This is the Categories.Functors module of the Agda Universal Algebra Library.

Recall, a *functor* `F`

is a function that maps the objects and morphisms of one category 𝒞 to the objects and morphisms of a category 𝒟 in such a way that the following *functor laws* are satisfied:

- ∀ f g, F(f ∘ g) = F(f) ∘ F(g)
- ∀ A, F(id A) = id (F A) (where id X denotes the identity map on X)

The main reference for this section is Modular Type-Safety Proofs in Agda by Schwaab and Siek (PLPV ‘07).

An important class of functors for our domain is the class of so called *polynomial functors*. These are functors that are built up using the constant, identity, sum, and product functors. To be precise, the actions of the latter on objects are as follows: `∀ A B`

(objects), `∀ F G`

(functors),

`const B A = B`

`Id A = A`

`(F + G) A = F A + G A`

`(F × G) A = F A × G A`

{-# OPTIONS --without-K --exact-split --safe #-} module Categories.Functors where -- Imports from Agda and the Agda Standard Library --------------------------------------- open import Agda.Primitive using ( _⊔_ ; lsuc ; Level ) renaming ( Set to Type ; lzero to ℓ₀ ) open import Data.Nat using ( ℕ ; zero ; suc ; _>_ ) open import Data.Sum.Base using ( _⊎_ ) renaming ( inj₁ to inl ; inj₂ to inr ) open import Data.Product using ( Σ-syntax ; _,_ ; _×_ ) open import Data.Unit using ( tt ) renaming ( ⊤ to ⊤₀ ) open import Data.Unit.Polymorphic using ( ⊤ ) open import Relation.Binary.PropositionalEquality using ( _≡_ ; refl ; _≢_ ) open import Level private variable α β : Level infixl 6 _⊕_ infixr 7 _⊗_ data Functor₀ : Type (lsuc ℓ₀) where Id : Functor₀ Const : Type → Functor₀ _⊕_ : Functor₀ → Functor₀ → Functor₀ _⊗_ : Functor₀ → Functor₀ → Functor₀ [_]₀ : Functor₀ → Type → Type [ Id ]₀ B = B [ Const C ]₀ B = C [ F ⊕ G ]₀ B = [ F ]₀ B ⊎ [ G ]₀ B [ F ⊗ G ]₀ B = [ F ]₀ B × [ G ]₀ B data Functor {ℓ : Level} : Type (lsuc ℓ) where Id : Functor Const : Type ℓ → Functor _⊕_ : Functor{ℓ} → Functor{ℓ} → Functor _⊗_ : Functor{ℓ} → Functor{ℓ} → Functor [_]_ : Functor → Type α → Type α [ Id ] B = B [ Const C ] B = C [ F ⊕ G ] B = [ F ] B ⊎ [ G ] B [ F ⊗ G ] B = [ F ] B × [ G ] B {- from Mimram's talk (MFPS 2021): record Poly (I J : Type) : Type (lsuc ℓ₀) where field Op : J → Type Pm : (i : I) → {j : J} → Op j → Type -}

The least fixed point of a polynomial function can then be defined in Agda with the following datatype declaration.

data μ_ (F : Functor) : Type where inn : [ F ] (μ F) → μ F

An important example is the polymorphic list datatype. The standard way to define this in Agda is as follows:

data list (A : Type) : Type ℓ₀ where [] : list A _∷_ : A → list A → list A

We could instead define a `List`

datatype by applying `μ`

to a polynomial functor `L`

as follows:

L : {ℓ : Level}(A : Type ℓ) → Functor{ℓ} L A = Const ⊤ ⊕ (Const A ⊗ Id) List : (A : Type) → Type ℓ₀ List A = μ (L A)

To see some examples demonstrating that both formulations of the polymorphic list type give what we expect, see the Examples.Categories.Functors module. The examples will use “getter” functions, which take a list `l`

and a natural number `n`

and return the element of `l`

at index `n`

. (Since such an element doesn’t always exist, we first define the `Option`

type.)

data Option (A : Type) : Type where some : A → Option A none : Option A _[_] : {A : Type} → List A → ℕ → Option A inn (inl x) [ n ] = none inn (inr (x , xs)) [ zero ] = some x inn (inr (x , xs)) [ suc n ] = xs [ n ] _⟦_⟧ : {A : Type} → list A → ℕ → Option A [] ⟦ n ⟧ = none (x ∷ l) ⟦ zero ⟧ = some x (x ∷ l) ⟦ suc n ⟧ = l ⟦ n ⟧