Cauchy series of species of types in a subuniverse

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-04-27.
Last modified on 2024-02-06.

module species.cauchy-series-species-of-types-in-subuniverses where
open import foundation.cartesian-product-types
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.functoriality-cartesian-product-types
open import foundation.functoriality-dependent-pair-types
open import
open import foundation.postcomposition-functions
open import foundation.propositions
open import foundation.subuniverses
open import foundation.type-arithmetic-dependent-pair-types
open import foundation.universe-levels

open import species.cauchy-series-species-of-types
open import species.species-of-types-in-subuniverses


The Cauchy series of a species S of types in subuniverse from P to Q at X is defined as :

Σ (U : type-subuniverse P) (S(U) × (U → X))


module _
  {l1 l2 l3 l4 l5 : Level}
  (P : subuniverse l1 l2)
  (Q : subuniverse l3 l4)
  (S : species-subuniverse P Q)
  (X : UU l5)

  cauchy-series-species-subuniverse :
    UU (lsuc l1  l2  l3  l5)
  cauchy-series-species-subuniverse =
    Σ ( type-subuniverse P)
      ( λ U  inclusion-subuniverse Q (S U) × (inclusion-subuniverse P U  X))


Equivalent form with species of types

  equiv-cauchy-series-Σ-extension-species-subuniverse :
    cauchy-series-species-types (Σ-extension-species-subuniverse P Q S) X
  equiv-cauchy-series-Σ-extension-species-subuniverse =
    ( equiv-tot
      ( λ U 
          ( type-Prop (P U))
          ( λ p  inclusion-subuniverse Q (S (U , p)))
          ( λ _  U  X))) ∘e
    ( associative-Σ
      ( UU l1)
      ( λ U  type-Prop (P U))
      ( λ U  Σ ( inclusion-subuniverse Q (S U))  _  pr1 U  X)))


module _
  {l1 l2 l3 l4 l5 : Level}
  (P : subuniverse l1 l2)
  (Q : global-subuniverse  l  l))
  (S : species-subuniverse P (subuniverse-global-subuniverse Q l3))
  (T : species-subuniverse P (subuniverse-global-subuniverse Q l4))
  (f :
    (F : type-subuniverse P) 
    inclusion-subuniverse (subuniverse-global-subuniverse Q l3) (S F) 
    inclusion-subuniverse (subuniverse-global-subuniverse Q l4) (T F))
  (X : UU l5)

  equiv-cauchy-series-equiv-species-subuniverse :
      ( P)
      ( subuniverse-global-subuniverse Q l3)
      ( S)
      ( X) 
      ( P)
      ( subuniverse-global-subuniverse Q l4)
      ( T)
      ( X)
  equiv-cauchy-series-equiv-species-subuniverse =
    equiv-tot λ X  equiv-product (f X) id-equiv

module _
  {l1 l2 l3 l4 l5 l6 : Level}
  (P : subuniverse l1 l2)
  (Q : subuniverse l3 l4)
  (S : species-subuniverse P Q)
  (X : UU l5)
  (Y : UU l6)
  (e : X  Y)

  equiv-cauchy-series-species-subuniverse :
    cauchy-series-species-subuniverse P Q S X 
    cauchy-series-species-subuniverse P Q S Y
  equiv-cauchy-series-species-subuniverse =
      ( λ F 
        equiv-product id-equiv (equiv-postcomp (inclusion-subuniverse P F) e))

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