open import Cat.Diagram.Limit.Base
open import Cat.Diagram.Initial
open import Cat.Prelude

import Cat.Reasoning as Cat

open make-is-limit
open Initial

module Cat.Diagram.Limit.Initial where

Initial objects as limits🔗

This module provides a characterisation of initial objects as limits, rather than as colimits. Namely, while an initial object is the colimit of the empty diagram, it is the limit of the identity functor, considered as a diagram.

Since the identity functor on $\mathcal{C}$ has $\mathcal{C}$ itself as a domain, computing its limit by general means is usually infeasible. However, if we have such a limit handy, then we know that $\mathcal{C}$ has an initial object. Conversely, if $\mathcal{C}$ has an initial object, then, as a triviality, we can say that it admits at least one “large” limit.

module _ {o ℓ} {C : Precategory o ℓ} (L : Limit (Id {C = C})) where
  open Cat C
  private
    module L = Limit L

    rem₁ : L.ψ L.apex ≡ id
    rem₁ = L.unique₂ (λ j → L.ψ j) (λ f → L.commutes f) (λ j → L.commutes _) (λ j → idr _)

  Id-limit→Initial : Initial C
  Id-limit→Initial .bot = L.apex
  Id-limit→Initial .has⊥ x = λ where
    .centre → L.ψ x
    .paths h → sym (intror rem₁ ∙ L.commutes h)

module _ {o ℓ} {C : Precategory o ℓ} (L : Initial C) where
  open Cat C
  private
    module L = Initial L

  Initial→Id-limit : Limit (Id {C = C})
  Initial→Id-limit = to-limit (to-is-limit mk) where
    mk : make-is-limit Id L.bot
    mk .ψ j = L.¡
    mk .commutes f = L.¡-unique₂ _ _
    mk .universal eps x = eps L.bot
    mk .factors eps p = p _
    mk .unique eps p other x = introl (L.¡-unique₂ _ _) ∙ x L.bot