This is an archived version pinned as of the submission of my master's thesis. An up-to-date version may be found online.

Dependent pair types

module foundation.dependent-pair-types where
Imports 
open import foundation.universe-levels

Idea

Consider a type family B over A. The dependent pair type Σ A B is the type consisting of (dependent) pairs (a , b) where a : A and b : B a. Such pairs are sometimes called dependent pairs because the type of b depends on the value of the first component a.

Definitions

The dependent pair type

record Σ {l1 l2 : Level} (A : UU l1) (B : A → UU l2) : UU (l1 ⊔ l2) where
  constructor pair
  field
    pr1 : A
    pr2 : B pr1

open Σ public

{-# BUILTIN SIGMA Σ #-}
{-# INLINE pair #-}

infixr 3 _,_
pattern _,_ a b = pair a b

The induction principle for dependent pair types

ind-Σ :
  {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : Σ A B → UU l3} →
  ((x : A) (y : B x) → C (x , y)) → (t : Σ A B) → C t
ind-Σ f (x , y) = f x y

The recursion principle for dependent pair types

rec-Σ :
  {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : UU l3} →
  ((x : A) → B x → C) → Σ A B → C
rec-Σ = ind-Σ

The evaluation function for dependent pairs

ev-pair :
  {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : Σ A B → UU l3} →
  ((t : Σ A B) → C t) → (x : A) (y : B x) → C (x , y)
ev-pair f x y = f (x , y)

We show that ev-pair is the inverse to ind-Σ in foundation.universal-property-dependent-pair-types.

Iterated dependent pair constructors

triple :
  {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : (x : A) → B x → UU l3} →
  (a : A) (b : B a) → C a b → Σ A (λ x → Σ (B x) (C x))
triple a b c = (a , b , c)

triple' :
  {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : Σ A B → UU l3} →
  (a : A) (b : B a) → C (pair a b) → Σ (Σ A B) C
triple' a b c = ((a , b) , c)

Families on dependent pair types

module _
  {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2}
  where

  fam-Σ : ((x : A) → B x → UU l3) → Σ A B → UU l3
  fam-Σ C (x , y) = C x y