Investigations on graph-theoretical constructions in Homotopy type theory

{-# OPTIONS --without-K --exact-split #-}

module foundations.FibreType where
open import foundations.BasicTypes

module
  _ {ℓ₁ ℓ₂ : Level} {A : Type ℓ₁}{B : Type ℓ₂}
  where

  fibre hfiber
    : (f : A → B)
    → (b : B)
    ---------------
    → Type (ℓ₁ ⊔ ℓ₂)

  fibre f b = Σ A (λ a → f a ≡ b)

  hfiber = fibre
  fib    = fibre
  fiber  = fibre

  syntax fibre f b = f // b

A function applied over the fiber returns the original point

  fib-eq
    : ∀ {f : A → B} {b : B}
    → (h : fib f b)
    ------------------
    → f (proj₁ h) ≡ b

  fib-eq (a , α) = α

Each point is on the fiber of its image.

  fib-image
    :  ∀ {f : A → B} → {a : A}
    → fib f (f a)

  fib-image {f} {a} = a , refl (f a)

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