{-# OPTIONS --without-K --safe #-}
open import Categories.Category
module Categories.Object.Terminal {o ℓ e} (C : Category o ℓ e) where
open import Level
open import Relation.Binary using (IsEquivalence; Setoid)
open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
open import Categories.Morphism C
open import Categories.Morphism.IsoEquiv C using (_≃_; ⌞_⌟)
open import Categories.Morphism.Reasoning C
open Category C
open HomReasoning
record IsTerminal (⊤ : Obj) : Set (o ⊔ ℓ ⊔ e) where
field
! : {A : Obj} → (A ⇒ ⊤)
!-unique : ∀ {A} → (f : A ⇒ ⊤) → ! ≈ f
!-unique₂ : ∀ {A} {f g : A ⇒ ⊤} → f ≈ g
!-unique₂ {A} {f} {g} = begin
f ≈˘⟨ !-unique f ⟩
! ≈⟨ !-unique g ⟩
g ∎
where open HomReasoning
⊤-id : (f : ⊤ ⇒ ⊤) → f ≈ id
⊤-id _ = !-unique₂
record Terminal : Set (o ⊔ ℓ ⊔ e) where
field
⊤ : Obj
⊤-is-terminal : IsTerminal ⊤
open IsTerminal ⊤-is-terminal public
open Terminal
from-⊤-is-Mono : ∀ {A : Obj} {t : Terminal} → (f : ⊤ t ⇒ A) → Mono f
from-⊤-is-Mono {_} {t} _ = λ _ _ _ → !-unique₂ t
up-to-iso : (t₁ t₂ : Terminal) → ⊤ t₁ ≅ ⊤ t₂
up-to-iso t₁ t₂ = record
{ from = ! t₂
; to = ! t₁
; iso = record { isoˡ = ⊤-id t₁ _; isoʳ = ⊤-id t₂ _ }
}
transport-by-iso : (t : Terminal) → ∀ {X} → ⊤ t ≅ X → Terminal
transport-by-iso t {X} t≅X = record
{ ⊤ = X
; ⊤-is-terminal = record
{ ! = from ∘ ! t
; !-unique = λ h → begin
from ∘ ! t ≈⟨ refl⟩∘⟨ !-unique t (to ∘ h) ⟩
from ∘ to ∘ h ≈⟨ cancelˡ isoʳ ⟩
h ∎
}
}
where open _≅_ t≅X
up-to-iso-unique : ∀ t t′ → (i : ⊤ t ≅ ⊤ t′) → up-to-iso t t′ ≃ i
up-to-iso-unique t t′ i = ⌞ !-unique t′ _ ⌟
up-to-iso-invˡ : ∀ {t X} {i : ⊤ t ≅ X} → up-to-iso t (transport-by-iso t i) ≃ i
up-to-iso-invˡ {t} {i = i} = up-to-iso-unique t (transport-by-iso t i) i
up-to-iso-invʳ : ∀ {t t′} → ⊤ (transport-by-iso t (up-to-iso t t′)) ≡ ⊤ t′
up-to-iso-invʳ {t} {t′} = ≡.refl