6.2 KiB
open import Cat.Prelude
open import Cat.Diagram.Terminal
open import Fams
module CwF where
Categories with families
A Category with families, written as CwF, is a category in which we can interpret the synax of a type theory. The objects of a CwF are contexts, and morphisms are a list of judgements, which capture the notions of substitution and weakening. For a more detailed introduction to CwFs see the introductory paper by Hoffmann. [1]
This file defines a CwF, as well as type formers for pi types, sigma types, identity types and universes. Here CwFs are defined as a GAT (generalised algebraic theory), as is presented in a paper by Abel, Coquand and Dybjer. [2]
A CwF is defined as a category \mathcal C equipped with a contravariant functor \mathcal F : \mathcal C^{op} \to \mathcal Fam, fufilling the rules of the GAT.
record is-CwF {o h f : Level} (𝓒 : Precategory o h) : Type (lsuc (o ⊔ h ⊔ f)) where
field
𝓕 : Functor (𝓒 ^op) (Fams f)
Two functions, Ty and Tm are defined, respectively giving the set of syntactic types, and terms in a context.
open Precategory 𝓒
open Functor 𝓕
Ty : Ob → Set f
Ty Γ = F₀ Γ .fst
Tr : (Γ : Ob) → ∣ Ty Γ ∣ → Set f
Tr Γ σ = F₀ Γ .snd σ
Which support substitution in syntactic types and terms.
_⟦_⟧ : {Δ Γ : Ob} → ∣ Ty Γ ∣ → Hom Δ Γ → ∣ Ty Δ ∣
A ⟦ γ ⟧ = F₁ γ .fst A
_[_] : {Δ Γ : Ob} {A : ∣ Ty Γ ∣} (a : ∣ Tr Γ A ∣)
(γ : Hom Δ Γ) → ∣ Tr Δ (A ⟦ γ ⟧) ∣
_[_] {_} {_} {A} a γ = F₁ γ .snd A a
Being defined from a functor, these substitution operators play nicely with composed homs and identities.
⟦⟧-compose : {Δ Γ Θ : Ob} {γ : Hom Δ Γ} {δ : Hom Θ Δ} {A : ∣ Ty Γ ∣}
→ A ⟦ γ ∘ δ ⟧ ≡ A ⟦ γ ⟧ ⟦ δ ⟧
⟦⟧-compose {_} {_} {_} {γ} {δ} {A} i = F-∘ δ γ i .fst A
⟦⟧-id : {Γ : Ob} {A : ∣ Ty Γ ∣}
→ A ⟦ id ⟧ ≡ A
⟦⟧-id {Γ} {A} i = F-id i .fst A
Because the type of the syntactic terms depends upon the syntactic types a path over paths is needed for stating that substitution is well behaved on terms.
[]-compose : {Δ Γ Θ : Ob} {γ : Hom Δ Γ} {δ : Hom Θ Δ} {A : ∣ Ty Γ ∣} {a : ∣ Tr Γ A ∣}
→ PathP (λ i → ∣ Tr Θ (F-∘ δ γ i .fst A) ∣)
(a [ γ ∘ δ ])
(a [ γ ] [ δ ])
[]-compose {_} {_} {_} {γ} {δ} {A} {a} i = F-∘ δ γ i .snd A a
[]-id : {Γ : Ob} {A : ∣ Ty Γ ∣} {a : ∣ Tr Γ A ∣}
→ PathP (λ i → ∣ Tr Γ (F-id i .fst A) ∣) (a [ id ]) a
[]-id {Γ} {A} {a} i = F-id i .snd A a
A CwF has a terminal object representing the empty context.
field
terminal : Terminal 𝓒
For any context \Gamma and A \in Ty(\Gamma) there is a context \Gamma,A, extending \Gamma by A.
_;_ : (Γ : Ob) → ∣ Ty Γ ∣ → Ob
For every \gamma : \Delta \to \Gamma and a \in Tr(\Delta, A \llbracket \gamma \rrbracket) there is a weakened map \langle \gamma , a \rangle : \Delta → \Gamma ; A
⟨_,_⟩ : {Γ Δ : Ob} {A : ∣ Ty Γ ∣} (γ : Hom Δ Γ) (a : ∣ Tr Δ (A ⟦ γ ⟧) ∣ )
→ Hom Δ (Γ ; A)
For every context \Gamma and A ∈ Ty(\Gamma) there is a projection map p_{\ \Gamma;A} : \Gamma;A → \Gamma and a syntactic term q_{\ \Gamma;A} : Tr(\Gamma;A, A \llbracket p_{\ \Gamma;A}\rrbracket).
p : {Γ : Ob} {A : ∣ Ty Γ ∣} → Hom (Γ ; A) Γ
q : {Γ : Ob} {A : ∣ Ty Γ ∣} → ∣ Tr (Γ ; A) (A ⟦ p ⟧) ∣
The meaning of q might be a bit confusing, but it’s essentialy an inverse of p, in the sense made precise below.
field
pq-id : {Γ : Ob} {A : ∣ Ty Γ ∣} → ⟨ p , q ⟩ ≡ id {Γ ; A}
There are some additional requirments placed upon p and q to ensure they play nice.
p-∘ : {Δ Γ : Ob} {γ : Hom Δ Γ} {A : ∣ Ty Γ ∣} (a : ∣ Tr Δ (A ⟦ γ ⟧) ∣)
→ p ∘ ⟨ γ , a ⟩ ≡ γ
The condition for q, depending on the condition for p holding, needs to be expressed with a rather annoying path over path.
q-∘-pathp : {Δ Γ : Ob} {γ : Hom Δ Γ} {A : ∣ Ty Γ ∣} (a : ∣ Tr Δ (A ⟦ γ ⟧) ∣)
→ ∣ Tr Δ (F₁ ⟨ γ , a ⟩ .fst (F₁ p .fst A)) ∣ ≡ ∣ Tr Δ (F₁ γ .fst A) ∣
q-∘-pathp {Δ} {Γ} {γ} {A} a =
∣ Tr Δ (F₁ ⟨ γ , a ⟩ .fst (F₁ p .fst A)) ∣ ≡⟨ (λ i → ∣ Tr Δ (F-∘ ⟨ γ , a ⟩ p (~ i) .fst A) ∣) ⟩
∣ Tr Δ (F₁ (p ∘ ⟨ γ , a ⟩) .fst A) ∣ ≡⟨ (λ i → ∣ Tr Δ (F₁ (p-∘ a i) .fst A) ∣) ⟩
∣ Tr Δ (F₁ γ .fst A) ∣ ∎
field
q-∘ : {Δ Γ : Ob} {γ : Hom Δ Γ} {A : ∣ Ty Γ ∣} {a : ∣ Tr Δ (A ⟦ γ ⟧) ∣}
→ PathP (λ i → q-∘-pathp a i)
(q [ ⟨ γ , a ⟩ ])
a
The weakening map is also required to play nice with composition.
⟨⟩-∘ : {Δ Γ Θ : Ob} {γ : Hom Γ Δ} {δ : Hom Δ Θ} {A : ∣ Ty Θ ∣} {a : ∣ Tr Δ (A ⟦ δ ⟧) ∣}
→ ⟨ δ , a ⟩ ∘ γ
≡ ⟨ δ ∘ γ , transport (λ i → ∣ Tr Γ (F-∘ γ δ (~ i) .fst A) ∣) (a [ γ ]) ⟩