module Cat.Displayed.Instances.CT-Structure
{o ℓ} (B : Precategory o ℓ)
(has-prods : ∀ X Y → Product B X Y)
where
open Precategory B
open Binary-products B has-prods
open Simple B has-prodsCT structures🔗
CT-Structures are a refinement of simple fibrations, which
allow us to model type theories where the contexts aren’t necessarily
representable as types. This is done by defining a CT-Structure (short for Context and Type
Structure) on our category of contexts, which allows us to distinguish
certain contexts as denoting types (for instance, singleton contexts, or
possibly the empty context if we have unit types). A CT-structure is
quite simple, consisting of only a predicate on contexts meant to denote
which ones are types, along with the restriction that there is at least
one type, to prevent the entire structure from becoming degenerate.
record CT-Structure (s : Level) : Type (o ⊔ lsuc s) where
field
is-tp : Ob → Prop s
∃-tp : ∃[ tp ∈ Ob ] (tp ∈ is-tp)
open CT-StructureWe can construct a displayed category over our category of contexts in much the same manner as the simple fibration; the only difference is that we restrict the displayed object to objects that the CT-structure distinguishes as types.
Simple-ct : ∀ {s} → CT-Structure s → Displayed B (o ⊔ s) ℓ
Simple-ct ct .Displayed.Ob[_] Γ = Σ[ X ∈ Ob ] ∣ is-tp ct X ∣
Simple-ct ct .Displayed.Hom[_] {Γ} {Δ} u X Y = Hom (Γ ⊗₀ X .fst) (Y .fst)
Simple-ct ct .Displayed.Hom[_]-set {Γ} {Δ} u X Y = Hom-set (Γ ⊗₀ X .fst) (Y .fst)
Simple-ct ct .Displayed.id' = π₂
Simple-ct ct .Displayed._∘'_ {f = u} {g = v} f g = f ∘ ⟨ v ∘ π₁ , g ⟩
Simple-ct ct .Displayed.idr' {f = u} f =
f ∘ ⟨ (id ∘ π₁) , π₂ ⟩ ≡⟨ products! has-prods ⟩
f ∎
Simple-ct ct .Displayed.idl' {f = u} f =
π₂ ∘ ⟨ u ∘ π₁ , f ⟩ ≡⟨ products! has-prods ⟩
f ∎
Simple-ct ct .Displayed.assoc' {f = u} {g = v} {h = w} f g h =
f ∘ ⟨ (v ∘ w) ∘ π₁ , g ∘ ⟨ w ∘ π₁ , h ⟩ ⟩ ≡⟨ products! has-prods ⟩
(f ∘ ⟨ v ∘ π₁ , g ⟩) ∘ ⟨ w ∘ π₁ , h ⟩ ∎Cartesian maps🔗
If a map is cartesian in the simple fibration, then it is cartesian in a simple CT fibration.
Simple-cartesian→Simple-ct-cartesian
: ∀ {s} {Γ Δ x y} {σ : Hom Γ Δ} {f : Hom (Γ ⊗₀ x) y}
→ (ct : CT-Structure s)
→ (x-tp : x ∈ ct .is-tp) → (y-tp : y ∈ ct .is-tp)
→ is-cartesian Simple σ f
→ is-cartesian (Simple-ct ct) {a' = x , x-tp} {b' = y , y-tp} σ f
Simple-cartesian→Simple-ct-cartesian ct x-tp y-tp cart = ct-cart where
open is-cartesian
ct-cart : is-cartesian (Simple-ct ct) _ _
ct-cart .universal = cart .universal
ct-cart .commutes = cart .commutes
ct-cart .unique = cart .uniqueFibration structure🔗
Much like the simple fibration, the simple fibration associated to a CT-structure also deserves its name.
open Cartesian-lift
open is-cartesian
Simple-ct-fibration : ∀ {s} (ct : CT-Structure s) → Cartesian-fibration (Simple-ct ct)
Simple-ct-fibration ct u Y .x' = Y
Simple-ct-fibration ct u Y .lifting = π₂
Simple-ct-fibration ct u Y .cartesian .universal _ h = h
Simple-ct-fibration ct u Y .cartesian .commutes g h = π₂∘⟨⟩
Simple-ct-fibration ct u Y .cartesian .unique {m = g} {h' = h} h' p =
h' ≡˘⟨ π₂∘⟨⟩ ⟩
π₂ ∘ ⟨ g ∘ π₁ , h' ⟩ ≡⟨ p ⟩
h ∎Embeddings🔗
There is an evident embedding of the simple fibration associated with a CT-structure into the simple fibration.
Simple-ct→Simple
: ∀ {s} → (ct : CT-Structure s)
→ Vertical-functor (Simple-ct ct) Simple
Simple-ct→Simple ct .Vertical-functor.F₀' = fst
Simple-ct→Simple ct .Vertical-functor.F₁' f = f
Simple-ct→Simple ct .Vertical-functor.F-id' = refl
Simple-ct→Simple ct .Vertical-functor.F-∘' = reflFurthermore, if is (merely) inhabited, then we can construct a CT-Structure that considers every context a type.
All-types : ∥ Ob ∥ → CT-Structure lzero
All-types X .is-tp _ = el ⊤ (hlevel 1)
All-types X .∃-tp = ∥-∥-map (λ x → x , tt) X
Simple→Simple-ct : ∀ {X} → Vertical-functor Simple (Simple-ct (All-types X))
Simple→Simple-ct .Vertical-functor.F₀' X = X , tt
Simple→Simple-ct .Vertical-functor.F₁' f = f
Simple→Simple-ct .Vertical-functor.F-id' = refl
Simple→Simple-ct .Vertical-functor.F-∘' = refl