open import Cat.Univalent open import Cat.Prelude module Cat.Diagram.Initial {o h} (C : Precategory o h) where

# Initial objects🔗

An object $\bot$ of a category $\mathcal{C}$ is said to be **initial** if there exists a *unique* map to any other object:

is-initial : Ob → Type _ is-initial ob = ∀ x → is-contr (Hom ob x) record Initial : Type (o ⊔ h) where field bot : Ob has⊥ : is-initial bot

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

¡ : ∀ {x} → Hom bot x ¡ = has⊥ _ .centre ¡-unique : ∀ {x} (h : Hom bot x) → ¡ ≡ h ¡-unique = has⊥ _ .paths ¡-unique₂ : ∀ {x} (f g : Hom bot x) → f ≡ g ¡-unique₂ = is-contr→is-prop (has⊥ _) open Initial

## Intuition🔗

The intuition here is that we ought to think about an initial object as having “the least amount of structure possible”, insofar that it can be mapped *into* any other object. For the category of `Sets`

, this is the empty set; there is no required structure beyond “being a set”, so the empty set sufficies.

In more structured categories, the situation becomes a bit more interesting. Once our category has enough structure that we can’t build maps from a totally trivial thing, the initial object begins to behave like a notion of **Syntax** for our category. The idea here is that we have a *unique* means of interpreting our syntax into any other object, which is exhibited by the universal map ¡

## Uniqueness🔗

One important fact about initial objects is that they are **unique** up to isomorphism:

⊥-unique : (i i′ : Initial) → bot i ≅ bot i′ ⊥-unique i i′ = make-iso (¡ i) (¡ i′) (¡-unique₂ i′ _ _) (¡-unique₂ i _ _)

Additionally, if $C$ is a category, then the space of initial objects is a proposition:

⊥-contractible : is-category C → is-prop Initial ⊥-contractible ccat x1 x2 i .bot = iso→path C ccat (⊥-unique x1 x2) i ⊥-contractible ccat x1 x2 i .has⊥ ob = is-prop→pathp (λ i → is-contr-is-prop {A = Hom (iso→path C ccat (⊥-unique x1 x2) i) _}) (x1 .has⊥ ob) (x2 .has⊥ ob) i