Hasse-Weil species

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

Created on 2023-05-25.
Last modified on 2024-02-07.

module species.hasse-weil-species where
Imports 
open import finite-algebra.commutative-finite-rings
open import finite-algebra.products-commutative-finite-rings

open import foundation.cartesian-product-types
open import foundation.equivalences
open import foundation.universe-levels

open import univalent-combinatorics.finite-types

Idea

Let S be a function from the type of commutative finite rings to the finite types that preserves cartesian products. The Hasse-Weil species is a species of finite inhabited types defined for any finite inhabited type k as

Σ (p : structure-semisimple-commutative-ring-𝔽 k) ; S (commutative-finite-ring-finite-semisimple-commutative-ring-structure-semisimple-commutative-ring-𝔽 k p)

Definitions

is-closed-under-products-function-from-Commutative-Ring-𝔽 :
  {l1 l2 : Level} → (Commutative-Ring-𝔽 l1 → 𝔽 l2) → UU (lsuc l1 ⊔ l2)
is-closed-under-products-function-from-Commutative-Ring-𝔽 {l1} {l2} S =
  (R1 R2 : Commutative-Ring-𝔽 l1) →
  ( type-𝔽 (S (product-Commutative-Ring-𝔽 R1 R2))) ≃
  ( type-𝔽 (S R1) × type-𝔽 (S R2))

module _
  {l1 l2 : Level}
  (l3 l4 : Level)
  (S : Commutative-Ring-𝔽 l1 → 𝔽 l2)
  (C : is-closed-under-products-function-from-Commutative-Ring-𝔽 S)
  where

  hasse-weil-species-Inhabited-𝔽 :
    species-Inhabited-𝔽 l1 (l1 ⊔ l2 ⊔ lsuc l3 ⊔ lsuc l4)
  hasse-weil-species-Inhabited-𝔽 ( k , (f , i)) =
    Σ-𝔽 {!!}
        ( λ p →
          S
            ( commutative-finite-ring-Semisimple-Commutative-Ring-𝔽
              ( finite-semisimple-commutative-ring-structure-semisimple-commutative-ring-𝔽
                ( l3)
                ( l4)
                ( k , f)
                ( p))))

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