Ian Ray, 07/02/2024

Singleton Properties. Of course there are alot more we can add to this file.
For now we will show that singletons are closed under retracts and Σ types.

\begin{code}

{-# OPTIONS --safe --without-K #-}

open import MLTT.Spartan
open import UF.Equiv
open import UF.EquivalenceExamples
open import UF.Retracts
open import UF.Subsingletons

module UF.Singleton-Properties where

singleton-closed-under-retract : (X : 𝓤 ̇ ) (Y : 𝓥 ̇ )
                                retract X of Y
                                is-singleton Y
                                is-singleton X
singleton-closed-under-retract X Y (r , s , H) (c , C) = (r c , C')
 where
  C' : is-central X (r c)
  C' x = r c      =⟨ ap r (C (s x)) 
         r (s x)  =⟨ H x 
         x         

Σ-is-singleton : {X : 𝓤 ̇ } {A : X  𝓥 ̇ }
                is-singleton X
                ((x : X)  is-singleton (A x))
                is-singleton (Σ A)
Σ-is-singleton {X = X} {A = A} (c , C) h = ((c , center (h c)) , Σ-centrality)
 where
  Σ-centrality : is-central (Σ A) (c , center (h c))
  Σ-centrality (x , a) =  Σ-=-≃ ⌝⁻¹ (C x , p)
   where
    p = transport A (C x) (center (h c)) =⟨ centrality (h x)
                                                (transport A (C x)
                                                     (center (h c))) ⁻¹ 
        center (h x)                     =⟨ centrality (h x) a 
        a                                

\end{code}