open import Cat.Displayed.Cartesian.Weak open import Cat.Functor.Hom.Displayed open import Cat.Displayed.Cartesian open import Cat.Displayed.Total.Op open import Cat.Instances.Functor open import Cat.Instances.Product open import Cat.Displayed.Fibre open import Cat.Displayed.Base open import Cat.Functor.Hom open import Cat.Prelude import Cat.Displayed.Cocartesian.Indexing as Indexing import Cat.Displayed.Morphism.Duality import Cat.Displayed.Cocartesian as Cocart import Cat.Displayed.Cocartesian as Cocart import Cat.Displayed.Reasoning import Cat.Displayed.Morphism import Cat.Reasoning as CR module Cat.Displayed.Cocartesian.Weak {o ℓ o′ ℓ′} {ℬ : Precategory o ℓ} (ℰ : Displayed ℬ o′ ℓ′) where

# Weak Cocartesian Morphisms🔗

We can define a weaker notion of cocartesian morphism much like we can with their cartesian counterparts.

record is-weak-cocartesian {a b a′ b′} (f : Hom a b) (f′ : Hom[ f ] a′ b′) : Type (o ⊔ ℓ ⊔ o′ ⊔ ℓ′) where no-eta-equality field universal : ∀ {x′} → (g′ : Hom[ f ] a′ x′) → Hom[ id ] b′ x′ commutes : ∀ {x′} → (g′ : Hom[ f ] a′ x′) → universal g′ ∘′ f′ ≡[ idl _ ] g′ unique : ∀ {x′} {g′ : Hom[ f ] a′ x′} → (h′ : Hom[ id ] b′ x′) → h′ ∘′ f′ ≡[ idl _ ] g′ → h′ ≡ universal g′

## Duality🔗

Weak cartesian maps in the total opposite category are equivalent to weak cocartesian maps.

weak-co-cartesian→weak-cocartesian : ∀ {x y} {f : Hom x y} {x′ y′} {f′ : Hom[ f ] x′ y′} → is-weak-cartesian (ℰ ^total-op) f f′ → is-weak-cocartesian f f′ weak-cocartesian→weak-co-cartesian : ∀ {x y} {f : Hom x y} {x′ y′} {f′ : Hom[ f ] x′ y′} → is-weak-cocartesian f f′ → is-weak-cartesian (ℰ ^total-op) f f′

## These functions just shuffle data around, so we omit their definitions.

weak-co-cartesian→weak-cocartesian wcart .is-weak-cocartesian.universal = is-weak-cartesian.universal wcart weak-co-cartesian→weak-cocartesian wcart .is-weak-cocartesian.commutes = is-weak-cartesian.commutes wcart weak-co-cartesian→weak-cocartesian wcart .is-weak-cocartesian.unique = is-weak-cartesian.unique wcart weak-cocartesian→weak-co-cartesian wcocart .is-weak-cartesian.universal = is-weak-cocartesian.universal wcocart weak-cocartesian→weak-co-cartesian wcocart .is-weak-cartesian.commutes = is-weak-cocartesian.commutes wcocart weak-cocartesian→weak-co-cartesian wcocart .is-weak-cartesian.unique = is-weak-cocartesian.unique wcocart

Weak cocartesian maps satisfy the dual properties of weak cartesian maps.

weak-cocartesian-codomain-unique : ∀ {x y} {f : Hom x y} → ∀ {x′ y′ y″} {f′ : Hom[ f ] x′ y′} {f″ : Hom[ f ] x′ y″} → is-weak-cocartesian f f′ → is-weak-cocartesian f f″ → y′ ≅↓ y″ cocartesian→weak-cocartesian : ∀ {x y x′ y′} {f : Hom x y} {f′ : Hom[ f ] x′ y′} → is-cocartesian f f′ → is-weak-cocartesian f f′ weak-cocartesian→cocartesian : ∀ {x y x′ y′} {f : Hom x y} {f′ : Hom[ f ] x′ y′} → Cocartesian-fibration → is-weak-cocartesian f f′ → is-cocartesian f f′

## The proofs consist of tedious applications of duality.

weak-cocartesian-codomain-unique f′-cocart f″-cocart = vertical-co-iso→vertical-iso $ weak-cartesian-domain-unique (ℰ ^total-op) (weak-cocartesian→weak-co-cartesian f″-cocart) (weak-cocartesian→weak-co-cartesian f′-cocart) cocartesian→weak-cocartesian cocart = weak-co-cartesian→weak-cocartesian $ cartesian→weak-cartesian (ℰ ^total-op) $ cocartesian→co-cartesian cocart weak-cocartesian→cocartesian opfib wcocart = co-cartesian→cocartesian $ weak-cartesian→cartesian (ℰ ^total-op) (opfibration→op-fibration opfib) (weak-cocartesian→weak-co-cartesian wcocart)

Notably, if $\mathcal{E}$ is a cartesian fibration, then all weak cocartesian morphisms are cocartesian.

fibration+weak-cocartesian→cocartesian : ∀ {x y x′ y′} {f : Hom x y} {f′ : Hom[ f ] x′ y′} → Cartesian-fibration ℰ → is-weak-cocartesian f f′ → is-cocartesian f f′ fibration+weak-cocartesian→cocartesian {x} {y} {x′} {y′} {f} {f′} fib weak = cocart where open Cartesian-fibration fib module weak = is-weak-cocartesian weak

To see show this, we need to construct a unique factorization of some morphism $h' : x' \to_{mf} u'$, as depicted in the following diagram

We start by taking the cartesian lift of $m$ to obtain the map $m^{*}$, which we have highlighted in red.

module Morphisms {u} {u′ : Ob[ u ]} (m : Hom y u) (h′ : Hom[ m ∘ f ] x′ u′) where y* : Ob[ y ] y* = Cartesian-lift.x′ (has-lift m u′) m* : Hom[ m ] y* u′ m* = Cartesian-lift.lifting (has-lift m u′) module m* = is-cartesian (Cartesian-lift.cartesian (has-lift m u′))

Next, we can construct the morphism $h^{*}$ (highlighted in red) as the universal factorisation of $h'$ through $m^{*}$.

h* : Hom[ f ] x′ y* h* = m*.universal f h′

Finally, we can construct a vertical morphism $h^{**} : y' \to y^{*}$, as $f'$ is weakly cartesian.

h** : Hom[ id ] y′ y* h** = weak.universal h*

Composing $m^{*}$ and $h^{**}$ gives the desired factorisation.

cocart : is-cocartesian f f′ cocart .is-cocartesian.universal m h′ = hom[ idr _ ] (m* ∘′ h**) where open Morphisms m h′

Showing that $m^{*} \cdot h^{**} = h'$ is best understood diagramatically; both the $m^{*} \cdot h^{*} = h'$ and $h^{**} \cdot f' = h^{*}$ cells commute.

cocart .is-cocartesian.commutes m h′ = hom[] (m* ∘′ h**) ∘′ f′ ≡˘⟨ yank _ _ _ ⟩≡˘ m* ∘′ hom[] (h** ∘′ f′) ≡⟨ ap (m* ∘′_) (from-pathp (weak.commutes _)) ⟩≡ m* ∘′ m*.universal f h′ ≡⟨ m*.commutes f h′ ⟩≡ h′ ∎ where open Morphisms m h′

Uniqueness is somewhat more delicate. We need to show that the blue cell in the following diagram commutes.

As a general fact, every morphism in a cartesian fibration factors into a composite of a cartesian and vertical morphism, obtained by taking the universal factorisation of $m' : y' \to{m \cdot i} u'$. We shall denote this morphism as $id*$.

However,
$h^{**}$
is the *unique* vertical map that factorises
$f'$
through
$h^{*}$,
so it suffices to show that the cell highlighted in blue commutes.

$h^{*}$ is the unique vertical map that factorises $h'$ through $m'$, and $h' = m' \cdot f'$ by our hypothesis, so it suffices to show that $m^{*} \cdot id^{*} \cdot f' = m' \cdot f'$. This commutes because $m^{*}$ is cartesian, thus finishing the proof.

cocart .is-cocartesian.unique {m = m} {h′ = h′} m′ p = m′ ≡⟨ from-pathp⁻ (symP (m*.commutesp (idr _) m′)) ⟩≡ hom[] (m* ∘′ id*) ≡⟨ hom[]⟩⟨ ap (m* ∘′_) (weak.unique _ (to-pathp $ m*.unique _ path )) ⟩≡ hom[] (m* ∘′ h**) ∎ where open Morphisms m h′ id* : Hom[ id ] y′ y* id* = m*.universalv m′ path : m* ∘′ hom[ idl _ ] (id* ∘′ f′) ≡ h′ path = m* ∘′ hom[] (id* ∘′ f′) ≡⟨ whisker-r _ ⟩≡ hom[] (m* ∘′ id* ∘′ f′) ≡⟨ cancel _ (ap (m ∘_) (idl _)) (pulll′ (idr _) (m*.commutesv m′)) ⟩≡ m′ ∘′ f′ ≡⟨ p ⟩≡ h′ ∎

## Weak cocartesian lifts🔗

We can also define the dual to weak cartesian lifts.

record Weak-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′ weak-cocartesian : is-weak-cocartesian f lifting open is-weak-cocartesian weak-cocartesian public

As expected, weak cocartesian lifts are dual to weak cartesian lifts.

weak-co-cartesian-lift→weak-cocartesian-lift : ∀ {x y} {f : Hom x y} {x′ : Ob[ x ]} → Weak-cartesian-lift (ℰ ^total-op) f x′ → Weak-cocartesian-lift f x′ weak-cocartesian-lift→weak-co-cartesian-lift : ∀ {x y} {f : Hom x y} {x′ : Ob[ x ]} → Weak-cocartesian-lift f x′ → Weak-cartesian-lift (ℰ ^total-op) f x′

## We omit the proofs, as they are even more applications of duality.

weak-co-cartesian-lift→weak-cocartesian-lift wlift .Weak-cocartesian-lift.y′ = Weak-cartesian-lift.x′ wlift weak-co-cartesian-lift→weak-cocartesian-lift wlift .Weak-cocartesian-lift.lifting = Weak-cartesian-lift.lifting wlift weak-co-cartesian-lift→weak-cocartesian-lift wlift .Weak-cocartesian-lift.weak-cocartesian = weak-co-cartesian→weak-cocartesian (Weak-cartesian-lift.weak-cartesian wlift) weak-cocartesian-lift→weak-co-cartesian-lift wlift .Weak-cartesian-lift.x′ = Weak-cocartesian-lift.y′ wlift weak-cocartesian-lift→weak-co-cartesian-lift wlift .Weak-cartesian-lift.lifting = Weak-cocartesian-lift.lifting wlift weak-cocartesian-lift→weak-co-cartesian-lift wlift .Weak-cartesian-lift.weak-cartesian = weak-cocartesian→weak-co-cartesian (Weak-cocartesian-lift.weak-cocartesian wlift)

A displayed category with all weak cocartesian lifts is called a
**weak cocartesian fibration**, though we will often refer
to them as **weak opfibrations** These are also sometimes
called **preopfibred categories**, though we avoid this
terminology, as it conflicts with the precategory/category
distinction.

record is-weak-cocartesian-fibration : Type (o ⊔ ℓ ⊔ o′ ⊔ ℓ′) where no-eta-equality field weak-lift : ∀ {x y} → (f : Hom x y) → (x′ : Ob[ x ]) → Weak-cocartesian-lift f x′ module weak-lift {x y} (f : Hom x y) (x′ : Ob[ x ]) = Weak-cocartesian-lift (weak-lift f x′)Weak opfibrations are dual to [weak fibrations].

weak-op-fibration→weak-opfibration : is-weak-cartesian-fibration (ℰ ^total-op) → is-weak-cocartesian-fibration weak-opfibration→weak-op-fibration : is-weak-cocartesian-fibration → is-weak-cartesian-fibration (ℰ ^total-op)

## As usual, we omit the duality proofs, as they are quite tedious.

weak-op-fibration→weak-opfibration wlift .is-weak-cocartesian-fibration.weak-lift f x′ = weak-co-cartesian-lift→weak-cocartesian-lift $ is-weak-cartesian-fibration.weak-lift wlift f x′ weak-opfibration→weak-op-fibration wlift .is-weak-cartesian-fibration.weak-lift f y′ = weak-cocartesian-lift→weak-co-cartesian-lift $ is-weak-cocartesian-fibration.weak-lift wlift f y′

Every opfibration is a weak opfibration.

cocartesian-lift→weak-cocartesian-lift : ∀ {x y} {f : Hom x y} {x′ : Ob[ x ]} → Cocartesian-lift f x′ → Weak-cocartesian-lift f x′ opfibration→weak-opfibration : Cocartesian-fibration → is-weak-cocartesian-fibration

A weak opfibration is an opfibration when weak cocartesian morphisms are closed under composition. This follows via duality.

weak-opfibration→opfibration : is-weak-cocartesian-fibration → (∀ {x y z x′ y′ z′} {f : Hom y z} {g : Hom x y} → {f′ : Hom[ f ] y′ z′} {g′ : Hom[ g ] x′ y′} → is-weak-cocartesian f f′ → is-weak-cocartesian g g′ → is-weak-cocartesian (f ∘ g) (f′ ∘′ g′)) → Cocartesian-fibration weak-opfibration→opfibration wopfib weak-∘ = op-fibration→opfibration $ weak-fibration→fibration (ℰ ^total-op) (weak-opfibration→weak-op-fibration wopfib) (λ f g → weak-cocartesian→weak-co-cartesian $ weak-∘ (weak-co-cartesian→weak-cocartesian g) (weak-co-cartesian→weak-cocartesian f))

If $\mathcal{E}$ is cartesian, we can drop the requirement that weak cocartesian maps are closed under composition, thanks to fibration+weak-cocartesian→cocartesian.

cartesian+weak-opfibration→opfibration : Cartesian-fibration ℰ → is-weak-cocartesian-fibration → Cocartesian-fibration cartesian+weak-opfibration→opfibration fib wlifts = weak-opfibration→opfibration wlifts λ f-weak g-weak → cocartesian→weak-cocartesian $ cocartesian-∘ (fibration+weak-cocartesian→cocartesian fib f-weak) (fibration+weak-cocartesian→cocartesian fib g-weak)

# Weak Opfibrations and Equivalence of Hom Sets🔗

If $\mathcal{E}$ is a weak opfibration, then the hom sets $x' \to_f y'$ and $f^{*}(x') \to_{id} y'$ are equivalent, where $f^{*}(x')$ is the codomain of the lift of $f$ along $y'$.

module _ (wopfib : is-weak-cocartesian-fibration) where open is-weak-cocartesian-fibration wopfib weak-opfibration→universal-is-equiv : ∀ {x y y′ x′} → (u : Hom x y) → is-equiv (weak-lift.universal u x′ {y′}) weak-opfibration→universal-is-equiv {x′ = x′} u = is-iso→is-equiv $ iso (λ u′ → hom[ idl u ] (u′ ∘′ weak-lift.lifting u x′)) (λ u′ → sym $ weak-lift.unique u x′ u′ (to-pathp refl)) (λ u′ → cancel _ _ (weak-lift.commutes u x′ u′)) weak-opfibration→vertical-equiv : ∀ {x y x′ y′} → (u : Hom x y) → Hom[ u ] x′ y′ ≃ Hom[ id ] (weak-lift.y′ u x′) y′ weak-opfibration→vertical-equiv {x′ = x′} u = weak-lift.universal u x′ , weak-opfibration→universal-is-equiv u

Furthermore, this equivalence is natural.

weak-opfibration→hom-iso-from : ∀ {x y x′} (u : Hom x y) → natural-iso (Hom-over-from ℰ u x′) (Hom-from (Fibre ℰ y) (weak-lift.y′ u x′)) weak-opfibration→hom-iso-from {y = y} {x′ = x′} u = to-natural-iso mi where open make-natural-iso u*x′ : Ob[ y ] u*x′ = weak-lift.y′ u x′ mi : make-natural-iso (Hom-over-from ℰ u x′) (Hom-from (Fibre ℰ y) u*x′) mi .eta x u′ = weak-lift.universal u x′ u′ mi .inv x v′ = hom[ idl u ] (v′ ∘′ weak-lift.lifting u x′) mi .eta∘inv _ = funext λ v′ → sym $ weak-lift.unique u _ _ (to-pathp refl) mi .inv∘eta _ = funext λ u′ → from-pathp $ weak-lift.commutes u _ _ mi .natural _ _ v′ = funext λ u′ → weak-lift.unique _ _ _ $ to-pathp $ smashl _ _ ∙ weave _ (ap (_∘ u) (idl id)) _ (pullr′ _ (weak-lift.commutes _ _ _))

As in the weak
cartesian case, the converse is also true: if there is a lifting of
objects `Ob[ x ] → Ob[ y ]`

for every morphism
$f : x \to y$
in
$\mathcal{B}$,
along with a equivalence of homs as above, then
$\mathcal{E}$
is a weak opfibration.

module _ (_*₀_ : ∀ {x y} → Hom x y → Ob[ x ] → Ob[ y ]) where private vertical-equiv-iso-natural : (∀ {x y x′ y′} {f : Hom x y} → Hom[ f ] x′ y′ → Hom[ id ] (f *₀ x′) y′) → Type _ vertical-equiv-iso-natural to = ∀ {x y x′ y′ y″} {g : Hom x y} → (f′ : Hom[ id ] y′ y″) (g′ : Hom[ g ] x′ y′) → to (hom[ idl g ] (f′ ∘′ g′)) ≡[ sym (idl id) ] f′ ∘′ to g′ vertical-equiv→weak-opfibration : (to : ∀ {x y x′ y′} {f : Hom x y} → Hom[ f ] x′ y′ → Hom[ id ] (f *₀ x′) y′) → (eqv : ∀ {x y x′ y′} {f : Hom x y} → is-equiv (to {x} {y} {x′} {y′} {f})) → (natural : vertical-equiv-iso-natural to) → is-weak-cocartesian-fibration vertical-equiv→weak-opfibration to to-eqv natural = weak-op-fibration→weak-opfibration $ vertical-equiv→weak-fibration (ℰ ^total-op) _*₀_ to to-eqv λ f′ g′ → to-pathp (reindex _ _ ∙ from-pathp (natural g′ f′))