forked from UniMath/agda-unimath
-
Notifications
You must be signed in to change notification settings - Fork 0
Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
Define representations of monoids (UniMath#765)
- Loading branch information
1 parent
c813a06
commit 1fe0803
Showing
57 changed files
with
2,167 additions
and
348 deletions.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
203 changes: 203 additions & 0 deletions
203
src/category-theory/dependent-products-of-categories.lagda.md
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,203 @@ | ||
| # Dependent products of categories | ||
|
|
||
| ```agda | ||
| module category-theory.dependent-products-of-categories where | ||
| ``` | ||
|
|
||
| <details><summary>Imports</summary> | ||
|
|
||
| ```agda | ||
| open import category-theory.categories | ||
| open import category-theory.dependent-products-of-precategories | ||
| open import category-theory.isomorphisms-in-categories | ||
| open import category-theory.precategories | ||
|
|
||
| open import foundation.dependent-pair-types | ||
| open import foundation.equivalences | ||
| open import foundation.function-extensionality | ||
| open import foundation.function-types | ||
| open import foundation.functoriality-dependent-function-types | ||
| open import foundation.identity-types | ||
| open import foundation.sets | ||
| open import foundation.universe-levels | ||
| ``` | ||
|
|
||
| </details> | ||
|
|
||
| ## Idea | ||
|
|
||
| Given a family of [categories](category-theory.categories.md) `Cᵢ` indexed by | ||
| `i : I`, the dependent product type `Π(i : I), Cᵢ` is a category consisting of | ||
| functions taking `i : I` to an object of `Cᵢ`. Every component of the structure | ||
| is given pointwise. | ||
|
|
||
| ## Definition | ||
|
|
||
| ```agda | ||
| module _ | ||
| {l1 l2 l3 : Level} (I : UU l1) (C : I → Category l2 l3) | ||
| where | ||
|
|
||
| precategory-Π-Category : Precategory (l1 ⊔ l2) (l1 ⊔ l3) | ||
| precategory-Π-Category = Π-Precategory I (precategory-Category ∘ C) | ||
|
|
||
| abstract | ||
| is-category-precategory-Π-Category : | ||
| is-category-Precategory precategory-Π-Category | ||
| is-category-precategory-Π-Category x y = | ||
| is-equiv-htpy-equiv | ||
| ( equiv-iso-Π-fiberwise-iso-Precategory I (precategory-Category ∘ C) ∘e | ||
| equiv-Π-equiv-family | ||
| ( λ i → extensionality-obj-Category (C i) (x i) (y i)) ∘e | ||
| equiv-funext) | ||
| ( λ {refl → refl}) | ||
|
|
||
| Π-Category : Category (l1 ⊔ l2) (l1 ⊔ l3) | ||
| pr1 Π-Category = Π-Precategory I (precategory-Category ∘ C) | ||
| pr2 Π-Category = is-category-precategory-Π-Category | ||
|
|
||
| obj-Π-Category : UU (l1 ⊔ l2) | ||
| obj-Π-Category = obj-Category Π-Category | ||
|
|
||
| hom-Π-Category : | ||
| obj-Π-Category → obj-Π-Category → Set (l1 ⊔ l3) | ||
| hom-Π-Category = hom-Category Π-Category | ||
|
|
||
| type-hom-Π-Category : | ||
| obj-Π-Category → obj-Π-Category → UU (l1 ⊔ l3) | ||
| type-hom-Π-Category = type-hom-Category Π-Category | ||
|
|
||
| comp-hom-Π-Category : | ||
| {x y z : obj-Π-Category} → | ||
| type-hom-Π-Category y z → | ||
| type-hom-Π-Category x y → | ||
| type-hom-Π-Category x z | ||
| comp-hom-Π-Category = comp-hom-Category Π-Category | ||
|
|
||
| associative-comp-hom-Π-Category : | ||
| {x y z w : obj-Π-Category} | ||
| (h : type-hom-Π-Category z w) | ||
| (g : type-hom-Π-Category y z) | ||
| (f : type-hom-Π-Category x y) → | ||
| ( comp-hom-Π-Category (comp-hom-Π-Category h g) f) ＝ | ||
| ( comp-hom-Π-Category h (comp-hom-Π-Category g f)) | ||
| associative-comp-hom-Π-Category = | ||
| associative-comp-hom-Category Π-Category | ||
|
|
||
| associative-composition-structure-Π-Category : | ||
| associative-composition-structure-Set hom-Π-Category | ||
| associative-composition-structure-Π-Category = | ||
| associative-composition-structure-Category Π-Category | ||
|
|
||
| id-hom-Π-Category : | ||
| {x : obj-Π-Category} → type-hom-Π-Category x x | ||
| id-hom-Π-Category = id-hom-Category Π-Category | ||
|
|
||
| left-unit-law-comp-hom-Π-Category : | ||
| {x y : obj-Π-Category} | ||
| (f : type-hom-Π-Category x y) → | ||
| comp-hom-Π-Category id-hom-Π-Category f ＝ f | ||
| left-unit-law-comp-hom-Π-Category = | ||
| left-unit-law-comp-hom-Category Π-Category | ||
|
|
||
| right-unit-law-comp-hom-Π-Category : | ||
| {x y : obj-Π-Category} (f : type-hom-Π-Category x y) → | ||
| comp-hom-Π-Category f id-hom-Π-Category ＝ f | ||
| right-unit-law-comp-hom-Π-Category = | ||
| right-unit-law-comp-hom-Category Π-Category | ||
|
|
||
| is-unital-Π-Category : | ||
| is-unital-composition-structure-Set | ||
| hom-Π-Category | ||
| associative-composition-structure-Π-Category | ||
| is-unital-Π-Category = is-unital-composition-structure-Category Π-Category | ||
|
|
||
| extensionality-obj-Π-Category : | ||
| (x y : obj-Category Π-Category) → (x ＝ y) ≃ iso-Category Π-Category x y | ||
| extensionality-obj-Π-Category = extensionality-obj-Category Π-Category | ||
| ``` | ||
|
|
||
| ## Properties | ||
|
|
||
| ### Isomorphisms in the dependent product category are fiberwise isomorphisms | ||
|
|
||
| ```agda | ||
| module _ | ||
| {l1 l2 l3 : Level} (I : UU l1) (C : I → Category l2 l3) | ||
| {x y : obj-Π-Category I C} | ||
| where | ||
|
|
||
| is-fiberwise-iso-is-iso-Π-Category : | ||
| (f : type-hom-Π-Category I C x y) → | ||
| is-iso-Category (Π-Category I C) f → | ||
| (i : I) → is-iso-Category (C i) (f i) | ||
| is-fiberwise-iso-is-iso-Π-Category = | ||
| is-fiberwise-iso-is-iso-Π-Precategory I (precategory-Category ∘ C) | ||
|
|
||
| fiberwise-iso-iso-Π-Category : | ||
| iso-Category (Π-Category I C) x y → | ||
| (i : I) → iso-Category (C i) (x i) (y i) | ||
| fiberwise-iso-iso-Π-Category = | ||
| fiberwise-iso-iso-Π-Precategory I (precategory-Category ∘ C) | ||
|
|
||
| is-iso-Π-is-fiberwise-iso-Category : | ||
| (f : type-hom-Π-Category I C x y) → | ||
| ((i : I) → is-iso-Category (C i) (f i)) → | ||
| is-iso-Category (Π-Category I C) f | ||
| is-iso-Π-is-fiberwise-iso-Category = | ||
| is-iso-Π-is-fiberwise-iso-Precategory I (precategory-Category ∘ C) | ||
|
|
||
| iso-Π-fiberwise-iso-Category : | ||
| ((i : I) → iso-Category (C i) (x i) (y i)) → | ||
| iso-Category (Π-Category I C) x y | ||
| iso-Π-fiberwise-iso-Category = | ||
| iso-Π-fiberwise-iso-Precategory I (precategory-Category ∘ C) | ||
|
|
||
| is-equiv-is-fiberwise-iso-is-iso-Π-Category : | ||
| (f : type-hom-Π-Category I C x y) → | ||
| is-equiv (is-fiberwise-iso-is-iso-Π-Category f) | ||
| is-equiv-is-fiberwise-iso-is-iso-Π-Category = | ||
| is-equiv-is-fiberwise-iso-is-iso-Π-Precategory I (precategory-Category ∘ C) | ||
|
|
||
| equiv-is-fiberwise-iso-is-iso-Π-Category : | ||
| (f : type-hom-Π-Category I C x y) → | ||
| ( is-iso-Category (Π-Category I C) f) ≃ | ||
| ( (i : I) → is-iso-Category (C i) (f i)) | ||
| equiv-is-fiberwise-iso-is-iso-Π-Category = | ||
| equiv-is-fiberwise-iso-is-iso-Π-Precategory I (precategory-Category ∘ C) | ||
|
|
||
| is-equiv-is-iso-Π-is-fiberwise-iso-Category : | ||
| (f : type-hom-Π-Category I C x y) → | ||
| is-equiv (is-iso-Π-is-fiberwise-iso-Category f) | ||
| is-equiv-is-iso-Π-is-fiberwise-iso-Category = | ||
| is-equiv-is-iso-Π-is-fiberwise-iso-Precategory I (precategory-Category ∘ C) | ||
|
|
||
| equiv-is-iso-Π-is-fiberwise-iso-Category : | ||
| ( f : type-hom-Π-Category I C x y) → | ||
| ( (i : I) → is-iso-Category (C i) (f i)) ≃ | ||
| ( is-iso-Category (Π-Category I C) f) | ||
| equiv-is-iso-Π-is-fiberwise-iso-Category = | ||
| equiv-is-iso-Π-is-fiberwise-iso-Precategory I (precategory-Category ∘ C) | ||
|
|
||
| is-equiv-fiberwise-iso-iso-Π-Category : | ||
| is-equiv fiberwise-iso-iso-Π-Category | ||
| is-equiv-fiberwise-iso-iso-Π-Category = | ||
| is-equiv-fiberwise-iso-iso-Π-Precategory I (precategory-Category ∘ C) | ||
|
|
||
| equiv-fiberwise-iso-iso-Π-Category : | ||
| ( iso-Category (Π-Category I C) x y) ≃ | ||
| ( (i : I) → iso-Category (C i) (x i) (y i)) | ||
| equiv-fiberwise-iso-iso-Π-Category = | ||
| equiv-fiberwise-iso-iso-Π-Precategory I (precategory-Category ∘ C) | ||
|
|
||
| is-equiv-iso-Π-fiberwise-iso-Category : | ||
| is-equiv iso-Π-fiberwise-iso-Category | ||
| is-equiv-iso-Π-fiberwise-iso-Category = | ||
| is-equiv-iso-Π-fiberwise-iso-Precategory I (precategory-Category ∘ C) | ||
|
|
||
| equiv-iso-Π-fiberwise-iso-Category : | ||
| ( (i : I) → iso-Category (C i) (x i) (y i)) ≃ | ||
| ( iso-Category (Π-Category I C) x y) | ||
| equiv-iso-Π-fiberwise-iso-Category = | ||
| equiv-iso-Π-fiberwise-iso-Precategory I (precategory-Category ∘ C) | ||
| ``` |
Oops, something went wrong.