Cycle index series of species

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

Created on 2023-03-21.
Last modified on 2023-10-09.

module species.cycle-index-series-species-of-types where
open import elementary-number-theory.natural-numbers

open import foundation.dependent-pair-types
open import foundation.universe-levels

open import univalent-combinatorics.cyclic-finite-types


The cycle index series of a species of types F is a type family indexed by finite families of cyclic types. Note that a finite family of cyclic types Cᵢ uniquely determines a permutation e on the disjoint union C := Σᵢ Cᵢ of the underlying types of the Cᵢ. This permutation determines an action F e on F C. The cycle index series of F at the family Cᵢ is the type Fix (F e) of fixed points of F e.


total-type-family-of-cyclic-types :
  {l1 l2 : Level} (X : UU l1) (C : X  Σ  (Cyclic-Type l2)) 
  UU (l1  l2)
total-type-family-of-cyclic-types X C =
  Σ X  x  type-Cyclic-Type (pr1 (C x)) (pr2 (C x)))

permutation-family-of-cyclic-types :
  {l1 l2 : Level} (X : 𝔽 l1) (C : type-𝔽 X → Σ ℕ (Cyclic-Type l2)) →
  Aut (total-type-family-of-cyclic-types X C)
permutation-family-of-cyclic-types X C = {!!}

cycle-index-series-species-types :
  {l1 l2 : Level} (F : species-types l1 l2) (X : 𝔽 l1) →
  (type-𝔽 X → Σ ℕ (Cyclic-Type {!!} ∘ succ-ℕ)) →
  UU {!!}
cycle-index-series-species-types F X C =
  Σ {!F (total-type-family-of-cyclic-types X C)!} {!!}

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