open import Cat.Univalent
open import Cat.Prelude

module Cat.Diagram.Terminal {o h} (C : Precategory o h) where

Terminal objects🔗

An object \top of a category C\mathcal{C} is said to be terminal if it admits a unique map from any other object:

is-terminal : Ob  Type _
is-terminal ob =  x  is-contr (Hom x ob)

record Terminal : Type (o  h) where
  field
    top : Ob
    has⊤ : is-terminal top

We refer to the centre of contraction as !. Since it inhabits a contractible type, it is unique.

  ! :  {x}  Hom x top
  ! = has⊤ _ .centre

  !-unique :  {x} (h : Hom x top)  !  h
  !-unique = has⊤ _ .paths

  !-unique₂ :  {x} (f g : Hom x top)  f  g
  !-unique₂ = is-contr→is-prop (has⊤ _)

open Terminal

Uniqueness🔗

If a category has two terminal objects t1t_1 and t2t_2, then there is a unique isomorphism t1t2t_1 \cong t_2. We first establish the isomorphism: Since t1t_1 (resp. t2t_2) is terminal, there is a unique map !1:t1t2!_1 : t_1 \to t_2 (resp. !2:t2t1!_2 : t_2 \to t_1). To show these maps are inverses, we must show that !1!2!_1 \circ !_2 is id\id{id}; But these morphisms inhabit a contractible space, namely the space of maps into t2t_2, so they are equal.

!-invertible : (t1 t2 : Terminal)  is-invertible (! t1 {top t2})
!-invertible t1 t2 = make-invertible (! t2) (!-unique₂ t1 _ _) (!-unique₂ t2 _ _)

⊤-unique : (t1 t2 : Terminal)  top t1  top t2
⊤-unique t1 t2 = invertible→iso (! t2) (!-invertible t2 t1)

Hence, if CC is additionally a category, it has a propositional space of terminal objects:

⊤-contractible : is-category C  is-prop Terminal
⊤-contractible ccat x1 x2 i .top =
  iso→path C ccat (⊤-unique x1 x2) i

⊤-contractible ccat x1 x2 i .has⊤ ob =
  is-prop→pathp
     i  is-contr-is-prop {A = Hom _
      (iso→path C ccat (⊤-unique x1 x2) i)})
    (x1 .has⊤ ob) (x2 .has⊤ ob) i

is-terminal-iso :  {A B}  A  B  is-terminal A  is-terminal B
is-terminal-iso isom term x = contr (isom .to  term x .centre) λ h 
  isom .to  term x .centre ≡⟨ ap (isom .to ∘_) (term x .paths _) 
  isom .to  isom .from  h ≡⟨ cancell (isom .invl) 
  h