Cartesian products of types equipped with endomorphisms

Content created by Fredrik Bakke, Egbert Rijke and Victor Blanchi.

Created on 2023-05-10.
Last modified on 2024-02-06.

module structured-types.cartesian-products-types-equipped-with-endomorphisms where
Imports 
open import foundation.cartesian-product-types
open import foundation.dependent-pair-types
open import foundation.functoriality-cartesian-product-types
open import foundation.universe-levels

open import structured-types.types-equipped-with-endomorphisms

Idea

The cartesian product of two types equipped with an endomorphism (A , f) and (B , g) is defined as (A × B , f × g)

Definitions

module _
  {l1 l2 : Level}
  (A : Type-With-Endomorphism l1) (B : Type-With-Endomorphism l2)
  where

  type-product-Type-With-Endomorphism : UU (l1 ⊔ l2)
  type-product-Type-With-Endomorphism =
    type-Type-With-Endomorphism A × type-Type-With-Endomorphism B

  endomorphism-product-Type-With-Endomorphism :
    type-product-Type-With-Endomorphism → type-product-Type-With-Endomorphism
  endomorphism-product-Type-With-Endomorphism =
    map-product
      ( endomorphism-Type-With-Endomorphism A)
      ( endomorphism-Type-With-Endomorphism B)

  product-Type-With-Endomorphism :
    Type-With-Endomorphism (l1 ⊔ l2)
  pr1 product-Type-With-Endomorphism =
    type-product-Type-With-Endomorphism
  pr2 product-Type-With-Endomorphism =
    endomorphism-product-Type-With-Endomorphism

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