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Equivalence injective type families (#1009)
I was checking out Agdas `INJECTIVE` pragma. Here are some edits I made while exploring.
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src/foundation/equivalence-injective-type-families.lagda.md
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# Equivalence injective type families | ||
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```agda | ||
module foundation.equivalence-injective-type-families where | ||
``` | ||
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<details><summary>Imports</summary> | ||
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```agda | ||
open import foundation.dependent-pair-types | ||
open import foundation.functoriality-dependent-function-types | ||
open import foundation.iterated-dependent-product-types | ||
open import foundation.univalence | ||
open import foundation.universal-property-equivalences | ||
open import foundation.universe-levels | ||
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open import foundation-core.equivalences | ||
open import foundation-core.function-types | ||
open import foundation-core.identity-types | ||
open import foundation-core.injective-maps | ||
open import foundation-core.propositions | ||
open import foundation-core.sets | ||
``` | ||
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</details> | ||
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## Idea | ||
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We say a type family `P` is | ||
{{#concept "equivalence injective" Disambiguation="type family" Agda=is-equivalence-injective}} | ||
if for every [equivalence of types](foundation-core.equivalences.md) `P x ≃ P y` | ||
we have `x = y `. By [univalence](foundation-core.univalence.md), the | ||
[structure](foundation.structure.md) of being equivalence injective is | ||
equivalent to being [injective as a map](foundation-core.injective-maps.md), but | ||
more generally every equivalence injective type family must always be injective | ||
as a map. | ||
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**Note.** The concept of equivalence injective type family as considered here is | ||
unrelated to the concept of "injective type" as studied by Martín Escardó in | ||
_Injective types in univalent mathematics_ | ||
([arXiv:1903.01211](https://arxiv.org/abs/1903.01211), | ||
[TypeTopology](https://www.cs.bham.ac.uk/~mhe/TypeTopology/InjectiveTypes.index.html)). | ||
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## Definition | ||
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### Equivalence injective type families | ||
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```agda | ||
is-equivalence-injective : | ||
{l1 l2 : Level} {A : UU l1} → (A → UU l2) → UU (l1 ⊔ l2) | ||
is-equivalence-injective {A = A} P = {x y : A} → P x ≃ P y → x = y | ||
``` | ||
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## Properties | ||
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### Equivalence injective type families are injective as maps | ||
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```agda | ||
module _ | ||
{l1 l2 : Level} {A : UU l1} {P : A → UU l2} | ||
where | ||
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is-injective-is-equivalence-injective : | ||
is-equivalence-injective P → is-injective P | ||
is-injective-is-equivalence-injective H = H ∘ equiv-eq | ||
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is-equivalence-injective-is-injective : | ||
is-injective P → is-equivalence-injective P | ||
is-equivalence-injective-is-injective H = H ∘ eq-equiv | ||
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is-equiv-is-injective-is-equivalence-injective : | ||
is-equiv is-injective-is-equivalence-injective | ||
is-equiv-is-injective-is-equivalence-injective = | ||
is-equiv-map-implicit-Π-is-fiberwise-equiv | ||
( λ x → | ||
is-equiv-map-implicit-Π-is-fiberwise-equiv | ||
( λ y → | ||
is-equiv-precomp-is-equiv | ||
( equiv-eq) | ||
( univalence (P x) (P y)) | ||
( x = y))) | ||
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equiv-is-injective-is-equivalence-injective : | ||
is-equivalence-injective P ≃ is-injective P | ||
pr1 equiv-is-injective-is-equivalence-injective = | ||
is-injective-is-equivalence-injective | ||
pr2 equiv-is-injective-is-equivalence-injective = | ||
is-equiv-is-injective-is-equivalence-injective | ||
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equiv-is-equivalence-injective-is-injective : | ||
is-injective P ≃ is-equivalence-injective P | ||
equiv-is-equivalence-injective-is-injective = | ||
inv-equiv equiv-is-injective-is-equivalence-injective | ||
``` | ||
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### For a type family over a set, being equivalence injective is a property | ||
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```agda | ||
module _ | ||
{l1 l2 : Level} {A : UU l1} (is-set-A : is-set A) (P : A → UU l2) | ||
where | ||
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is-prop-is-equivalence-injective : is-prop (is-equivalence-injective P) | ||
is-prop-is-equivalence-injective = | ||
is-prop-iterated-implicit-Π 2 (λ x y → is-prop-function-type (is-set-A x y)) | ||
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is-equivalence-injective-Prop : Prop (l1 ⊔ l2) | ||
pr1 is-equivalence-injective-Prop = is-equivalence-injective P | ||
pr2 is-equivalence-injective-Prop = is-prop-is-equivalence-injective | ||
``` |
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