# Large categories ```agda module category-theory.large-categories where ``` <details><summary>Imports</summary> ```agda open import category-theory.categories open import category-theory.isomorphisms-in-large-precategories open import category-theory.large-precategories open import category-theory.precategories open import foundation.action-on-identifications-binary-functions open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.homotopies open import foundation.identity-types open import foundation.sets open import foundation.universe-levels ``` </details> ## Idea A **large category** in Homotopy Type Theory is a [large precategory](category-theory.large-precategories.md) for which the [identities](foundation-core.identity-types.md) between the objects are the [isomorphisms](category-theory.isomorphisms-in-large-categories.md). More specifically, an equality between objects gives rise to an isomorphism between them, by the J-rule. A large precategory is a large category if this function is an equivalence. Note that being a large category is a [proposition](foundation-core.propositions.md) since `is-equiv` is a proposition. ## Definition ### The predicate on large precategories of being a large category ```agda is-large-category-Large-Precategory : {α : Level → Level} {β : Level → Level → Level} → (C : Large-Precategory α β) → UUω is-large-category-Large-Precategory C = {l : Level} (X Y : obj-Large-Precategory C l) → is-equiv (iso-eq-Large-Precategory C X Y) ``` ### The large type of large categories ```agda record Large-Category (α : Level → Level) (β : Level → Level → Level) : UUω where constructor make-Large-Category field large-precategory-Large-Category : Large-Precategory α β is-large-category-Large-Category : is-large-category-Large-Precategory large-precategory-Large-Category open Large-Category public ``` ```agda module _ {α : Level → Level} {β : Level → Level → Level} (C : Large-Category α β) where obj-Large-Category : (l : Level) → UU (α l) obj-Large-Category = obj-Large-Precategory (large-precategory-Large-Category C) hom-set-Large-Category : {l1 l2 : Level} → obj-Large-Category l1 → obj-Large-Category l2 → Set (β l1 l2) hom-set-Large-Category = hom-set-Large-Precategory (large-precategory-Large-Category C) hom-Large-Category : {l1 l2 : Level} (X : obj-Large-Category l1) (Y : obj-Large-Category l2) → UU (β l1 l2) hom-Large-Category X Y = type-Set (hom-set-Large-Category X Y) is-set-hom-Large-Category : {l1 l2 : Level} (X : obj-Large-Category l1) (Y : obj-Large-Category l2) → is-set (hom-Large-Category X Y) is-set-hom-Large-Category X Y = is-set-type-Set (hom-set-Large-Category X Y) comp-hom-Large-Category : {l1 l2 l3 : Level} {X : obj-Large-Category l1} {Y : obj-Large-Category l2} {Z : obj-Large-Category l3} → hom-Large-Category Y Z → hom-Large-Category X Y → hom-Large-Category X Z comp-hom-Large-Category = comp-hom-Large-Precategory (large-precategory-Large-Category C) id-hom-Large-Category : {l1 : Level} {X : obj-Large-Category l1} → hom-Large-Category X X id-hom-Large-Category = id-hom-Large-Precategory (large-precategory-Large-Category C) associative-comp-hom-Large-Category : {l1 l2 l3 l4 : Level} {X : obj-Large-Category l1} {Y : obj-Large-Category l2} {Z : obj-Large-Category l3} {W : obj-Large-Category l4} → (h : hom-Large-Category Z W) (g : hom-Large-Category Y Z) (f : hom-Large-Category X Y) → ( comp-hom-Large-Category (comp-hom-Large-Category h g) f) = ( comp-hom-Large-Category h (comp-hom-Large-Category g f)) associative-comp-hom-Large-Category = associative-comp-hom-Large-Precategory (large-precategory-Large-Category C) inv-associative-comp-hom-Large-Category : {l1 l2 l3 l4 : Level} {X : obj-Large-Category l1} {Y : obj-Large-Category l2} {Z : obj-Large-Category l3} {W : obj-Large-Category l4} → (h : hom-Large-Category Z W) (g : hom-Large-Category Y Z) (f : hom-Large-Category X Y) → ( comp-hom-Large-Category h (comp-hom-Large-Category g f)) = ( comp-hom-Large-Category (comp-hom-Large-Category h g) f) inv-associative-comp-hom-Large-Category = inv-associative-comp-hom-Large-Precategory ( large-precategory-Large-Category C) left-unit-law-comp-hom-Large-Category : {l1 l2 : Level} {X : obj-Large-Category l1} {Y : obj-Large-Category l2} (f : hom-Large-Category X Y) → ( comp-hom-Large-Category id-hom-Large-Category f) = f left-unit-law-comp-hom-Large-Category = left-unit-law-comp-hom-Large-Precategory ( large-precategory-Large-Category C) right-unit-law-comp-hom-Large-Category : {l1 l2 : Level} {X : obj-Large-Category l1} {Y : obj-Large-Category l2} (f : hom-Large-Category X Y) → ( comp-hom-Large-Category f id-hom-Large-Category) = f right-unit-law-comp-hom-Large-Category = right-unit-law-comp-hom-Large-Precategory ( large-precategory-Large-Category C) ap-comp-hom-Large-Category : {l1 l2 l3 : Level} {X : obj-Large-Category l1} {Y : obj-Large-Category l2} {Z : obj-Large-Category l3} {g g' : hom-Large-Category Y Z} (p : g = g') {f f' : hom-Large-Category X Y} (q : f = f') → comp-hom-Large-Category g f = comp-hom-Large-Category g' f' ap-comp-hom-Large-Category p q = ap-binary comp-hom-Large-Category p q comp-hom-Large-Category' : {l1 l2 l3 : Level} {X : obj-Large-Category l1} {Y : obj-Large-Category l2} {Z : obj-Large-Category l3} → hom-Large-Category X Y → hom-Large-Category Y Z → hom-Large-Category X Z comp-hom-Large-Category' f g = comp-hom-Large-Category g f ``` ### Categories obtained from large categories ```agda module _ {α : Level → Level} {β : Level → Level → Level} (C : Large-Precategory α β) (is-large-category-C : is-large-category-Large-Precategory C) where is-category-is-large-category-Large-Precategory : (l : Level) → is-category-Precategory (precategory-Large-Precategory C l) is-category-is-large-category-Large-Precategory l X Y = is-equiv-htpy ( iso-eq-Large-Precategory C X Y) ( compute-iso-eq-Large-Precategory C X Y) ( is-large-category-C X Y) module _ {α : Level → Level} {β : Level → Level → Level} (C : Large-Category α β) where precategory-Large-Category : (l : Level) → Precategory (α l) (β l l) precategory-Large-Category = precategory-Large-Precategory (large-precategory-Large-Category C) is-category-Large-Category : (l : Level) → is-category-Precategory (precategory-Large-Category l) is-category-Large-Category = is-category-is-large-category-Large-Precategory ( large-precategory-Large-Category C) ( is-large-category-Large-Category C) category-Large-Category : (l : Level) → Category (α l) (β l l) pr1 (category-Large-Category l) = precategory-Large-Category l pr2 (category-Large-Category l) = is-category-Large-Category l ``` ### Equalities induce morphisms ```agda module _ {α : Level → Level} {β : Level → Level → Level} (C : Large-Category α β) {l1 : Level} (X Y : obj-Large-Category C l1) where hom-eq-Large-Category : X = Y → hom-Large-Category C X Y hom-eq-Large-Category = hom-eq-Large-Precategory (large-precategory-Large-Category C) X Y hom-inv-eq-Large-Category : X = Y → hom-Large-Category C Y X hom-inv-eq-Large-Category = hom-inv-eq-Large-Precategory (large-precategory-Large-Category C) X Y compute-hom-eq-Large-Category : hom-eq-Category (category-Large-Category C l1) X Y ~ hom-eq-Large-Category compute-hom-eq-Large-Category = compute-hom-eq-Large-Precategory (large-precategory-Large-Category C) X Y ``` ### Pre- and postcomposing by a morphism ```agda module _ {α : Level → Level} {β : Level → Level → Level} (C : Large-Category α β) {l1 l2 l3 : Level} where precomp-hom-Large-Category : {X : obj-Large-Category C l1} {Y : obj-Large-Category C l2} (f : hom-Large-Category C X Y) → (Z : obj-Large-Category C l3) → hom-Large-Category C Y Z → hom-Large-Category C X Z precomp-hom-Large-Category f Z g = comp-hom-Large-Category C g f postcomp-hom-Large-Category : (X : obj-Large-Category C l1) {Y : obj-Large-Category C l2} {Z : obj-Large-Category C l3} (f : hom-Large-Category C Y Z) → hom-Large-Category C X Y → hom-Large-Category C X Z postcomp-hom-Large-Category X f g = comp-hom-Large-Category C f g ```