Dirichlet series of species of types in subuniverses
Content created by Fredrik Bakke, Victor Blanchi and Egbert Rijke.
Created on 2023-05-22.
Last modified on 2025-08-30.
module species.dirichlet-series-species-of-types-in-subuniverses where
Imports
open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.subuniverses open import foundation.universe-levels open import species.species-of-types-in-subuniverses
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 in subuniverses.
Let P
and Q
two subuniverse such that P
is closed under
cartesian products. Let H : P → 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¶
to any species of types in subuniverses 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 l5 l6 : Level} (P : subuniverse l1 l2) (Q : subuniverse l3 l4) (C1 : is-closed-under-products-subuniverse P) (H : species-subuniverse-domain l5 P) (C2 : preserves-product-species-subuniverse-domain P C1 H) (T : species-subuniverse P Q) (S : UU l6) where dirichlet-series-species-subuniverse : UU (lsuc l1 ⊔ l2 ⊔ l3 ⊔ l5 ⊔ l6) dirichlet-series-species-subuniverse = Σ (type-subuniverse P) (λ X → inclusion-subuniverse Q (T X) × (S → H (X)))
Recent changes
- 2025-08-30. Fredrik Bakke. Some cleanup of species (#1502).
- 2025-02-14. Fredrik Bakke. Rename
UU-Fin
toType-With-Cardinality-ℕ
(#1316). - 2025-02-11. Fredrik Bakke. Switch from
𝔽
toFinite-*
(#1312). - 2023-06-09. Fredrik Bakke. Remove unused imports (#648).
- 2023-06-08. Victor Blanchi, Fredrik Bakke and Egbert Rijke. Product of Dirichlet series (#643).