Martin Escardo 8th May 2020.

An old version of this file is at UF.Classifiers-Old.

This version is ported from the Midlands Graduate School 2019 lecture notes

 https://www.cs.bham.ac.uk/~mhe/HoTT-UF.in-Agda-Lecture-Notes/HoTT-UF-Agda.html
 https://github.com/martinescardo/HoTT-UF.Agda-Lecture-Notes

but with the universe levels generalized and Ξ£-fibers added in
September 2022.

   * Universes are map classifiers.
   * Ξ© 𝓀 is the embedding classifier.
   * The type of pointed types is the retraction classifier.
   * The type inhabited types is the surjection classifier.
   * The fiber of Ξ£ are non-dependent function types.

\begin{code}

{-# OPTIONS --without-K --exact-split --safe --auto-inline #-}

module UF.Classifiers where

open import MLTT.Spartan
open import UF.Base
open import UF.Embeddings
open import UF.Equiv
open import UF.Univalence
open import UF.FunExt
open import UF.UA-FunExt
open import UF.Subsingletons
open import UF.Subsingletons-FunExt
open import UF.Powerset hiding (𝕋)
open import UF.EquivalenceExamples
open import UF.Retracts

\end{code}

The slice type:

\begin{code}

_/_ : (𝓀 : Universe) β†’ π“₯ Μ‡ β†’ 𝓀 ⁺ βŠ” π“₯ Μ‡
𝓀 / Y = Ξ£ X κž‰ 𝓀 Μ‡ , (X β†’ Y)

\end{code}

A modification of the slice type, where P doesn't need to be
proposition-valued and so can add structure. An example is P = id,
which is studied below in connection with retractions.

\begin{code}

_/[_]_ : (𝓀 : Universe) β†’ (𝓀 βŠ” π“₯ Μ‡ β†’ 𝓦 Μ‡ ) β†’ π“₯ Μ‡ β†’ 𝓀 ⁺ βŠ” π“₯ βŠ” 𝓦 Μ‡
𝓀 /[ P ] Y = Ξ£ X κž‰ 𝓀 Μ‡ , Ξ£ f κž‰ (X β†’ Y) , ((y : Y) β†’ P (fiber f y))

\end{code}

We first consider the original situation of the MGS development with a
single universe for comparison.

\begin{code}

module classifier-single-universe (𝓀 : Universe) where

 Ο‡ : (Y : 𝓀 Μ‡ ) β†’ 𝓀 / Y  β†’ (Y β†’ 𝓀 Μ‡ )
 Ο‡ Y (X , f) = fiber f

 universe-is-classifier : 𝓀 ⁺ Μ‡
 universe-is-classifier = (Y : 𝓀 Μ‡ ) β†’ is-equiv (Ο‡ Y)

 𝕋 : (Y : 𝓀 Μ‡ ) β†’ (Y β†’ 𝓀 Μ‡ ) β†’ 𝓀 / Y
 𝕋 Y A = Ξ£ A , pr₁

 χη : is-univalent 𝓀
    β†’ (Y : 𝓀 Μ‡ ) (Οƒ : 𝓀 / Y) β†’ 𝕋 Y (Ο‡ Y Οƒ) ＝ Οƒ
 χη ua Y (X , f) = r
  where
   e : Ξ£ (fiber f) ≃ X
   e = total-fiber-is-domain f

   p : Σ (fiber f) ＝ X
   p = eqtoid ua (Ξ£ (fiber f)) X e

   NB : ⌜ e ⌝⁻¹ ＝ (Ξ» x β†’ f x , x , refl)
   NB = refl

   q = transport (Ξ» - β†’ - β†’ Y) p pr₁ ＝⟨ transport-is-pre-comp' ua e pr₁ ⟩
       pr₁ ∘ ⌜ e ⌝⁻¹                 ＝⟨ refl ⟩
       f                             ∎

   r : (Ξ£ (fiber f) , pr₁) ＝ (X , f)
   r = to-Σ-＝ (p , q)

 χΡ : is-univalent 𝓀
    β†’ funext 𝓀 (𝓀 ⁺)
    β†’ (Y : 𝓀 Μ‡ ) (A : Y β†’ 𝓀 Μ‡ ) β†’ Ο‡ Y (𝕋 Y A) ＝ A
 χΡ ua fe Y A = dfunext fe Ξ³
  where
   f : βˆ€ y β†’ fiber pr₁ y β†’ A y
   f y ((y , a) , refl) = a

   g : βˆ€ y β†’ A y β†’ fiber pr₁ y
   g y a = (y , a) , refl

   Ξ· : βˆ€ y Οƒ β†’ g y (f y Οƒ) ＝ Οƒ
   Ξ· y ((y , a) , refl) = refl

   Ξ΅ : βˆ€ y a β†’ f y (g y a) ＝ a
   Ξ΅ y a = refl

   Ξ³ : βˆ€ y β†’ fiber pr₁ y ＝ A y
   Ξ³ y = eqtoid ua _ _ (qinveq (f y) (g y , Ξ· y , Ξ΅ y))

 universes-are-classifiers : is-univalent 𝓀
                           β†’ funext 𝓀 (𝓀 ⁺)
                           β†’ universe-is-classifier
 universes-are-classifiers ua fe Y = qinvs-are-equivs (Ο‡ Y)
                                      (𝕋 Y , χη ua Y , χΡ ua fe Y)

 classification : is-univalent 𝓀
                β†’ funext 𝓀 (𝓀 ⁺)
                β†’ (Y : 𝓀 Μ‡ ) β†’ 𝓀 / Y ≃ (Y β†’ 𝓀 Μ‡ )
 classification ua fe Y = Ο‡ Y , universes-are-classifiers ua fe Y

module special-classifier-single-universe (𝓀 : Universe) where

 open classifier-single-universe 𝓀

 Ο‡-special : (P : 𝓀 Μ‡ β†’ π“₯ Μ‡ ) (Y : 𝓀 Μ‡ ) β†’ 𝓀 /[ P ] Y  β†’ (Y β†’ Ξ£ P)
 Ο‡-special P Y (X , f , Ο†) y = fiber f y , Ο† y

 universe-is-special-classifier : (𝓀 Μ‡ β†’ π“₯ Μ‡ ) β†’ 𝓀 ⁺ βŠ” π“₯ Μ‡
 universe-is-special-classifier P = (Y : 𝓀 Μ‡ ) β†’ is-equiv (Ο‡-special P Y)

 classifier-gives-special-classifier : universe-is-classifier
                                     β†’ (P : 𝓀 Μ‡ β†’ π“₯ Μ‡ )
                                     β†’ universe-is-special-classifier P
 classifier-gives-special-classifier s P Y = Ξ³
  where
   e = (𝓀 /[ P ] Y)                               β‰ƒβŸ¨ a ⟩
       (Ξ£ Οƒ κž‰ 𝓀 / Y , ((y : Y) β†’ P ((Ο‡ Y) Οƒ y)))  β‰ƒβŸ¨ b ⟩
       (Ξ£ A κž‰ (Y β†’ 𝓀 Μ‡ ), ((y : Y) β†’ P (A y)))     β‰ƒβŸ¨ c ⟩
       (Y β†’ Ξ£ P)                                  β– 
    where
     a = ≃-sym Ξ£-assoc
     b = Ξ£-change-of-variable (Ξ» A β†’ Ξ  (P ∘ A)) (Ο‡ Y) (s Y)
     c = ≃-sym Ξ Ξ£-distr-≃

   NB : Ο‡-special P Y ＝ ⌜ e ⌝
   NB = refl

   Ξ³ : is-equiv (Ο‡-special P Y)
   γ = ⌜⌝-is-equiv e

 Ο‡-special-is-equiv : is-univalent 𝓀
                    β†’ funext 𝓀 (𝓀 ⁺)
                    β†’ (P : 𝓀 Μ‡ β†’ π“₯ Μ‡ ) (Y : 𝓀 Μ‡ )
                    β†’ is-equiv (Ο‡-special P Y)
 Ο‡-special-is-equiv ua fe P Y = classifier-gives-special-classifier (universes-are-classifiers ua fe) P Y

 special-classification : is-univalent 𝓀
                        β†’ funext 𝓀 (𝓀 ⁺)
                        β†’ (P : 𝓀 Μ‡ β†’ π“₯ Μ‡ ) (Y : 𝓀 Μ‡ )
                        β†’ 𝓀 /[ P ] Y ≃ (Y β†’ Ξ£ P)
 special-classification ua fe P Y = Ο‡-special P Y , Ο‡-special-is-equiv ua fe P Y

\end{code}

Some examples of special classifiers follow.

The universe of pointed types classifies retractions:

\begin{code}

module retraction-classifier (𝓀 : Universe) where

 open special-classifier-single-universe 𝓀

 retractions-into : 𝓀 Μ‡ β†’ 𝓀 ⁺ Μ‡
 retractions-into Y = Ξ£ X κž‰ 𝓀 Μ‡ , Y ◁ X

 pointed-types : (𝓀 : Universe) β†’ 𝓀 ⁺ Μ‡
 pointed-types 𝓀 = Ξ£ X κž‰ 𝓀 Μ‡ , X

 retraction-classifier : Univalence
                       β†’ (Y : 𝓀 Μ‡ ) β†’ retractions-into Y ≃ (Y β†’ pointed-types 𝓀)
 retraction-classifier ua Y =
  retractions-into Y                                              β‰ƒβŸ¨ i ⟩
  ((𝓀 /[ id ] Y))                                                 β‰ƒβŸ¨ ii ⟩
  (Y β†’ pointed-types 𝓀)                                           β– 
  where
   i  = ≃-sym (Ξ£-cong (Ξ» X β†’ Ξ£-cong (Ξ» f β†’ Ξ Ξ£-distr-≃)))
   ii = special-classification (ua 𝓀)
         (univalence-gives-funext' 𝓀 (𝓀 ⁺) (ua 𝓀) (ua (𝓀 ⁺)))
         id Y

\end{code}

The universe of inhabited types classifies surjections:

\begin{code}

module surjection-classifier (𝓀 : Universe) where

 open special-classifier-single-universe 𝓀

 open import UF.PropTrunc

 module _ (pt : propositional-truncations-exist) where

  open PropositionalTruncation pt public
  open import UF.ImageAndSurjection pt public

  surjections-into : 𝓀 Μ‡ β†’ 𝓀 ⁺ Μ‡
  surjections-into Y = Ξ£ X κž‰ 𝓀 Μ‡ , X β†  Y

  inhabited-types : 𝓀 ⁺ Μ‡
  inhabited-types = Ξ£ X κž‰ 𝓀 Μ‡ , βˆ₯ X βˆ₯

  surjection-classification : Univalence
                            β†’ (Y : 𝓀 Μ‡ )
                            β†’ surjections-into Y ≃ (Y β†’ inhabited-types)
  surjection-classification ua =
   special-classification (ua 𝓀)
     (univalence-gives-funext' 𝓀 (𝓀 ⁺) (ua 𝓀) (ua (𝓀 ⁺)))
     βˆ₯_βˆ₯

\end{code}

Added 11th September 2022. We now generalize the universe levels of
the classifier and special classifier modules.

\begin{code}

module classifier (𝓀 π“₯ : Universe) where

 Ο‡ : (Y : 𝓀 Μ‡ ) β†’ (𝓀 βŠ” π“₯) / Y  β†’ (Y β†’ 𝓀 βŠ” π“₯ Μ‡ )
 Ο‡ Y (X , f) = fiber f

\end{code}

Definition of when the given pair of universes is a classifier,

\begin{code}

 universe-is-classifier : (𝓀 βŠ” π“₯)⁺ Μ‡
 universe-is-classifier = (Y : 𝓀 Μ‡ ) β†’ is-equiv (Ο‡ Y)

 𝕋 : (Y : 𝓀 Μ‡ ) β†’ (Y β†’ 𝓀 βŠ” π“₯ Μ‡ ) β†’ (𝓀 βŠ” π“₯) / Y
 𝕋 Y A = Ξ£ A , pr₁

 χη : is-univalent (𝓀 βŠ” π“₯)
    β†’ (Y : 𝓀 Μ‡ ) (Οƒ : (𝓀 βŠ” π“₯) / Y) β†’ 𝕋 Y (Ο‡ Y Οƒ) ＝ Οƒ
 χη ua Y (X , f) = r
  where
   e : Ξ£ (fiber f) ≃ X
   e = total-fiber-is-domain f

   p : Σ (fiber f) ＝ X
   p = eqtoid ua (Ξ£ (fiber f)) X e

   NB : ⌜ e ⌝⁻¹ ＝ (Ξ» x β†’ f x , x , refl)
   NB = refl

   q = transport (Ξ» - β†’ - β†’ Y) p pr₁ ＝⟨ transport-is-pre-comp' ua e pr₁ ⟩
       pr₁ ∘ ⌜ e ⌝⁻¹                 ＝⟨ refl ⟩
       f                             ∎

   r : (Ξ£ (fiber f) , pr₁) ＝ (X , f)
   r = to-Σ-＝ (p , q)

 χΡ : is-univalent (𝓀 βŠ” π“₯)
    β†’ funext 𝓀 ((𝓀 βŠ” π“₯)⁺)
    β†’ (Y : 𝓀 Μ‡ ) (A : Y β†’ 𝓀 βŠ” π“₯ Μ‡ ) β†’ Ο‡ Y (𝕋 Y A) ＝ A
 χΡ ua fe Y A = dfunext fe Ξ³
  where
   f : βˆ€ y β†’ fiber pr₁ y β†’ A y
   f y ((y , a) , refl) = a

   g : βˆ€ y β†’ A y β†’ fiber pr₁ y
   g y a = (y , a) , refl

   Ξ· : βˆ€ y Οƒ β†’ g y (f y Οƒ) ＝ Οƒ
   Ξ· y ((y , a) , refl) = refl

   Ξ΅ : βˆ€ y a β†’ f y (g y a) ＝ a
   Ξ΅ y a = refl

   Ξ³ : βˆ€ y β†’ fiber pr₁ y ＝ A y
   Ξ³ y = eqtoid ua _ _ (qinveq (f y) (g y , Ξ· y , Ξ΅ y))

 universes-are-classifiers : is-univalent (𝓀 βŠ” π“₯)
                           β†’ funext 𝓀 ((𝓀 βŠ” π“₯)⁺)
                           β†’ universe-is-classifier
 universes-are-classifiers ua fe Y = qinvs-are-equivs (Ο‡ Y)
                                          (𝕋 Y , χη ua Y , χΡ ua fe Y)

 classification : is-univalent (𝓀 βŠ” π“₯)
                β†’ funext 𝓀 ((𝓀 βŠ” π“₯)⁺)
                β†’ (Y : 𝓀 Μ‡ ) β†’ (𝓀 βŠ” π“₯) / Y ≃ (Y β†’ 𝓀 βŠ” π“₯ Μ‡ )
 classification ua fe Y = Ο‡ Y , universes-are-classifiers ua fe Y

\end{code}

In the case of special classifiers we now need to assume a further
universe 𝓦.

\begin{code}

module special-classifier (𝓀 π“₯ 𝓦 : Universe) where

 open classifier 𝓀 π“₯ public

 Ο‡-special : (P : 𝓀 βŠ” π“₯ Μ‡ β†’ 𝓦 Μ‡ ) (Y : 𝓀 Μ‡ ) β†’ (𝓀 βŠ” π“₯) /[ P ] Y  β†’ (Y β†’ Ξ£ P)
 Ο‡-special P Y (X , f , Ο†) y = fiber f y , Ο† y

 universe-is-special-classifier : (𝓀 βŠ” π“₯ Μ‡ β†’ 𝓦 Μ‡ ) β†’ (𝓀 βŠ” π“₯)⁺ βŠ” 𝓦 Μ‡
 universe-is-special-classifier P = (Y : 𝓀 Μ‡ ) β†’ is-equiv (Ο‡-special P Y)

 classifier-gives-special-classifier : universe-is-classifier
                                     β†’ (P : 𝓀 βŠ” π“₯ Μ‡ β†’ 𝓦 Μ‡ )
                                     β†’ universe-is-special-classifier P
 classifier-gives-special-classifier s P Y = Ξ³
  where
   e = ((𝓀 βŠ” π“₯) /[ P ] Y)                               β‰ƒβŸ¨ a ⟩
       (Ξ£ Οƒ κž‰ (𝓀 βŠ” π“₯) / Y , ((y : Y) β†’ P ((Ο‡ Y) Οƒ y)))  β‰ƒβŸ¨ b ⟩
       (Ξ£ A κž‰ (Y β†’ 𝓀 βŠ” π“₯ Μ‡ ), ((y : Y) β†’ P (A y)))       β‰ƒβŸ¨ c ⟩
       (Y β†’ Ξ£ P)                                        β– 
    where
     a = ≃-sym Ξ£-assoc
     b = Ξ£-change-of-variable (Ξ» A β†’ Ξ  (P ∘ A)) (Ο‡ Y) (s Y)
     c = ≃-sym Ξ Ξ£-distr-≃

   NB : Ο‡-special P Y ＝ ⌜ e ⌝
   NB = refl

   Ξ³ : is-equiv (Ο‡-special P Y)
   γ = ⌜⌝-is-equiv e

 Ο‡-special-is-equiv : is-univalent (𝓀 βŠ” π“₯)
                    β†’ funext 𝓀 ((𝓀 βŠ” π“₯)⁺)
                    β†’ (P : 𝓀 βŠ” π“₯ Μ‡ β†’ 𝓦 Μ‡ ) (Y : 𝓀 Μ‡ )
                    β†’ is-equiv (Ο‡-special P Y)
 Ο‡-special-is-equiv ua fe P Y = classifier-gives-special-classifier
                                 (universes-are-classifiers ua fe) P Y

 special-classification : is-univalent (𝓀 βŠ” π“₯)
                        β†’ funext 𝓀 ((𝓀 βŠ” π“₯)⁺)
                        β†’ (P : 𝓀 βŠ” π“₯ Μ‡ β†’ 𝓦 Μ‡ ) (Y : 𝓀 Μ‡ )
                        β†’ (𝓀 βŠ” π“₯) /[ P ] Y ≃ (Y β†’ Ξ£ P)
 special-classification ua fe P Y = Ο‡-special P Y , Ο‡-special-is-equiv ua fe P Y

\end{code}

The subtype classifier with general universes:

\begin{code}

Ξ©-is-subtype-classifier' : is-univalent (𝓀 βŠ” π“₯)
                         β†’ funext 𝓀 ((𝓀 βŠ” π“₯)⁺)
                         β†’ (Y : 𝓀 Μ‡ )
                         β†’ Subtypes' (𝓀 βŠ” π“₯) Y ≃ (Y β†’ Ξ© (𝓀 βŠ” π“₯))
Ξ©-is-subtype-classifier' {𝓀} {π“₯} ua fe = special-classification ua fe
                                          is-subsingleton
 where
  open special-classifier 𝓀 π“₯ (𝓀 βŠ” π“₯)

Ξ©-is-subtype-classifier : is-univalent 𝓀
                        β†’ funext 𝓀 (𝓀 ⁺)
                        β†’ (Y : 𝓀 Μ‡ )
                        β†’ Subtypes Y ≃ (Y β†’ Ξ© 𝓀)
Ξ©-is-subtype-classifier {𝓀} = Ξ©-is-subtype-classifier' {𝓀} {𝓀}

subtypes-form-set : Univalence β†’ (Y : 𝓀 Μ‡ ) β†’ is-set (Subtypes' (𝓀 βŠ” π“₯) Y)
subtypes-form-set {𝓀} {π“₯} ua Y =
 equiv-to-set
  (Ξ©-is-subtype-classifier' {𝓀} {π“₯}
    (ua (𝓀 βŠ” π“₯))
    (univalence-gives-funext' 𝓀 ((𝓀 βŠ” π“₯)⁺)
      (ua 𝓀)
      (ua ((𝓀 βŠ” π“₯)⁺)))
    Y)
  (powersets-are-sets''
    (univalence-gives-funext' _ _ (ua 𝓀) (ua ((𝓀 βŠ” π“₯)⁺)))
    (univalence-gives-funext' _ _ (ua (𝓀 βŠ” π“₯)) (ua (𝓀 βŠ” π“₯)))
    (univalence-gives-propext (ua (𝓀 βŠ” π“₯))))

\end{code}

9th September 2022, with universe levels generalized 11
September. Here is an application of the above.

\begin{code}

Ξ£-fibers-≃ : {𝓀 π“₯ : Universe}
           β†’ is-univalent (𝓀 βŠ” π“₯)
           β†’ funext 𝓀 ((𝓀 βŠ” π“₯)⁺)
           β†’ {X : 𝓀 βŠ” π“₯ Μ‡ } {Y : 𝓀 Μ‡ }
           β†’ (Ξ£ A κž‰ (Y β†’ 𝓀 βŠ” π“₯ Μ‡ ) , Ξ£ A ≃ X) ≃ (X β†’ Y)
Ξ£-fibers-≃ {𝓀} {π“₯} ua fe⁺ {X} {Y} =
  (Ξ£ A κž‰ (Y β†’ 𝓀 βŠ” π“₯ Μ‡ ) , Ξ£ A ≃ X)                            β‰ƒβŸ¨ I ⟩
  (Ξ£ (Z , g) κž‰ (𝓀 βŠ” π“₯) / Y , (Ξ£ y κž‰ Y , fiber g y) ≃ X)      β‰ƒβŸ¨ II ⟩
  (Ξ£ Z κž‰ 𝓀 βŠ” π“₯ Μ‡ , Ξ£ g κž‰ (Z β†’ Y) , (Ξ£ y κž‰ Y , fiber g y) ≃ X) β‰ƒβŸ¨ III ⟩
  (Ξ£ Z κž‰ 𝓀 βŠ” π“₯ Μ‡ , (Z β†’ Y) Γ— (Z ≃ X))                         β‰ƒβŸ¨ IV ⟩
  (Ξ£ Z κž‰ 𝓀 βŠ” π“₯ Μ‡ , (Z ≃ X) Γ— (Z β†’ Y))                         β‰ƒβŸ¨ V ⟩
  (Ξ£ Z κž‰ 𝓀 βŠ” π“₯ Μ‡ , (X ≃ Z) Γ— (Z β†’ Y))                         β‰ƒβŸ¨ VI ⟩
  (Ξ£ (Z , _) κž‰ (Ξ£ Z κž‰ 𝓀 βŠ” π“₯ Μ‡ , X ≃ Z) , (Z β†’ Y))             β‰ƒβŸ¨ VII ⟩
  (X β†’ Y)                                                    β– 
 where
  open classifier 𝓀 π“₯
  open import UF.Equiv-FunExt
  open import UF.PropIndexedPiSigma
  open import UF.Yoneda

  fe : funext (𝓀 βŠ” π“₯) (𝓀 βŠ” π“₯)
  fe = univalence-gives-funext ua

  I   = ≃-sym (Ξ£-change-of-variable (Ξ» A β†’ Ξ£ A ≃ X) (Ο‡ Y)
               (universes-are-classifiers ua fe⁺ Y))
  II  = Ξ£-assoc
  III = Ξ£-cong (Ξ» Z β†’ Ξ£-cong (Ξ» g β†’ ≃-cong-left' fe fe fe fe fe
                                     (total-fiber-is-domain g)))
  IV  = Ξ£-cong (Ξ» Z β†’ Γ—-comm)
  V   = Ξ£-cong (Ξ» Z β†’ Γ—-cong (≃-Sym' fe fe fe fe) (≃-refl (Z β†’ Y)))
  VI  = ≃-sym Ξ£-assoc
  VII = prop-indexed-sum
         (singletons-are-props
           (univalence-via-singletons→ ua X))
         (X , ≃-refl X)

private
 βˆ‘ : {𝓀 π“₯ : Universe} (X : 𝓀 Μ‡ ) (Y : X β†’ π“₯ Μ‡ ) β†’ 𝓀 βŠ” π“₯ Μ‡
 βˆ‘ X Y = Ξ£ Y

\end{code}

The use of equality rather than equivalence prevents us from having
more general universes in the following:

\begin{code}

Ξ£-fibers : is-univalent 𝓀
         β†’ funext 𝓀 (𝓀 ⁺)
         β†’ {X : 𝓀 Μ‡ } {Y : 𝓀 Μ‡ }
         β†’ fiber (βˆ‘ Y) X ≃ (X β†’ Y)
Ξ£-fibers {𝓀} ua fe⁺ {X} {Y} =
  (Ξ£ A κž‰ (Y β†’ 𝓀 Μ‡ ) , Ξ£ A ＝ X) β‰ƒβŸ¨ Ξ£-cong (Ξ» A β†’ univalence-≃ ua (Ξ£ A) X) ⟩
  (Ξ£ A κž‰ (Y β†’ 𝓀 Μ‡ ) , Ξ£ A ≃ X)  β‰ƒβŸ¨ Ξ£-fibers-≃ {𝓀} {𝓀} ua fe⁺ ⟩
  (X β†’ Y)                       β–