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1 module ToposEx where
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2 open import CCC
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3 open import Level
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4 open import Category
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5 open import cat-utility
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6 open import HomReasoning
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7
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8 open Topos
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9 open Equalizer
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10
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11 -- ○ b
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12 -- b -----------→ 1
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13 -- | |
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14 -- m | | ⊤
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15 -- ↓ char m ↓
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16 -- a -----------→ Ω
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17 -- h
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18 --
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19 -- Ker t h : Equalizer A h (A [ ⊤ o (○ a) ])
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20
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21 topos-pullback : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( １ : Obj A) (○ : (a : Obj A ) → Hom A a １)
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22 → (e2 : {a : Obj A} → ∀ { f : Hom A a １ } → A [ f ≈ ○ a ] )
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23 → (t : Topos A １ ○ ) → {a : Obj A} → (h : Hom A a (Ω t)) → Pullback A h (⊤ t)
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24 topos-pullback A １ ○ e2 t {a} h = record {
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25 ab = equalizer-c (Ker t h) -- b
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26 ; π1 = equalizer (Ker t h) -- m
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27 ; π2 = ○ ( equalizer-c (Ker t h) ) -- ○ b
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28 ; isPullback = record {
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29 commute = comm
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30 ; pullback = λ {d} {p1} {p2} eq → IsEqualizer.k (isEqualizer (Ker t h)) p1 (lemma1 p1 p2 eq )
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31 ; π1p=π1 = IsEqualizer.ek=h (isEqualizer (Ker t h))
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32 ; π2p=π2 = λ {d} {p1'} {p2'} {eq} → lemma2 eq
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33 ; uniqueness = uniq
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34 }
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35 } where
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36 open ≈-Reasoning A
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37 comm : A [ A [ h o equalizer (Ker t h) ] ≈ A [ ⊤ t o ○ (equalizer-c (Ker t h)) ] ]
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38 comm = begin
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39 h o equalizer (Ker t h) ≈⟨ IsEqualizer.fe=ge (isEqualizer (Ker t h)) ⟩
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40 (⊤ t o ○ a ) o equalizer (Ker t h) ≈↑⟨ assoc ⟩
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41 ⊤ t o (○ a o equalizer (Ker t h)) ≈⟨ cdr e2 ⟩
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42 ⊤ t o ○ (equalizer-c (Ker t h)) ∎
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43 lemma1 : {d : Obj A} (p1 : Hom A d a) (p2 : Hom A d １) (eq : A [ A [ h o p1 ] ≈ A [ ⊤ t o p2 ] ] )
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44 → A [ A [ h o p1 ] ≈ A [ A [ ⊤ t o ○ a ] o p1 ] ]
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45 lemma1 {d} p1 p2 eq = begin
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46 h o p1 ≈⟨ eq ⟩
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47 ⊤ t o p2 ≈⟨ cdr e2 ⟩
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48 ⊤ t o (○ d) ≈↑⟨ cdr e2 ⟩
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49 ⊤ t o ( ○ a o p1 ) ≈⟨ assoc ⟩
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50 (⊤ t o ○ a ) o p1 ∎
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51 lemma2 : {d : Obj A} {p1' : Hom A d a} {p2' : Hom A d １} (eq : A [ A [ h o p1' ] ≈ A [ ⊤ t o p2' ] ] )
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52 → A [ A [ ○ (equalizer-c (Ker t h)) o IsEqualizer.k (isEqualizer (Ker t h)) p1'(lemma1 p1' p2' eq) ] ≈ p2' ]
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53 lemma2 {d} {p1'} {p2'} eq = begin
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54 ○ (equalizer-c (Ker t h)) o IsEqualizer.k (isEqualizer (Ker t h)) p1'(lemma1 p1' p2' eq) ≈⟨ e2 ⟩
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55 ○ d ≈↑⟨ e2 ⟩
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56 p2' ∎
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57 uniq : {d : Obj A} (p' : Hom A d (equalizer-c (Ker t h))) {π1' : Hom A d a} {π2' : Hom A d １} {eq : A [ A [ h o π1' ] ≈ A [ ⊤ t o π2' ] ]}
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58 {π1p=π1' : A [ A [ equalizer (Ker t h) o p' ] ≈ π1' ]} {π2p=π2' : A [ A [ ○ (equalizer-c (Ker t h)) o p' ] ≈ π2' ]}
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59 → A [ IsEqualizer.k (isEqualizer (Ker t h)) π1' (lemma1 π1' π2' eq) ≈ p' ]
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60 uniq {d} (p') {p1'} {p2'} {eq} {pe1} {pe2} = begin
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61 IsEqualizer.k (isEqualizer (Ker t h)) p1' (lemma1 p1' p2' eq) ≈⟨ IsEqualizer.uniqueness (isEqualizer (Ker t h)) pe1 ⟩
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62 p' ∎
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63
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64 topos-m-pullback : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( １ : Obj A) (○ : (a : Obj A ) → Hom A a １)
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65 → (e2 : {a : Obj A} → ∀ { f : Hom A a １ } → A [ f ≈ ○ a ] )
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66 → (t : Topos A １ ○ ) → {a b : Obj A} → (m : Hom A b a) → (mono : Mono A m ) → Pullback A (char t m mono ) (⊤ t)
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67 topos-m-pullback A １ ○ e2 t {a} {b} m mono = topos-pullback A １ ○ e2 t {a} (char t m mono)
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68
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69 ---
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70 ---
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71 ---
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72
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73 -- SubObjectFunctor : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) → ( １ : Obj A) (○ : (a : Obj A ) → Hom A a １)
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74 -- → (e2 : {a : Obj A} → ∀ { f : Hom A a １ } → A [ f ≈ ○ a ] ) → (t : Topos A １ ○ ) → Functor A A
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75 -- SubObjectFunctor A １ ○ e2 t = record {
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76 -- FObj = λ x → Ω t
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77 -- ; FMap = {!!}
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78 -- ; isFunctor = record {
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79 -- identity = {!!}
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80 -- ; distr = {!!}
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81 -- ; ≈-cong = {!!}
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82 -- }
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83 -- }
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84
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85 module _ {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (c : CCC A) (t : Topos A (CCC.１ c) (CCC.○ c) ) where
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86 open ≈-Reasoning A
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87 open CCC.CCC c
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88
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89 δmono : {b : Obj A } → Mono A < id1 A b , id1 A b >
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90 δmono = record {
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91 isMono = m1
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92 } where
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93 m1 : {d b : Obj A} (f g : Hom A d b) → A [ A [ < id1 A b , id1 A b > o f ] ≈ A [ < id1 A b , id1 A b > o g ] ] → A [ f ≈ g ]
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94 m1 {d} {b} f g = {!!}
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95
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96 prop32→ : {a b : Obj A}→ (f g : Hom A a b )
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97 → A [ f ≈ g ] → A [ A [ char t < id1 A b , id1 A b > δmono o < f , g > ] ≈ A [ ⊤ t o ○ a ] ]
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98 prop32→ = {!!}
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99
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100 prop23→ : {a b : Obj A}→ (f g : Hom A a b )
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101 → A [ A [ char t < id1 A b , id1 A b > δmono o < f , g > ] ≈ A [ ⊤ t o ○ a ] ] → A [ f ≈ g ]
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102 prop23→ = {!!}
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103
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104 N : (n : ToposNat A １ ) → Obj A
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105 N n = NatD.Nat (ToposNat.TNat n)
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106
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107 record prop33 (n : ToposNat A １ ) {a : Obj A} (f : Hom A １ a ) ( h : Hom A (N n ∧ a) a ) : Set ( suc c₁ ⊔ suc c₂ ⊔ suc ℓ ) where
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108 field
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109 g : Hom A (N n) a
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110 g0=f : A [ A [ g o NatD.nzero (ToposNat.TNat n) ] ≈ f ]
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111 gs=h : A [ A [ g o NatD.nsuc (ToposNat.TNat n) ] ≈ A [ h o < id1 A _ , g > ] ]
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112
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113 cor33 : (n : ToposNat A １ )
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114 → ( {a : Obj A} (f : Hom A １ a ) ( h : Hom A (N n ∧ a) a ) → prop33 n f h )
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115 → coProduct A １ ( NatD.Nat (ToposNat.TNat n) ) -- N ≅ N + 1
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116 cor33 n p = record {
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117 coproduct = N n
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118 ; κ1 = NatD.nzero (ToposNat.TNat n)
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119 ; κ2 = id1 A (N n)
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120 ; isProduct = record {
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121 _+_ = λ {a} f g → prop33.g ( p {a} f π' )
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122 ; κ1fxg=f = λ {a} {f} {g} → prop33.g0=f (p f π' )
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123 ; κ2fxg=g = λ {a} {f} {g} → ?
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124 ; uniqueness = {!!}
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125 ; +-cong = {!!}
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126
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127 }
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128 }
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