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Let being an isomorphism be a property #1071
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src/category-theory/invertible-morphisms-in-precategories.lagda.md
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| Original file line number | Diff line number | Diff line change |
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| # Retractions in categories | ||
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| ```agda | ||
| module category-theory.retractions-in-categories where | ||
| ``` | ||
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| <details><summary>Imports</summary> | ||
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| ```agda | ||
| open import category-theory.categories | ||
| open import category-theory.retractions-in-precategories | ||
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| open import foundation.dependent-pair-types | ||
| open import foundation.identity-types | ||
| open import foundation.universe-levels | ||
| ``` | ||
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| </details> | ||
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| ## Idea | ||
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| Consider a morphism `f : A → B` in a [category](category-theory.categories.md) `𝒞`. A {{#concept "retraction" Disambiguation="morphism in a category" Agda=retraction-hom-Category}} of `f` consists of a morphism `g : B → A` equipped with an [identification](foundation-core.identifications.md) | ||
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| ```text | ||
| g ∘ f ＝ id. | ||
| ``` | ||
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| ## Definitions | ||
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| ### The predicate on morphisms in a category of being a retraction | ||
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| ```agda | ||
| module _ | ||
| {l1 l2 : Level} (𝒞 : Category l1 l2) | ||
| {A B : obj-Category 𝒞} (f : hom-Category 𝒞 A B) | ||
| where | ||
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| is-retraction-hom-Category : hom-Category 𝒞 B A → UU l2 | ||
| is-retraction-hom-Category = | ||
| is-retraction-hom-Precategory (precategory-Category 𝒞) f | ||
| ``` | ||
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| ### Retractions of a morphism in a category | ||
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| ```agda | ||
| module _ | ||
| {l1 l2 : Level} (𝒞 : Category l1 l2) | ||
| {A B : obj-Category 𝒞} (f : hom-Category 𝒞 A B) | ||
| where | ||
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| retraction-hom-Category : UU l2 | ||
| retraction-hom-Category = | ||
| retraction-hom-Precategory (precategory-Category 𝒞) f | ||
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| module _ | ||
| (r : retraction-hom-Category) | ||
| where | ||
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| hom-retraction-hom-Category : hom-Category 𝒞 B A | ||
| hom-retraction-hom-Category = | ||
| hom-retraction-hom-Precategory (precategory-Category 𝒞) f r | ||
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| is-retraction-retraction-hom-Category : | ||
| is-retraction-hom-Category 𝒞 f hom-retraction-hom-Category | ||
| is-retraction-retraction-hom-Category = | ||
| is-retraction-retraction-hom-Precategory (precategory-Category 𝒞) f r | ||
| ``` |
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src/category-theory/retractions-in-large-categories.lagda.md
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| # Retractions in large categories | ||
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| ```agda | ||
| module category-theory.retractions-in-large-categories where | ||
| ``` | ||
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| <details><summary>Imports</summary> | ||
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| ```agda | ||
| open import category-theory.large-categories | ||
| open import category-theory.retractions-in-large-precategories | ||
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| open import foundation.dependent-pair-types | ||
| open import foundation.identity-types | ||
| open import foundation.universe-levels | ||
| ``` | ||
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| </details> | ||
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| ## Idea | ||
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| Consider a morphism `f : A → B` in a [large category](category-theory.large-categories.md) `𝒞`. A {{#concept "retraction" Disambiguation="morphism in a large category" Agda=retraction-hom-Large-Category}} of `f` consists of a morphism `g : B → A` equipped with an [identification](foundation-core.identifications.md) | ||
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| ```text | ||
| g ∘ f ＝ id. | ||
| ``` | ||
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| ## Definitions | ||
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| ### The predicate on morphisms in a large category of being a retraction | ||
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| ```agda | ||
| module _ | ||
| {α : Level → Level} {β : Level → Level → Level} (𝒞 : Large-Category α β) | ||
| {l1 l2 : Level} {A : obj-Large-Category 𝒞 l1} | ||
| {B : obj-Large-Category 𝒞 l2} (f : hom-Large-Category 𝒞 A B) | ||
| where | ||
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| is-retraction-hom-Large-Category : | ||
| hom-Large-Category 𝒞 B A → UU (β l1 l1) | ||
| is-retraction-hom-Large-Category = | ||
| is-retraction-hom-Large-Precategory (large-precategory-Large-Category 𝒞) f | ||
| ``` | ||
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| ### Retractions of a morphism in a large category | ||
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| ```agda | ||
| module _ | ||
| {α : Level → Level} {β : Level → Level → Level} (𝒞 : Large-Category α β) | ||
| {l1 l2 : Level} {A : obj-Large-Category 𝒞 l1} | ||
| {B : obj-Large-Category 𝒞 l2} (f : hom-Large-Category 𝒞 A B) | ||
| where | ||
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| retraction-hom-Large-Category : UU (β l1 l1 ⊔ β l2 l1) | ||
| retraction-hom-Large-Category = | ||
| retraction-hom-Large-Precategory (large-precategory-Large-Category 𝒞) f | ||
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| module _ | ||
| (r : retraction-hom-Large-Category) | ||
| where | ||
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| hom-retraction-hom-Large-Category : hom-Large-Category 𝒞 B A | ||
| hom-retraction-hom-Large-Category = | ||
| hom-retraction-hom-Large-Precategory | ||
| ( large-precategory-Large-Category 𝒞) | ||
| ( f) | ||
| ( r) | ||
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| is-retraction-retraction-hom-Large-Category : | ||
| is-retraction-hom-Large-Category 𝒞 f | ||
| ( hom-retraction-hom-Large-Category) | ||
| is-retraction-retraction-hom-Large-Category = | ||
| is-retraction-retraction-hom-Large-Precategory | ||
| ( large-precategory-Large-Category 𝒞) | ||
| ( f) | ||
| ( r) | ||
| ``` | ||
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Considering that morphism types in a category are sets, I'd propose we rather use the following definition
for isomorphisms, as this type has a strictly involutive inverse operator. In fact, it would be even better to use the pairing trick that Anders showed us (I'm sorry, this means backtracking on my initiative to use the strictly involutive identity type in set-level category theory), as that definition is predicative, something that might be important for large categories. The only caveat being that it only works for set-level things.