Equivalences between large precategories

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-09-13.
Last modified on 2023-10-17.

module category-theory.equivalences-of-large-precategories where
Imports 
open import category-theory.functors-large-precategories
open import category-theory.large-precategories
open import category-theory.natural-isomorphisms-functors-large-precategories

open import foundation.universe-levels

Idea

Two large precategories C and D are said to be equivalent if there are functors F : C → D and G : D → C such that

  • G ∘ F is naturally isomorphic to the identity functor on C,
  • F ∘ G is naturally isomorphic to the identity functor on D.

Definition

module _
  {αC αD : Level → Level} {βC βD : Level → Level → Level}
  (C : Large-Precategory αC βC) (D : Large-Precategory αD βD)
  where

  record
    equivalence-Large-Precategory (γ γ' : Level → Level) : UUω where
    constructor
      make-equivalence-Large-Precategory
    field
      functor-equivalence-Large-Precategory :
        functor-Large-Precategory γ C D
      functor-inv-equivalence-Large-Precategory :
        functor-Large-Precategory γ' D C
      is-section-functor-inv-equivalence-Large-Precategory :
        natural-isomorphism-Large-Precategory
          ( comp-functor-Large-Precategory D C D
            functor-equivalence-Large-Precategory
            functor-inv-equivalence-Large-Precategory)
          ( id-functor-Large-Precategory D)
      is-retraction-functor-inv-equivalence-Large-Precategory :
        natural-isomorphism-Large-Precategory
          ( comp-functor-Large-Precategory C D C
            functor-inv-equivalence-Large-Precategory
            functor-equivalence-Large-Precategory)
          ( id-functor-Large-Precategory C)

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