Cauchy products of species of types

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2023-04-27.
Last modified on 2023-05-16.

module species.cauchy-products-species-of-types where
Imports 
open import foundation.cartesian-product-types
open import foundation.coproduct-decompositions
open import foundation.dependent-pair-types
open import foundation.universe-levels

open import species.species-of-types

Idea

The Cauchy product of two species of types S and T on X is defined as

  Σ (k : UU) (Σ (k' : UU) (Σ (e : k + k' ≃ X) S(k) × T(k')))

Definition

module _
  {l1 l2 l3 : Level}
  (S : species-types l1 l2)
  (T : species-types l1 l3)
  where

  cauchy-product-species-types : species-types l1 (lsuc l1 ⊔ l2 ⊔ l3)
  cauchy-product-species-types X =
    Σ ( binary-coproduct-Decomposition l1 l1 X)
      ( λ d →
        S (left-summand-binary-coproduct-Decomposition d) ×
        T (right-summand-binary-coproduct-Decomposition d))

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