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Rachel Lambda Samuelsson 2022-10-08 13:03:08 +02:00
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src/CwF.lagda.md
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@ -43,11 +43,11 @@ module CwF where
``` ```
# Categories with families # 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. [@hoffmann97] A Category with families, a CwF, is a category in which we can interpret the synax of a type theory. The objects of a CwF are contexts and the morphisms are a list of judgements. The morphisms, being lists of judgements, are able to describe substitution and weakening. For a more detailed introduction to CwFs see the introductory paper by Hoffmann. [@hoffmann97]
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. [@abd2008] This file defines a CwF, without any particular type formers. Here CwFs are thought of as GATs (generalised algebraic theories), as is presented in a paper by Abel, Coquand and Dybjer. [@abd2008]
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. A CwF is defined as a precategory $\mathcal C$ equipped with a contravariant functor $\mathcal F : \mathcal C^{op} \to \mathcal Fam$, fufilling the properties listed below.
``` ```
record is-CwF {o h f : Level} (𝓒 : Precategory o h) : Type (lsuc (o ⊔ h ⊔ f)) where record is-CwF {o h f : Level} (𝓒 : Precategory o h) : Type (lsuc (o ⊔ h ⊔ f)) where
@ -55,7 +55,7 @@ record is-CwF {o h f : Level} (𝓒 : Precategory o h) : Type (lsuc (o ⊔ h ⊔
𝓕 : Functor (𝓒 ^op) (Fams f) 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. Two functions, $Ty$ and $Tm$ are defined in terms of $\mathcal F$, giving a contexts set of syntactic types, and terms respectively.
``` ```
open Precategory 𝓒 open Precategory 𝓒
open Functor 𝓕 open Functor 𝓕
@ -67,7 +67,7 @@ Two functions, $Ty$ and $Tm$ are defined, respectively giving the set of syntact
Tr Γ σ = F₀ Γ .snd σ Tr Γ σ = F₀ Γ .snd σ
``` ```
Which support substitution in syntactic types and terms. From functoriality of $\mathcal F$ it can be derived that these support substitution in both types and terms.
``` ```
_⟦_⟧ : {Δ Γ : Ob} → ∣ Ty Γ ∣ → Hom Δ Γ → ∣ Ty Δ ∣ _⟦_⟧ : {Δ Γ : Ob} → ∣ Ty Γ ∣ → Hom Δ Γ → ∣ Ty Δ ∣
A ⟦ γ ⟧ = F₁ γ .fst A A ⟦ γ ⟧ = F₁ γ .fst A
@ -77,27 +77,39 @@ Which support substitution in syntactic types and terms.
_[_] {_} {_} {A} a γ = F₁ γ .snd A a _[_] {_} {_} {A} a γ = F₁ γ .snd A a
``` ```
Being defined from a functor, these substitution operators play nicely with composed homs and identities. Likewise, it follows that the substitution operators respect composition and identity.
``` ```
⟦⟧-compose : {Δ Γ Θ : Ob} {γ : Hom Δ Γ} {δ : Hom Θ Δ} {A : ∣ Ty Γ ∣} ⟦⟧-compose : {Δ Γ Θ : Ob} (γ : Hom Δ Γ) (δ : Hom Θ Δ) (A : ∣ Ty Γ ∣)
→ A ⟦ γ ∘ δ ⟧ ≡ A ⟦ γ ⟧ ⟦ δ ⟧ → A ⟦ γ ∘ δ ⟧ ≡ A ⟦ γ ⟧ ⟦ δ ⟧
⟦⟧-compose {_} {_} {_} {γ} {δ} {A} i = F-∘ δ γ i .fst A ⟦⟧-compose {_} {_} {_} γ δ A i = F-∘ δ γ i .fst A
⟦⟧-id : {Γ : Ob} {A : ∣ Ty Γ ∣} ⟦⟧-id : {Γ : Ob} (A : ∣ Ty Γ ∣)
→ A ⟦ id ⟧ ≡ A → A ⟦ id ⟧ ≡ A
⟦⟧-id {Γ} {A} i = F-id i .fst 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. Since the terms are indexed by types there will be some commonly used paths between types.
``` ```
[]-compose : {Δ Γ Θ : Ob} {γ : Hom Δ Γ} {δ : Hom Θ Δ} {A : ∣ Ty Γ ∣} {a : ∣ Tr Γ A ∣} ⟦⟧-tr-comp : {Δ Γ Θ : Ob} (γ : Hom Δ Γ) (δ : Hom Θ Δ) (A : ∣ Ty Γ ∣)
→ PathP (λ i → ∣ Tr Θ (F-∘ δ γ i .fst A) ∣) → ∣ Tr Θ (A ⟦ γ ∘ δ ⟧) ∣ ≡ ∣ Tr Θ (A ⟦ γ ⟧ ⟦ δ ⟧) ∣
(a [ γ ∘ δ ]) ⟦⟧-tr-comp {_} {_} {Θ} γ δ A i = ∣ Tr Θ (⟦⟧-compose γ δ A i) ∣
(a [ γ ] [ δ ])
[]-compose {_} {_} {_} {γ} {δ} {A} {a} i = F-∘ δ γ i .snd A a ⟦⟧-tr-id : {Γ : Ob} (A : ∣ Ty Γ ∣)
→ ∣ Tr Γ (A ⟦ id ⟧) ∣ ≡ ∣ Tr Γ A ∣
⟦⟧-tr-id {Γ} A i = ∣ Tr Γ (⟦⟧-id A i) ∣
```
Using the paths defined above, composition and identity laws for terms can be stated and proven.
```
[]-compose : {Δ Γ Θ : Ob} (γ : Hom Δ Γ) (δ : Hom Θ Δ) (A : ∣ Ty Γ ∣)
{a : ∣ Tr Γ A ∣}
→ PathP (λ i → ⟦⟧-tr-comp γ δ A i)
(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 ∣} []-id : {Γ : Ob} {A : ∣ Ty Γ ∣} {a : ∣ Tr Γ A ∣}
→ PathP (λ i → ∣ Tr Γ (F-id i .fst A) ∣) (a [ id ]) a → PathP (λ i → ⟦⟧-tr-id A i) (a [ id ]) a
[]-id {Γ} {A} {a} i = F-id i .snd A a []-id {Γ} {A} {a} i = F-id i .snd A a
``` ```
@ -107,32 +119,28 @@ A CwF has a terminal object representing the empty context.
terminal : Terminal 𝓒 terminal : Terminal 𝓒
``` ```
For any context $\Gamma$ and $A \in Ty(\Gamma)$ there is a context $\Gamma,A$, extending $\Gamma$ by $A$. For any context $\Gamma$ and $A \in Ty(\Gamma)$ there is a context $\Gamma;A$, extending $\Gamma$ by $A$.
``` ```
_;_ : (Γ : Ob) → ∣ Ty Γ ∣ → Ob _;_ : (Γ : Ob) → ∣ Ty Γ ∣ → Ob
``` ```
Thus contexts can be built by repeatedly extending the empty context.
For every $\gamma : \Delta \to \Gamma$ and $a \in Tr(\Delta, A \llbracket \gamma \rrbracket)$ there is a weakened map $\langle \gamma , a \rangle : \Delta → \Gamma ; A$ For any list of judgements between $\Delta$ and $\Gamma$, there should be a weakened version from $\Delta$ to $\Gamma;A$.
``` ```
⟨_,_⟩ : {Γ Δ : Ob} {A : ∣ Ty Γ ∣} (γ : Hom Δ Γ) (a : ∣ Tr Δ (A ⟦ γ ⟧) ∣ ) ⟨_,_⟩ : {Γ Δ : Ob} {A : ∣ Ty Γ ∣} (γ : Hom Δ Γ) (a : ∣ Tr Δ (A ⟦ γ ⟧) ∣ )
→ Hom Δ (Γ ; A) → 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 $q_{\ \Gamma;A} : Tr(\Gamma;A, A \llbracket p_{\ \Gamma;A}\rrbracket)$. For every context $\Gamma$ and $A ∈ Ty(\Gamma)$ there is a projection map $p_{\ \Gamma;A} : \Gamma;A → \Gamma$ and a term $q_{\ \Gamma;A} : Tr(\Gamma;A, A \llbracket p_{\ \Gamma;A}\rrbracket)$.
``` ```
p : {Γ : Ob} {A : ∣ Ty Γ ∣} → Hom (Γ ; A) Γ p : {Γ : Ob} {A : ∣ Ty Γ ∣} → Hom (Γ ; A) Γ
q : {Γ : Ob} {A : ∣ Ty Γ ∣} → ∣ Tr (Γ ; A) (A ⟦ p ⟧) ∣ q : {Γ : Ob} {A : ∣ Ty Γ ∣} → ∣ Tr (Γ ; A) (A ⟦ p ⟧) ∣
``` ```
It's required that the map $p$ weakened by $q$ is the identity.
```
field
pq-id : {Γ : Ob} {A : ∣ Ty Γ ∣} → ⟨ p , q ⟩ ≡ id {Γ ; A}
```
Dubbing $p$ a projection is no coincidence, it projects a map out of a weakened map. Dubbing $p$ a projection is no coincidence, it projects a map out of a weakened map.
``` ```
field
p-∘ : {Δ Γ : Ob} {γ : Hom Δ Γ} {A : ∣ Ty Γ ∣} (a : ∣ Tr Δ (A ⟦ γ ⟧) ∣) p-∘ : {Δ Γ : Ob} {γ : Hom Δ Γ} {A : ∣ Ty Γ ∣} (a : ∣ Tr Δ (A ⟦ γ ⟧) ∣)
→ p ∘ ⟨ γ , a ⟩ ≡ γ → p ∘ ⟨ γ , a ⟩ ≡ γ
``` ```
@ -140,11 +148,11 @@ Dubbing $p$ a projection is no coincidence, it projects a map out of a weakened
Likewise, $q$ "projects" an object out of a weakened substitution. Likewise, $q$ "projects" an object out of a weakened substitution.
``` ```
q-∘-pathp : {Δ Γ : Ob} {γ : Hom Δ Γ} {A : ∣ Ty Γ ∣} (a : ∣ Tr Δ (A ⟦ γ ⟧) ∣) 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) ∣ → ∣ Tr Δ (A ⟦ p ⟧ ⟦ ⟨ γ , a ⟩ ⟧) ∣ ≡ ∣ Tr Δ (A ⟦ γ ⟧) ∣
q-∘-pathp {Δ} {Γ} {γ} {A} 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 (A ⟦ p ⟧ ⟦ ⟨ γ , a ⟩ ⟧) ∣ ≡˘⟨ ⟦⟧-tr-comp p ⟨ γ , a ⟩ A ⟩
∣ Tr Δ (F₁ (p ∘ ⟨ γ , a ⟩) .fst A) ∣ ≡⟨ (λ i → ∣ Tr Δ (F₁ (p-∘ a i) .fst A) ∣) ⟩ ∣ Tr Δ (A ⟦ p ∘ ⟨ γ , a ⟩ ⟧) ∣ ≡⟨ (λ i → ∣ Tr Δ (A ⟦ p-∘ a i ⟧) ∣) ⟩
∣ Tr Δ (F₁ γ .fst A) ∣ ∎ ∣ Tr Δ (A ⟦ γ ⟧) ∣ ∎
field field
q-∘ : {Δ Γ : Ob} {γ : Hom Δ Γ} {A : ∣ Ty Γ ∣} {a : ∣ Tr Δ (A ⟦ γ ⟧) ∣} q-∘ : {Δ Γ : Ob} {γ : Hom Δ Γ} {A : ∣ Ty Γ ∣} {a : ∣ Tr Δ (A ⟦ γ ⟧) ∣}
@ -153,9 +161,15 @@ Likewise, $q$ "projects" an object out of a weakened substitution.
a a
``` ```
The weakening map is also required to play nice with composition. Further motivating the use of the word projection, pairing up the two projections in a weakened map must result in the identity.
``` ```
⟨⟩-∘ : {Δ Γ Θ : Ob} {γ : Hom Γ Δ} {δ : Hom Δ Θ} {A : ∣ Ty Θ ∣} {a : ∣ Tr Δ (A ⟦ δ ⟧) ∣} pq-id : {Γ : Ob} {A : ∣ Ty Γ ∣} → ⟨ p , q ⟩ ≡ id {Γ ; A}
```
Lastly, it is required that the weakening map behaves as expected under 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 , a ⟩ ∘ γ
≡ ⟨ δ ∘ γ , transport (λ i → ∣ Tr Γ (F-∘ γ δ (~ i) .fst A) ∣) (a [ γ ]) ⟩ ≡ ⟨ δ ∘ γ , transport (sym (⟦⟧-tr-comp δ γ A)) (a [ γ ]) ⟩
``` ```