open import Cat.Displayed.Cartesian.Discrete
open import Cat.Displayed.Functor
open import Cat.Displayed.Base
open import Cat.Prelude

import Cat.Displayed.Cartesian
import Cat.Displayed.Reasoning
import Cat.Displayed.Morphism
import Cat.Reasoning

module Cat.Displayed.Cartesian.Right
  {o β„“ o' β„“'}
  {ℬ : Precategory o β„“}
  (β„° : Displayed ℬ o' β„“')
  where

open Cat.Reasoning ℬ
open Displayed β„°
open Cat.Displayed.Cartesian β„°
open Cat.Displayed.Morphism β„°
open Cat.Displayed.Reasoning β„°

Right fibrationsπŸ”—

A cartesian fibration is said to be a right fibration if every morphism in is cartesian.

record Right-fibration : Type (o βŠ” β„“ βŠ” o' βŠ” β„“') where
  no-eta-equality
  field
    is-fibration : Cartesian-fibration
    cartesian
      : βˆ€ {x y} {f : Hom x y}
      β†’ βˆ€ {x' y'} (f' : Hom[ f ] x' y')
      β†’ is-cartesian f f'

  open Cartesian-fibration is-fibration public

One notable fact is every vertical morphism of a right fibrations is invertible. In other words, if is a right fibration, then all of its fibre categories are groupoids. This follows directly from the fact that vertical cartesian maps are invertible.

right-fibration→vertical-invertible
  : Right-fibration
  β†’ βˆ€ {x} {x' x'' : Ob[ x ]} β†’ (f' : Hom[ id ] x' x'') β†’ is-invertible↓ f'
right-fibration→vertical-invertible rfib f' =
  vertical+cartesian→invertible (Right-fibration.cartesian rfib f')

More notably, this is an exact characterization of categories fibred in groupoids! If is a cartesian fibration where all vertical morphisms are invertible, then it must be a right fibration.

vertical-invertible+fibration→right-fibration
  : Cartesian-fibration
  β†’ (βˆ€ {x} {x' x'' : Ob[ x ]} β†’ (f' : Hom[ id ] x' x'') β†’ is-invertible↓ f')
  β†’ Right-fibration
vertical-invertible+fibration→right-fibration fib vert-inv
  .Right-fibration.is-fibration = fib
vertical-invertible+fibration→right-fibration fib vert-inv
  .Right-fibration.cartesian {x = x} {f = f} {x' = x'} {y' = y'} f' = f-cart where
    open Cartesian-fibration fib
    open Cartesian-lift renaming (x' to x-lift)

To see this, recall that cartesian morphisms are stable under vertical retractions. The cartesian lift of is obviously cartesian, so it suffices to show that there is a vertical retraction To construct this retraction, we shall factorize over which yields a vertical morphism By our hypotheses, is invertible, and thus is a retraction. What remains to be shown is that the inverse to factors and this follows from the factorisation of and the fact that is invertible.

    x* : Ob[ x ]
    x* = has-lift f y' .x-lift

    f* : Hom[ f ] x* y'
    f* = has-lift f y' .lifting

    module f* = is-cartesian (has-lift f y' .cartesian)

    i* : Hom[ id ] x' x*
    i* = f*.universal' (idr f) f'

    module i*-inv = is-invertible[_] (vert-inv i*)

    i*⁻¹ : Hom[ id ] x* x'
    i*⁻¹ = i*-inv.inv'

    factors : f' ∘' i*⁻¹ ≑[ idr f ] f*
    factors = to-pathp⁻ $
      f' ∘' i*⁻¹               β‰‘βŸ¨ shiftr _ (pushl' _ (symP $ f*.commutesp (idr f) f') {q = ap (f ∘_) (sym (idl _))}) βŸ©β‰‘
      hom[] (f* ∘' i* ∘' i*⁻¹) β‰‘βŸ¨ weave _ (elimr (idl id)) _ (elimr' _ i*-inv.invl') βŸ©β‰‘
      hom[] f* ∎

    f-cart : is-cartesian f f'
    f-cart = cartesian-vertical-retraction-stable
      (has-lift f y' .cartesian)
      (inverses[]β†’from-has-section[] i*-inv.inverses')
      factors

As a corollary, we get that all discrete fibrations are right fibrations. Intuitively, this is true, as sets are 0-groupoids.

discrete→right-fibration
  : Discrete-fibration β„°
  β†’ Right-fibration
discrete→right-fibration dfib =
  vertical-invertible+fibration→right-fibration
    (discrete→cartesian ℰ dfib)
    (discrete→vertical-invertible ℰ dfib)

Fibred functors and right fibrationsπŸ”—

As every map in a right fibration is cartesian, every displayed functor into a right fibration is automatically fibred.

functor+right-fibration→fibred
  : βˆ€ {oβ‚‚ β„“β‚‚ oβ‚‚' β„“β‚‚'}
  β†’ {π’Ÿ : Precategory oβ‚‚ β„“β‚‚}
  β†’ {β„± : Displayed π’Ÿ oβ‚‚' β„“β‚‚'}
  β†’ {F : Functor π’Ÿ ℬ}
  β†’ Right-fibration
  β†’ (F' : Displayed-functor β„± β„° F)
  β†’ Fibred-functor β„± β„° F
functor+right-fibration→fibred rfib F' .Fibred-functor.disp =
  F'
functor+right-fibration→fibred rfib F' .Fibred-functor.F-cartesian f' _ =
  Right-fibration.cartesian rfib (F₁' f')
  where open Displayed-functor F'

Specifically, this implies that all displayed functors into a discrete fibrations are fibred. This means that we can drop the fibred condition when working with functors between discrete fibrations.

functor+discrete→fibred
  : βˆ€ {oβ‚‚ β„“β‚‚ oβ‚‚' β„“β‚‚'}
  β†’ {π’Ÿ : Precategory oβ‚‚ β„“β‚‚}
  β†’ {β„± : Displayed π’Ÿ oβ‚‚' β„“β‚‚'}
  β†’ {F : Functor π’Ÿ ℬ}
  β†’ Discrete-fibration β„°
  β†’ (F' : Displayed-functor β„± β„° F)
  β†’ Fibred-functor β„± β„° F
functor+discrete→fibred disc F' =
  functor+right-fibration→fibred (discrete→right-fibration disc) F'