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

open import Cat.Diagram.Initial C
open import Cat.Diagram.Terminal C

Zero objects🔗

In some categories, Initial and Terminal objects coincide. When this occurs, we call the object a zero object.

record is-zero (ob : Ob) : Type (o ⊔ h) where
  field
    has-is-initial  : is-initial ob
    has-is-terminal : is-terminal ob

record Zero : Type (o ⊔ h) where
  field
    ∅       : Ob
    has-is-zero : is-zero ∅

  open is-zero has-is-zero public

  terminal : Terminal
  terminal = record { top = ∅ ; has⊤ = has-is-terminal }

  initial : Initial
  initial = record { bot = ∅ ; has⊥ = has-is-initial }

  open Terminal terminal public hiding (top)
  open Initial initial public hiding (bot)

A curious fact about zero objects is that their existence implies that every hom set is inhabited!

  zero→ : ∀ {x y} → Hom x y
  zero→ = ¡ ∘ !

  zero-∘l : ∀ {x y z} → (f : Hom y z) → f ∘ zero→ {x} {y} ≡ zero→
  zero-∘l f = pulll (sym (¡-unique (f ∘ ¡)))

  zero-∘r : ∀ {x y z} → (f : Hom x y) → zero→ {y} {z} ∘ f ≡ zero→
  zero-∘r f = pullr (sym (!-unique (! ∘ f)))

  zero-comm : ∀ {x y z} → (f : Hom y z) → (g : Hom x y) → f ∘ zero→  ≡ zero→ ∘ g
  zero-comm f g = zero-∘l f ∙ sym (zero-∘r g)

  zero-comm-sym : ∀ {x y z} → (f : Hom y z) → (g : Hom x y) → zero→ ∘ f  ≡ g ∘ zero→
  zero-comm-sym f g = zero-∘r f ∙ sym (zero-∘l g)

Intuition🔗

Most categories that have zero objects have enough structure to rule out totally trivial structures like the empty set, but not enough structure to cause the initial and terminal objects to “separate”. The canonical example here is the category of groups: the unit rules out a completely trivial group, yet there’s nothing else that would require the initial object to have any more structure.

Another point of interest is that any category with zero objects is canonically enriched in pointed sets: the zero→ morphism from earlier acts as the designated basepoint for each of the hom sets.