Dirichlet series of species of types

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

Created on 2023-06-08.
Last modified on 2025-08-30.

module species.dirichlet-series-species-of-types where
Imports 
open import foundation.cartesian-product-types
open import foundation.dependent-pair-types
open import foundation.universe-levels

open import species.species-of-types

Idea

In classical mathematics, the Dirichlet series of a species of finite inhabited types T is the formal series in s:

  Σ (n : ℕ∖{0}), (|T({1,...,n})| n^(-s) / n!).

If s is a negative integer, the categorified version of this formula is

  Σ (F : Finite-Type∖{∅}), T(F) × (S → F).

We can generalize it to species of types as

  Σ (X : UU), (T(X) × (S → X)).

The interesting case is when s is a positive number. The categorified version of this formula then becomes

  Σ ( n : ℕ∖{0}),
    ( Σ ( F : Type-With-Cardinality-ℕ n),
        ( T(F) × (S → cycle-prime-decomposition-ℕ n)).

We can generalize the two notions to species of types. Let H : UU → UU be a species such that for every X , Y : P the following equivalence is satisfied H (X × Y) ≃ H X × H Y. Then we can define the H-Dirichlet series of any species of types T by

  Σ (X : P), (T(X) × (S → H(X))).

The condition on H ensure that all the usual properties of the Dirichlet series are satisfied.

Definition

module _
  {l1 l2 l3 l4 : Level}
  (H : species-types l1 l2)
  (C1 : preserves-product-species-types H)
  (T : species-types l1 l3)
  (S : UU l4)
  where

  dirichlet-series-species-types : UU (lsuc l1 ⊔ l2 ⊔ l3 ⊔ l4)
  dirichlet-series-species-types = Σ (UU l1) (λ X → (T X) × (S → H (X)))

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