module Cat.Displayed.Cocartesian {o ℓ o′ ℓ′} {ℬ : Precategory o ℓ} (ℰ : Displayed ℬ o′ ℓ′) where open Cat.Reasoning ℬ open Displayed ℰ open Cat.Displayed.Morphism ℰ open Cat.Displayed.Morphism.Duality ℰ open DR ℰ
Cocartesian morphisms and opfibrations🔗
Cartesian fibrations provide a way of describing pseudofunctorial families of categories purely in terms of displayed structure. It’s then natural to ask: what about covariant pseudofunctorial families of categories? Such pseudofunctors can be encoded by dualising cartesian fibrations.
To do this, we must first dualise the notion of a cartesian map to a cocartesian map. Fix a map in , objects and displayed over and resp., and a map over . We say that is cocartesian if it has the shape of a “pushout diagram”, in contrast to the “pullback diagrams” shape associated with cartesian maps.
record is-cocartesian {a b a′ b′} (f : Hom a b) (f′ : Hom[ f ] a′ b′) : Type (o ⊔ ℓ ⊔ o′ ⊔ ℓ′) where no-eta-equality
Concretely, let and be displayed over . Furthermore, suppose we have a map (highlighted in violet below), along with a map displayed over (highlighted in red). For to be cocartesian, every such situation must give rise to a unique universal factorisation of through a map (highlighted in green).
field universal : ∀ {u u′} (m : Hom b u) (h′ : Hom[ m ∘ f ] a′ u′) → Hom[ m ] b′ u′ commutes : ∀ {u u′} (m : Hom b u) (h′ : Hom[ m ∘ f ] a′ u′) → universal m h′ ∘′ f′ ≡ h′ unique : ∀ {u u′} {m : Hom b u} {h′ : Hom[ m ∘ f ] a′ u′} → (m′ : Hom[ m ] b′ u′) → m′ ∘′ f′ ≡ h′ → m′ ≡ universal m h′
universal′ : ∀ {u u′} {m : Hom b u} {k : Hom a u} → (p : m ∘ f ≡ k) (h′ : Hom[ k ] a′ u′) → Hom[ m ] b′ u′ universal′ {u′ = u′} p h′ = universal _ (coe1→0 (λ i → Hom[ p i ] a′ u′) h′) commutesp : ∀ {u u′} {m : Hom b u} {k : Hom a u} → (p : m ∘ f ≡ k) (h′ : Hom[ k ] a′ u′) → universal′ p h′ ∘′ f′ ≡[ p ] h′ commutesp {u′ = u′} p h′ = to-pathp⁻ (commutes _ (coe1→0 (λ i → Hom[ p i ] a′ u′) h′)) universalp : ∀ {u u′} {m₁ m₂ : Hom b u} {k : Hom a u} → (p : m₁ ∘ f ≡ k) (q : m₁ ≡ m₂) (r : m₂ ∘ f ≡ k) → (h′ : Hom[ k ] a′ u′) → universal′ p h′ ≡[ q ] universal′ r h′ universalp {u = u} p q r h′ i = universal′ (is-set→squarep (λ _ _ → Hom-set a u) (ap (_∘ f) q) p r refl i) h′ uniquep : ∀ {u u′} {m₁ m₂ : Hom b u} {k : Hom a u} → (p : m₁ ∘ f ≡ k) (q : m₁ ≡ m₂) (r : m₂ ∘ f ≡ k) → {h′ : Hom[ k ] a′ u′} → (m′ : Hom[ m₁ ] b′ u′) → m′ ∘′ f′ ≡[ p ] h′ → m′ ≡[ q ] universal′ r h′ uniquep p q r {h′ = h′} m′ s = to-pathp⁻ (unique m′ (from-pathp⁻ s) ∙ from-pathp⁻ (universalp p q r h′)) uniquep₂ : ∀ {u u′} {m₁ m₂ : Hom b u} {k : Hom a u} → (p : m₁ ∘ f ≡ k) (q : m₁ ≡ m₂) (r : m₂ ∘ f ≡ k) → {h′ : Hom[ k ] a′ u′} → (m₁′ : Hom[ m₁ ] b′ u′) → (m₂′ : Hom[ m₂ ] b′ u′) → m₁′ ∘′ f′ ≡[ p ] h′ → m₂′ ∘′ f′ ≡[ r ] h′ → m₁′ ≡[ q ] m₂′ uniquep₂ p q r {h′ = h′} m₁′ m₂′ α β = to-pathp⁻ $ unique m₁′ (from-pathp⁻ α) ·· from-pathp⁻ (universalp p q r _) ·· ap hom[] (sym (unique m₂′ (from-pathp⁻ β))) universalv : ∀ {b″} (f″ : Hom[ f ] a′ b″) → Hom[ id ] b′ b″ universalv f″ = universal′ (idl _) f″ commutesv : ∀ {x′} (g′ : Hom[ f ] a′ x′) → universalv g′ ∘′ f′ ≡[ idl _ ] g′ commutesv = commutesp (idl _) uniquev : ∀ {x′} {g′ : Hom[ f ] a′ x′} → (h′ : Hom[ id ] b′ x′) → h′ ∘′ f′ ≡[ idl _ ] g′ → h′ ≡ universalv g′ uniquev h′ p = uniquep (idl _) refl (idl _) h′ p uniquev₂ : ∀ {x′} {g′ : Hom[ f ] a′ x′} → (h′ h″ : Hom[ id ] b′ x′) → h′ ∘′ f′ ≡[ idl _ ] g′ → h″ ∘′ f′ ≡[ idl _ ] g′ → h′ ≡ h″ uniquev₂ h′ h″ p q = uniquep₂ (idl _) refl (idl _) h′ h″ p q record Cocartesian-morphism {x y : Ob} (f : Hom x y) (x′ : Ob[ x ]) (y′ : Ob[ y ]) : Type (o ⊔ ℓ ⊔ o′ ⊔ ℓ′) where no-eta-equality field hom′ : Hom[ f ] x′ y′ cocartesian : is-cocartesian f hom′ open is-cocartesian cocartesian public
Duality🔗
As noted before, cocartesian maps the are duals to cartesian maps. We can make this correspondence precise by showing that cartesian maps in the total opposite of are cocartesian maps, and vice versa.
co-cartesian→cocartesian : ∀ {x y} {f : Hom x y} {x′ y′} {f′ : Hom[ f ] x′ y′} → is-cartesian (ℰ ^total-op) f f′ → is-cocartesian f f′ cocartesian→co-cartesian : ∀ {x y} {f : Hom x y} {x′ y′} {f′ : Hom[ f ] x′ y′} → is-cocartesian f f′ → is-cartesian (ℰ ^total-op) f f′
These functions just unpack and repack data, they are not particularly interesting.
co-cartesian→cocartesian cart^op .is-cocartesian.universal m h′ = is-cartesian.universal cart^op m h′ co-cartesian→cocartesian cart^op .is-cocartesian.commutes m h′ = is-cartesian.commutes cart^op m h′ co-cartesian→cocartesian cart^op .is-cocartesian.unique m′ p = is-cartesian.unique cart^op m′ p cocartesian→co-cartesian cocart .is-cartesian.universal m h′ = is-cocartesian.universal cocart m h′ cocartesian→co-cartesian cocart .is-cartesian.commutes m h′ = is-cocartesian.commutes cocart m h′ cocartesian→co-cartesian cocart .is-cartesian.unique m′ p = is-cocartesian.unique cocart m′ p
Furthermore, these 2 functions form an equivalence, which we can extend to a path via univalence.
co-cartesian→cocartesian-is-equiv : ∀ {x y} {f : Hom x y} {x′ y′} {f′ : Hom[ f ] x′ y′} → is-equiv (co-cartesian→cocartesian {f′ = f′}) co-cartesian→cocartesian-is-equiv {f′ = f′} = is-iso→is-equiv $ iso cocartesian→co-cartesian cocart-invl cocart-invr where cocart-invl : ∀ f → co-cartesian→cocartesian {f′ = f′} (cocartesian→co-cartesian f) ≡ f cocart-invl f i .is-cocartesian.universal m h′ = is-cocartesian.universal f m h′ cocart-invl f i .is-cocartesian.commutes m h′ = is-cocartesian.commutes f m h′ cocart-invl f i .is-cocartesian.unique m′ p = is-cocartesian.unique f m′ p cocart-invr : ∀ f → cocartesian→co-cartesian {f′ = f′} (co-cartesian→cocartesian f) ≡ f cocart-invr f i .is-cartesian.universal m h′ = is-cartesian.universal f m h′ cocart-invr f i .is-cartesian.commutes m h′ = is-cartesian.commutes f m h′ cocart-invr f i .is-cartesian.unique m′ p = is-cartesian.unique f m′ p co-cartesian≡cocartesian : ∀ {x y} {f : Hom x y} {x′ y′} {f′ : Hom[ f ] x′ y′} → is-cartesian (ℰ ^total-op) f f′ ≡ is-cocartesian f f′ co-cartesian≡cocartesian = ua (co-cartesian→cocartesian , co-cartesian→cocartesian-is-equiv)
Properties of Cocartesian Morphisms🔗
We shall now prove the following properties of cocartesian morphisms.
cocartesian-∘ : ∀ {x y z} {f : Hom y z} {g : Hom x y} → ∀ {x′ y′ z′} {f′ : Hom[ f ] y′ z′} {g′ : Hom[ g ] x′ y′} → is-cocartesian f f′ → is-cocartesian g g′ → is-cocartesian (f ∘ g) (f′ ∘′ g′) cocartesian-id : ∀ {x x′} → is-cocartesian id (id′ {x} {x′}) invertible→cocartesian : ∀ {x y} {f : Hom x y} {x′ y′} {f′ : Hom[ f ] x′ y′} → (f-inv : is-invertible f) → is-invertible[ f-inv ] f′ → is-cocartesian f f′ cocartesian→weak-epic : ∀ {x y} {f : Hom x y} → ∀ {x′ y′} {f′ : Hom[ f ] x′ y′} → is-cocartesian f f′ → is-weak-epic f′ cocartesian-codomain-unique : ∀ {x y} {f : Hom x y} → ∀ {x′ y′ y″} {f′ : Hom[ f ] x′ y′} {f″ : Hom[ f ] x′ y″} → is-cocartesian f f′ → is-cocartesian f f″ → y′ ≅↓ y″ cocartesian-vertical-section-stable : ∀ {x y} {f : Hom x y} → ∀ {x′ y′ y″} {f′ : Hom[ f ] x′ y′} {f″ : Hom[ f ] x′ y″} {ϕ : Hom[ id ] y″ y′} → is-cocartesian f f′ → has-retract↓ ϕ → ϕ ∘′ f″ ≡[ idl _ ] f′ → is-cocartesian f f″ cocartesian-pasting : ∀ {x y z} {f : Hom y z} {g : Hom x y} → ∀ {x′ y′ z′} {f′ : Hom[ f ] y′ z′} {g′ : Hom[ g ] x′ y′} → is-cocartesian g g′ → is-cocartesian (f ∘ g) (f′ ∘′ g′) → is-cocartesian f f′ vertical+cocartesian→invertible : ∀ {x} {x′ x″ : Ob[ x ]} {f′ : Hom[ id ] x′ x″} → is-cocartesian id f′ → is-invertible↓ f′
The proofs are all applications of duality, as we have already done the hard work of proving these properties for cartesian morphisms.
cocartesian-∘ f-cocart g-cocart = co-cartesian→cocartesian $ cartesian-∘ _ (cocartesian→co-cartesian g-cocart) (cocartesian→co-cartesian f-cocart) cocartesian-id = co-cartesian→cocartesian (cartesian-id _) invertible→cocartesian f-inv f′-inv = co-cartesian→cocartesian $ invertible→cartesian _ _ (invertible[]→co-invertible[] f′-inv) cocartesian→weak-epic cocart = cartesian→weak-monic (ℰ ^total-op) (cocartesian→co-cartesian cocart) cocartesian-codomain-unique f′-cocart f″-cocart = vertical-co-iso→vertical-iso $ cartesian-domain-unique (ℰ ^total-op) (cocartesian→co-cartesian f″-cocart) (cocartesian→co-cartesian f′-cocart) cocartesian-vertical-section-stable cocart ret factor = co-cartesian→cocartesian $ cartesian-vertical-retraction-stable (ℰ ^total-op) (cocartesian→co-cartesian cocart) (vertical-retract→vertical-co-section ret) factor cocartesian-pasting g-cocart fg-cocart = co-cartesian→cocartesian $ cartesian-pasting (ℰ ^total-op) (cocartesian→co-cartesian g-cocart) (cocartesian→co-cartesian fg-cocart) vertical+cocartesian→invertible cocart = vertical-co-invertible→vertical-invertible $ vertical+cartesian→invertible (ℰ ^total-op) (cocartesian→co-cartesian cocart)
Furthermore, is cocartesian if and only if the function is an equivalence.
precompose-equiv→cocartesian : ∀ {x y x′ y′} {f : Hom x y} → (f′ : Hom[ f ] x′ y′) → (∀ {z z′} {g : Hom y z} → is-equiv {A = Hom[ g ] y′ z′} (_∘′ f′)) → is-cocartesian f f′ precompose-equiv→cocartesian f′ cocart = co-cartesian→cocartesian $ postcompose-equiv→cartesian (ℰ ^total-op) f′ cocart cocartesian→precompose-equiv : ∀ {x y z x′ y′ z′} {g : Hom y z} {f : Hom x y} {f′ : Hom[ f ] x′ y′} → is-cocartesian f f′ → is-equiv {A = Hom[ g ] y′ z′} (_∘′ f′) cocartesian→precompose-equiv cocart = cartesian→postcompose-equiv (ℰ ^total-op) $ cocartesian→co-cartesian cocart
Cocartesian Lifts🔗
We call an object over together with a cartesian arrow a cocartesian lift of .
record Cocartesian-lift {x y} (f : Hom x y) (x′ : Ob[ x ]) : Type (o ⊔ ℓ ⊔ o′ ⊔ ℓ′) where no-eta-equality field {y′} : Ob[ y ] lifting : Hom[ f ] x′ y′ cocartesian : is-cocartesian f lifting open is-cocartesian cocartesian public
We also can apply duality to cocartesian lifts.
co-cartesian-lift→cocartesian-lift : ∀ {x y} {f : Hom x y} {x′ : Ob[ x ]} → Cartesian-lift (ℰ ^total-op) f x′ → Cocartesian-lift f x′ cocartesian-lift→co-cartesian-lift : ∀ {x y} {f : Hom x y} {x′ : Ob[ x ]} → Cocartesian-lift f x′ → Cartesian-lift (ℰ ^total-op) f x′
The proofs are simply applying duality, so they are not particularly interesting.
co-cartesian-lift→cocartesian-lift cart .Cocartesian-lift.y′ = Cartesian-lift.x′ cart co-cartesian-lift→cocartesian-lift cart .Cocartesian-lift.lifting = Cartesian-lift.lifting cart co-cartesian-lift→cocartesian-lift cart .Cocartesian-lift.cocartesian = co-cartesian→cocartesian (Cartesian-lift.cartesian cart) cocartesian-lift→co-cartesian-lift cocart .Cartesian-lift.x′ = Cocartesian-lift.y′ cocart cocartesian-lift→co-cartesian-lift cocart .Cartesian-lift.lifting = Cocartesian-lift.lifting cocart cocartesian-lift→co-cartesian-lift cocart .Cartesian-lift.cartesian = cocartesian→co-cartesian (Cocartesian-lift.cocartesian cocart)
We can use this notion to define cocartesian fibrations (sometimes referred to as opfibrations).
record Cocartesian-fibration : Type (o ⊔ ℓ ⊔ o′ ⊔ ℓ′) where no-eta-equality field has-lift : ∀ {x y} (f : Hom x y) (x′ : Ob[ x ]) → Cocartesian-lift f x′ module has-lift {x y} (f : Hom x y) (x′ : Ob[ x ]) = Cocartesian-lift (has-lift f x′)
rebase : ∀ {x y x′ x″} → (f : Hom x y) → Hom[ id ] x′ x″ → Hom[ id ] (has-lift.y′ f x′) (has-lift.y′ f x″) rebase f vert = has-lift.universalv f _ (hom[ idr _ ] (has-lift.lifting f _ ∘′ vert))
op-fibration→opfibration : Cartesian-fibration (ℰ ^total-op) → Cocartesian-fibration opfibration→op-fibration : Cocartesian-fibration → Cartesian-fibration (ℰ ^total-op)
The proofs here are just further applications of duality, so we omit them.
op-fibration→opfibration fib .Cocartesian-fibration.has-lift f x′ = co-cartesian-lift→cocartesian-lift (Cartesian-fibration.has-lift fib f x′) opfibration→op-fibration opfib .Cartesian-fibration.has-lift f y′ = cocartesian-lift→co-cartesian-lift (Cocartesian-fibration.has-lift opfib f y′)