Dirichlet series of species of finite inhabited types
Content created by Fredrik Bakke and Victor Blanchi.
Created on 2023-05-22.
Last modified on 2025-02-14.
module species.dirichlet-series-species-of-finite-inhabited-types where
Imports
open import elementary-number-theory.natural-numbers open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.universe-levels open import species.species-of-finite-inhabited-types open import univalent-combinatorics.cycle-prime-decomposition-natural-numbers open import univalent-combinatorics.finite-types open import univalent-combinatorics.inhabited-finite-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 (U : UU) (T (U) × (S → U))
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 have picked the concrete group cycle-prime-decomposition-ℕ (n)
because it
is closed under cartesian product and also because its groupoid cardinality is
equal to 1/n
.
Definition
dirichlet-series-species-Inhabited-Finite-Type : {l1 l2 l3 : Level} → species-Inhabited-Finite-Type l1 l2 → UU l3 → UU (lsuc l1 ⊔ l2 ⊔ l3) dirichlet-series-species-Inhabited-Finite-Type {l1} T S = Σ ( ℕ) ( λ n → Σ ( Type-With-Cardinality-ℕ l1 (succ-ℕ n)) ( λ F → type-Finite-Type ( T ( type-Type-With-Cardinality-ℕ (succ-ℕ n) F , is-finite-and-inhabited-type-Type-With-Cardinality-ℕ-succ-ℕ n F)) × S → cycle-prime-decomposition-ℕ (succ-ℕ n) _))
Recent changes
- 2025-02-14. Fredrik Bakke. Rename
UU-Fin
toType-With-Cardinality-ℕ
(#1316). - 2025-02-11. Fredrik Bakke. Switch from
𝔽
toFinite-*
(#1312). - 2023-05-22. Victor Blanchi and Fredrik Bakke. Definition of dirichlet series (#626).