|
| 1 | +# Equivalences in the slice over a type |
| 2 | + |
| 3 | +```agda |
| 4 | +module foundation.equivalences-slice where |
| 5 | +``` |
| 6 | + |
| 7 | +<details><summary>Imports</summary> |
| 8 | + |
| 9 | +```agda |
| 10 | +open import foundation.dependent-pair-types |
| 11 | +open import foundation.dependent-products-propositions |
| 12 | +open import foundation.equivalences |
| 13 | +open import foundation.fundamental-theorem-of-identity-types |
| 14 | +open import foundation.homotopies |
| 15 | +open import foundation.homotopy-induction |
| 16 | +open import foundation.logical-equivalences |
| 17 | +open import foundation.morphisms-slice |
| 18 | +open import foundation.slice |
| 19 | +open import foundation.structure-identity-principle |
| 20 | +open import foundation.type-arithmetic-dependent-pair-types |
| 21 | +open import foundation.univalence |
| 22 | +open import foundation.universe-levels |
| 23 | +
|
| 24 | +open import foundation-core.embeddings |
| 25 | +open import foundation-core.families-of-equivalences |
| 26 | +open import foundation-core.fibers-of-maps |
| 27 | +open import foundation-core.function-types |
| 28 | +open import foundation-core.functoriality-dependent-pair-types |
| 29 | +open import foundation-core.identity-types |
| 30 | +open import foundation-core.injective-maps |
| 31 | +open import foundation-core.propositions |
| 32 | +open import foundation-core.torsorial-type-families |
| 33 | +open import foundation-core.type-theoretic-principle-of-choice |
| 34 | +``` |
| 35 | + |
| 36 | +</details> |
| 37 | + |
| 38 | +## Idea |
| 39 | + |
| 40 | +The slice of a category over an object X is the category of morphisms into X. A |
| 41 | +{{#concept "morphism"}} in the slice from `f : A → X` to `g : B → X` consists of |
| 42 | +a function `h : A → B` such that the triangle `f ~ g ∘ h` commutes. We make |
| 43 | +these definitions for types. |
| 44 | + |
| 45 | +## Definitions |
| 46 | + |
| 47 | +### The predicate on a morphism in the slice of being an equivalence |
| 48 | + |
| 49 | +```agda |
| 50 | +is-equiv-hom-slice : |
| 51 | + {l1 l2 l3 : Level} {X : UU l1} {A : UU l2} {B : UU l3} |
| 52 | + (f : A → X) (g : B → X) → hom-slice f g → UU (l2 ⊔ l3) |
| 53 | +is-equiv-hom-slice f g h = is-equiv (map-hom-slice f g h) |
| 54 | +``` |
| 55 | + |
| 56 | +### The type of equivalences in the slice |
| 57 | + |
| 58 | +```agda |
| 59 | +equiv-slice : |
| 60 | + {l1 l2 l3 : Level} {X : UU l1} {A : UU l2} {B : UU l3} → |
| 61 | + (A → X) → (B → X) → UU (l1 ⊔ l2 ⊔ l3) |
| 62 | +equiv-slice {A = A} {B = B} f g = Σ (A ≃ B) (λ e → f ~ g ∘ map-equiv e) |
| 63 | +
|
| 64 | +module _ |
| 65 | + {l1 l2 l3 : Level} {X : UU l1} {A : UU l2} {B : UU l3} |
| 66 | + (f : A → X) (g : B → X) (e : equiv-slice f g) |
| 67 | + where |
| 68 | +
|
| 69 | + equiv-equiv-slice : A ≃ B |
| 70 | + equiv-equiv-slice = pr1 e |
| 71 | +
|
| 72 | + map-equiv-slice : A → B |
| 73 | + map-equiv-slice = map-equiv equiv-equiv-slice |
| 74 | +
|
| 75 | + is-equiv-map-equiv-slice : is-equiv map-equiv-slice |
| 76 | + is-equiv-map-equiv-slice = is-equiv-map-equiv equiv-equiv-slice |
| 77 | +
|
| 78 | + coh-equiv-slice : f ~ g ∘ map-equiv-slice |
| 79 | + coh-equiv-slice = pr2 e |
| 80 | +
|
| 81 | + hom-equiv-slice : hom-slice f g |
| 82 | + hom-equiv-slice = (map-equiv-slice , coh-equiv-slice) |
| 83 | +``` |
| 84 | + |
| 85 | +## Properties |
| 86 | + |
| 87 | +### A morphism in the slice over `X` is an equivalence if and only if the induced map between fibers is a fiberwise equivalence |
| 88 | + |
| 89 | +```agda |
| 90 | +module _ |
| 91 | + {l1 l2 l3 : Level} {X : UU l1} {A : UU l2} {B : UU l3} |
| 92 | + (f : A → X) (g : B → X) |
| 93 | + where |
| 94 | +
|
| 95 | + abstract |
| 96 | + is-fiberwise-equiv-fiberwise-equiv-equiv-slice : |
| 97 | + (t : hom-slice f g) → is-equiv (map-hom-slice f g t) → |
| 98 | + is-fiberwise-equiv (fiberwise-hom-hom-slice f g t) |
| 99 | + is-fiberwise-equiv-fiberwise-equiv-equiv-slice (h , H) = |
| 100 | + is-fiberwise-equiv-is-equiv-triangle f g h H |
| 101 | +
|
| 102 | + abstract |
| 103 | + is-equiv-hom-slice-is-fiberwise-equiv-fiberwise-hom-hom-slice : |
| 104 | + (t : hom-slice f g) → |
| 105 | + ((x : X) → is-equiv (fiberwise-hom-hom-slice f g t x)) → |
| 106 | + is-equiv (map-hom-slice f g t) |
| 107 | + is-equiv-hom-slice-is-fiberwise-equiv-fiberwise-hom-hom-slice (h , H) = |
| 108 | + is-equiv-triangle-is-fiberwise-equiv f g h H |
| 109 | +
|
| 110 | + equiv-fiberwise-equiv-equiv-slice : |
| 111 | + equiv-slice f g ≃ fiberwise-equiv (fiber f) (fiber g) |
| 112 | + equiv-fiberwise-equiv-equiv-slice = |
| 113 | + equiv-Σ is-fiberwise-equiv (equiv-fiberwise-hom-hom-slice f g) α ∘e |
| 114 | + equiv-right-swap-Σ |
| 115 | + where |
| 116 | + α : |
| 117 | + (h : hom-slice f g) → |
| 118 | + is-equiv (map-hom-slice f g h) ≃ |
| 119 | + is-fiberwise-equiv (map-equiv (equiv-fiberwise-hom-hom-slice f g) h) |
| 120 | + α h = |
| 121 | + equiv-iff-is-prop |
| 122 | + ( is-property-is-equiv _) |
| 123 | + ( is-prop-Π (λ _ → is-property-is-equiv _)) |
| 124 | + ( is-fiberwise-equiv-fiberwise-equiv-equiv-slice h) |
| 125 | + ( is-equiv-hom-slice-is-fiberwise-equiv-fiberwise-hom-hom-slice h) |
| 126 | +
|
| 127 | + equiv-equiv-slice-fiberwise-equiv : |
| 128 | + fiberwise-equiv (fiber f) (fiber g) ≃ equiv-slice f g |
| 129 | + equiv-equiv-slice-fiberwise-equiv = |
| 130 | + inv-equiv equiv-fiberwise-equiv-equiv-slice |
| 131 | +
|
| 132 | + fiberwise-equiv-equiv-slice : |
| 133 | + equiv-slice f g → fiberwise-equiv (fiber f) (fiber g) |
| 134 | + fiberwise-equiv-equiv-slice = |
| 135 | + map-equiv equiv-fiberwise-equiv-equiv-slice |
| 136 | +
|
| 137 | + equiv-fam-equiv-equiv-slice : |
| 138 | + equiv-slice f g ≃ fam-equiv (fiber f) (fiber g) |
| 139 | + equiv-fam-equiv-equiv-slice = |
| 140 | + inv-distributive-Π-Σ ∘e equiv-fiberwise-equiv-equiv-slice |
| 141 | +``` |
| 142 | + |
| 143 | +### Logically equivalent injections into a type are equivalent slices over that type |
| 144 | + |
| 145 | +```agda |
| 146 | +module _ |
| 147 | + {l1 l2 l3 : Level} {X : UU l1} {A : UU l2} {B : UU l3} |
| 148 | + where |
| 149 | +
|
| 150 | + abstract |
| 151 | + is-equiv-hom-slice-is-injective : |
| 152 | + {f : A → X} {g : B → X} → |
| 153 | + is-injective f → is-injective g → |
| 154 | + (h : hom-slice f g) → |
| 155 | + hom-slice g f → |
| 156 | + is-equiv-hom-slice f g h |
| 157 | + is-equiv-hom-slice-is-injective {f} {g} F G h i = |
| 158 | + is-equiv-is-invertible |
| 159 | + ( map-hom-slice g f i) |
| 160 | + ( λ y → |
| 161 | + G ( inv |
| 162 | + ( ( triangle-hom-slice g f i y) ∙ |
| 163 | + ( triangle-hom-slice f g h (map-hom-slice g f i y))))) |
| 164 | + ( λ x → |
| 165 | + F ( inv |
| 166 | + ( ( triangle-hom-slice f g h x) ∙ |
| 167 | + ( triangle-hom-slice g f i (map-hom-slice f g h x))))) |
| 168 | +
|
| 169 | + is-equiv-hom-slice-injection : |
| 170 | + (f : injection A X) (g : injection B X) |
| 171 | + (h : hom-slice (map-injection f) (map-injection g)) → |
| 172 | + hom-slice (map-injection g) (map-injection f) → |
| 173 | + is-equiv-hom-slice (map-injection f) (map-injection g) h |
| 174 | + is-equiv-hom-slice-injection (f , F) (g , G) = |
| 175 | + is-equiv-hom-slice-is-injective F G |
| 176 | +``` |
| 177 | + |
| 178 | +### Logically equivalent embeddings into a type are equivalent slices over that type |
| 179 | + |
| 180 | +```agda |
| 181 | +module _ |
| 182 | + {l1 l2 l3 : Level} {X : UU l1} {A : UU l2} {B : UU l3} |
| 183 | + where |
| 184 | +
|
| 185 | + is-equiv-hom-slice-emb : |
| 186 | + (f : A ↪ X) (g : B ↪ X) |
| 187 | + (h : hom-slice (map-emb f) (map-emb g)) → |
| 188 | + hom-slice (map-emb g) (map-emb f) → |
| 189 | + is-equiv-hom-slice (map-emb f) (map-emb g) h |
| 190 | + is-equiv-hom-slice-emb f g = |
| 191 | + is-equiv-hom-slice-injection (injection-emb f) (injection-emb g) |
| 192 | +``` |
| 193 | + |
| 194 | +### Characterization of the identity type of `Slice l A` |
| 195 | + |
| 196 | +```agda |
| 197 | +module _ |
| 198 | + {l1 l2 : Level} {A : UU l1} |
| 199 | + where |
| 200 | +
|
| 201 | + equiv-Slice : (f g : Slice l2 A) → UU (l1 ⊔ l2) |
| 202 | + equiv-Slice f g = equiv-slice (map-Slice f) (map-Slice g) |
| 203 | +
|
| 204 | + id-equiv-Slice : (f : Slice l2 A) → equiv-Slice f f |
| 205 | + id-equiv-Slice f = (id-equiv , refl-htpy) |
| 206 | +
|
| 207 | + equiv-eq-Slice : (f g : Slice l2 A) → f ＝ g → equiv-Slice f g |
| 208 | + equiv-eq-Slice f .f refl = id-equiv-Slice f |
| 209 | +``` |
| 210 | + |
| 211 | +### Univalence in a slice |
| 212 | + |
| 213 | +```agda |
| 214 | +module _ |
| 215 | + {l1 l2 : Level} {A : UU l1} |
| 216 | + where |
| 217 | +
|
| 218 | + abstract |
| 219 | + is-torsorial-equiv-Slice : (f : Slice l2 A) → is-torsorial (equiv-Slice f) |
| 220 | + is-torsorial-equiv-Slice (X , f) = |
| 221 | + is-torsorial-Eq-structure |
| 222 | + ( is-torsorial-equiv X) |
| 223 | + ( X , id-equiv) |
| 224 | + ( is-torsorial-htpy f) |
| 225 | +
|
| 226 | + abstract |
| 227 | + is-equiv-equiv-eq-Slice : (f g : Slice l2 A) → is-equiv (equiv-eq-Slice f g) |
| 228 | + is-equiv-equiv-eq-Slice f = |
| 229 | + fundamental-theorem-id |
| 230 | + ( is-torsorial-equiv-Slice f) |
| 231 | + ( equiv-eq-Slice f) |
| 232 | +
|
| 233 | + extensionality-Slice : (f g : Slice l2 A) → (f ＝ g) ≃ equiv-Slice f g |
| 234 | + extensionality-Slice f g = (equiv-eq-Slice f g , is-equiv-equiv-eq-Slice f g) |
| 235 | +
|
| 236 | + eq-equiv-Slice : (f g : Slice l2 A) → equiv-Slice f g → f ＝ g |
| 237 | + eq-equiv-Slice f g = map-inv-equiv (extensionality-Slice f g) |
| 238 | +``` |
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