Cauchy series of species of types in subuniverses

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-04-27.
Last modified on 2025-08-30.

module species.cauchy-series-species-of-types-in-subuniverses where
Imports 
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 foundation.global-subuniverses
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

Idea

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)).

Definition

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

  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))

Property

Equivalent form with species of types

  equiv-cauchy-series-Σ-extension-species-subuniverse :
    cauchy-series-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 → inv-associative-Σ _ _ _)) ∘e (associative-Σ _ _ _)

Equivalences

module _
  {α : Level → Level} {l1 l2 l3 l4 l5 : Level}
  (P : subuniverse l1 l2) (Q : global-subuniverse α)
  (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)
  where

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

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)
  where

  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 =
    equiv-tot
      ( λ F →
        equiv-product-right (equiv-postcomp (inclusion-subuniverse P F) e))

Recent changes