_iso f) := begin apply is_hprop.mk, intros (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_hprop.elim, apply is_hprop.elim, end /- iso objects -/ structure iso (a b : ob) := (to_hom : hom a b) [struct : is_iso to_hom] infix `≅`:50 := iso.iso attribute iso.struct [instance] [priority 400] namespace iso attribute to_hom [coercion] definition MK (f : a ⟶ b) (g : b ⟶ a) (H1 : g ∘ f = id) (H2 : f ∘ g = id) := @mk _ _ _ _ f (is_iso.mk H1 H2) definition to_inv (f : a ≅ b) : b ⟶ a := (to_hom f)⁻¹ protected definition refl (a : ob) : a ≅ a := mk (ID a) protected definition symm ⦃a b : ob⦄ (H : a ≅ b) : b ≅ a := mk (to_hom H)⁻¹ protected definition trans ⦃a b c : ob⦄ (H1 : a ≅ b) (H2 : b ≅ c) : a ≅ c := mk (to_hom H2 ∘ to_hom H1) protected definition eq_mk' {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_hprop.elim protected definition eq_mk {f f' : a ≅ b} (p : to_hom f = to_hom f') : f = f' := by (cases f; cases f'; apply (iso.eq_mk' p)) -- The structure for isomorphism can be characterized up to equivalence by a sigma type. 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 end iso -- The type of isomorphisms between two objects is a set definition is_hset_iso [instance] : is_hset (a ≅ b) := begin apply is_trunc_is_equiv_closed, apply (equiv.to_is_equiv (!iso.sigma_char)), end definition iso_of_eq (p : a = b) : a ≅ b := eq.rec_on p (iso.refl a) definition hom_of_eq [reducible] (p : a = b) : a ⟶ b := iso.to_hom (iso_of_eq p) definition inv_of_eq [reducible] (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 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_mk !id_comp⁻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) := eq.rec_on p !id_leftright⁻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) : {retr 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 ... = 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 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 definition comp.right_inverse : q ∘ q⁻¹ = id := !right_inverse definition comp.left_inverse : q⁻¹ ∘ q = id := !left_inverse definition inverse_comp_cancel_left : q⁻¹ ∘ (q ∘ p) = p := by rewrite [assoc, left_inverse, id_left] definition comp_inverse_cancel_left : q ∘ (q⁻¹ ∘ g) = g := by rewrite [assoc, right_inverse, id_left] definition comp_inverse_cancel_right : (r ∘ q) ∘ q⁻¹ = r := by rewrite [-assoc, right_inverse, id_right] definition inverse_comp_cancel_right : (f ∘ q⁻¹) ∘ q = f := by rewrite [-assoc, left_inverse, id_right] definition comp_inverse [Hp : is_iso p] [Hpq : is_iso (q ∘ p)] : (q ∘ p)⁻¹ʰ = p⁻¹ʰ ∘ q⁻¹ʰ := inverse_eq_left (show (p⁻¹ʰ ∘ q⁻¹ʰ) ∘ q ∘ p = id, from by rewrite [-assoc, inverse_comp_cancel_left, left_inverse]) definition inverse_comp_inverse_left [H' : is_iso g] : (q⁻¹ ∘ g)⁻¹ = g⁻¹ ∘ q := inverse_involutive q ▹ comp_inverse q⁻¹ g definition inverse_comp_inverse_right [H' : is_iso f] : (q ∘ f⁻¹)⁻¹ = f ∘ q⁻¹ := inverse_involutive f ▹ comp_inverse q f⁻¹ definition inverse_comp_inverse_inverse [H' : is_iso r] : (q⁻¹ ∘ r⁻¹)⁻¹ = r ∘ q := inverse_involutive r ▹ inverse_comp_inverse_left q r⁻¹ end section variables {ob : Type} {C : precategory ob} include C variables {d c b a : ob} {i : b ⟶ c} {f : b ⟶ a} {r : c ⟶ d} {q : b ⟶ c} {p : a ⟶ b} {g : d ⟶ c} {h : c ⟶ b} {x : b ⟶ d} {z : a ⟶ c} {y : d ⟶ b} {w : c ⟶ a} variable [Hq : is_iso q] include Hq definition comp_eq_of_eq_inverse_comp (H : y = q⁻¹ ∘ g) : q ∘ y = g := H⁻¹ ▹ comp_inverse_cancel_left q g definition comp_eq_of_eq_comp_inverse (H : w = f ∘ q⁻¹) : w ∘ q = f := H⁻¹ ▹ inverse_comp_cancel_right f q definition inverse_comp_eq_of_eq_comp (H : z = q ∘ p) : q⁻¹ ∘ z = p := H⁻¹ ▹ inverse_comp_cancel_left q p definition comp_inverse_eq_of_eq_comp (H : x = r ∘ q) : x ∘ q⁻¹ = r := H⁻¹ ▹ comp_inverse_cancel_right r q definition eq_comp_of_inverse_comp_eq (H : q⁻¹ ∘ g = y) : g = q ∘ y := (comp_eq_of_eq_inverse_comp H⁻¹)⁻¹ definition eq_comp_of_comp_inverse_eq (H : f ∘ q⁻¹ = w) : f = w ∘ q := (comp_eq_of_eq_comp_inverse H⁻¹)⁻¹ definition eq_inverse_comp_of_comp_eq (H : q ∘ p = z) : p = q⁻¹ ∘ z := (inverse_comp_eq_of_eq_comp H⁻¹)⁻¹ definition eq_comp_inverse_of_comp_eq (H : r ∘ q = x) : r = x ∘ q⁻¹ := (comp_inverse_eq_of_eq_comp H⁻¹)⁻¹ definition eq_inverse_of_comp_eq_id' (H : h ∘ q = id) : h = q⁻¹ := (inverse_eq_left H)⁻¹ definition eq_inverse_of_comp_eq_id (H : q ∘ h = id) : h = q⁻¹ := (inverse_eq_right H)⁻¹ definition eq_of_comp_inverse_eq_id (H : i ∘ q⁻¹ = id) : i = q := eq_inverse_of_comp_eq_id' H ⬝ inverse_involutive q definition eq_of_inverse_comp_eq_id (H : q⁻¹ ∘ i = id) : i = q := eq_inverse_of_comp_eq_id H ⬝ inverse_involutive q definition eq_of_id_eq_comp_inverse (H : id = i ∘ q⁻¹) : q = i := (eq_of_comp_inverse_eq_id H⁻¹)⁻¹ definition eq_of_id_eq_inverse_comp (H : id = q⁻¹ ∘ i) : q = i := (eq_of_inverse_comp_eq_id H⁻¹)⁻¹ definition inverse_eq_of_id_eq_comp (H : id = h ∘ q) : q⁻¹ = h := (eq_inverse_of_comp_eq_id' H⁻¹)⁻¹ definition inverse_eq_of_id_eq_comp' (H : id = q ∘ h) : q⁻¹ = h := (eq_inverse_of_comp_eq_id H⁻¹)⁻¹ end end iso