open import Cat.Displayed.Cartesian.Weak
open import Cat.Displayed.Cocartesian
open import Cat.Displayed.Cartesian
open import Cat.Functor.Equivalence
open import Cat.Functor.Properties
open import Cat.Diagram.Pullback
open import Cat.Displayed.Fibre
open import Cat.Instances.Slice
open import Cat.Displayed.Base
open import Cat.Prelude

import Cat.Reasoning as CR

module Cat.Displayed.Instances.Slice {o ℓ} (B : Precategory o ℓ) where

open Cartesian-lift
open Displayed
open is-cartesian
open is-weak-cartesian
open Functor
open CR B
open /-Obj

The canonical self-indexing🔗

There is a canonical way of viewing any category $\mathcal{B}$ as displayed over itself, given fibrewise by taking slice categories. Following (Sterling and Angiuli 2022), we refer to this construction as the canonical self-indexing of $\mathcal{B}$ and denote it $\underline{\mathcal{B}}\text{.}$ Recall that the objects in the slice over $y$ are pairs consisting of an object $x$ and a map $f:x\to y\text{.}$ The core idea is that any morphism lets us view an object $x$ as being “structure over” an object $y\text{;}$ the collection of all possible such structures, then, is the set of morphisms $x\to y\text{,}$ with domain allowed to vary.

Contrary to the maps in the slice category, the maps in the canonical self-indexing have an extra “adjustment” by a morphism $f:x\to y$ of the base category. Where maps in the ordinary slice are given by commuting triangles, maps in the canonical self-indexing are given by commuting squares, of the form

where the primed objects and dotted arrows are displayed.

record
  Slice-hom
    {x y} (f : Hom x y)
    (px : /-Obj {C = B} x) (py : /-Obj {C = B} y)
    : Type ℓ
  where
  constructor slice-hom
  field
    to      : Hom (px .domain) (py .domain)
    commute : f ∘ px .map ≡ py .map ∘ to

open Slice-hom

private unquoteDecl eqv = declare-record-iso eqv (quote Slice-hom)

The intuitive idea for the canonical self-indexing is possibly best obtained by considering the canonical self-indexing of ${\mathbf{Sets}}_{\kappa }\text{.}$ First, recall that an object $f:\mathbf{Sets}/X$ is equivalently a $X\text{-indexed}$ family of sets, with the value of the family at each point $x:X$ being the fibre $f^{\ast }(x)\text{.}$ A function $X\to Y$ of sets then corresponds to a reindexing, which takes an $Y\text{-family}$ of sets to a $X\text{-family}$ of sets (in a functorial way). A morphism $X^{\prime }\to Y^{\prime }$ in the canonical self-indexing of $\mathbf{Sets}$ lying over a map $f:X\to Y$ is then a function between the families $X^{\prime }\to Y^{\prime }$ which commutes with the reindexing given by $f\text{.}$

module _ {x y} {f g : Hom x y} {px : /-Obj x} {py : /-Obj y}
         {f' : Slice-hom f px py} {g' : Slice-hom g px py} where

  Slice-pathp : (p : f ≡ g) → (f' .to ≡ g' .to) → PathP (λ i → Slice-hom (p i) px py) f' g'
  Slice-pathp p p' i .to = p' i
  Slice-pathp p p' i .commute =
    is-prop→pathp
      (λ i → Hom-set _ _ (p i ∘ px .map) (py .map ∘ (p' i)))
      (f' .commute)
      (g' .commute)
      i

Slice-path
  : ∀ {x y} {f : Hom x y} {px : /-Obj x} {py : /-Obj y}
  → {f' g' : Slice-hom f px py}
  → (f' .to ≡ g' .to)
  → f' ≡ g'
Slice-path = Slice-pathp refl

unquoteDecl H-Level-Slice-hom = declare-record-hlevel  H-Level-Slice-hom (quote Slice-hom)

It’s straightforward to piece together the objects of the (ordinary) slice category and our displayed maps Slice-hom into a category displayed over $\mathcal{B}\text{.}$

Slices : Displayed B (o ⊔ ℓ) ℓ
Slices .Ob[_] = /-Obj {C = B}
Slices .Hom[_] = Slice-hom
Slices .Hom[_]-set _ _ _ = hlevel 
Slices .id' = slice-hom id id-comm-sym
Slices ._∘'_ {x = x} {y = y} {z = z} {f = f} {g = g} px py =
  slice-hom (px .to ∘ py .to) $
    (f ∘ g) ∘ x .map           ≡⟨ pullr (py .commute) ⟩≡
    f ∘ (y .map ∘ py .to)      ≡⟨ extendl (px .commute) ⟩≡
    z .map ∘ (px .to ∘ py .to) ∎
Slices .idr' {f = f} f' = Slice-pathp (idr f) (idr (f' .to))
Slices .idl' {f = f} f' = Slice-pathp (idl f) (idl (f' .to))
Slices .assoc' {f = f} {g = g} {h = h} f' g' h' =
  Slice-pathp (assoc f g h) (assoc (f' .to) (g' .to) (h' .to))

It’s only slightly more annoying to show that a vertical map in the canonical self-indexing is a map in the ordinary slice category which, since the objects displayed over $x$ are defined to be those of the slice category $\mathcal{B}/x\text{,}$ gives an equivalence of categories between the fibre ${\underline{\mathcal{B}}}^{\ast }(x)$ and the slice $\mathcal{B}/x\text{.}$

Fibre→slice : ∀ {x} → Functor (Fibre Slices x) (Slice B x)
Fibre→slice .F₀ x = x
Fibre→slice .F₁ f ./-Hom.map = f .to
Fibre→slice .F₁ f ./-Hom.commutes = sym (f .commute) ∙ eliml refl
Fibre→slice .F-id = ext refl
Fibre→slice .F-∘ f g = ext (transport-refl _)

Fibre→slice-is-ff : ∀ {x} → is-fully-faithful (Fibre→slice {x = x})
Fibre→slice-is-ff {_} {x} {y} = is-iso→is-equiv λ where
  .is-iso.from hom → slice-hom (hom ./-Hom.map) (eliml refl ∙ sym (hom ./-Hom.commutes))
  .is-iso.rinv x → ext refl
  .is-iso.linv x → Slice-pathp refl refl

Fibre→slice-is-equiv : ∀ {x} → is-equivalence (Fibre→slice {x})
Fibre→slice-is-equiv = is-precat-iso→is-equivalence $
  record { has-is-ff = Fibre→slice-is-ff
         ; has-is-iso = id-equiv
         }

Cartesian maps🔗

A map $f^{\prime }:x^{\prime }\to y^{\prime }$ over $f:x\to y$ in the codomain fibration is cartesian if and only if it forms a pullback square as below:

This follows by a series of relatively straightforward computations, so we do not comment too heavily on the proof.

cartesian→pullback
  : ∀ {x y x' y'} {f : Hom x y} {f' : Slice-hom f x' y'}
  → is-cartesian Slices f f'
  → is-pullback B (x' .map) f (f' .to) (y' .map)
cartesian→pullback {x} {y} {x'} {y'} {f} {f'} cart = pb where
  pb : is-pullback B (x' .map) f (f' .to) (y' .map)
  pb .is-pullback.square = f' .commute
  pb .is-pullback.universal p =
    cart .universal _ (slice-hom _ (idr _ ∙ p)) .to
  pb .is-pullback.p₁∘universal =
    sym (cart .universal _ _ .commute) ∙ idr _
  pb .is-pullback.p₂∘universal =
    ap Slice-hom.to (cart .commutes _ _)
  pb .is-pullback.unique p q =
    ap Slice-hom.to (cart .unique (slice-hom _ (idr _ ∙ sym p)) (Slice-pathp refl q))

pullback→cartesian
  : ∀ {x y x' y'} {f : Hom x y} {f' : Slice-hom f x' y'}
  → is-pullback B (x' .map) f (f' .to) (y' .map)
  → is-cartesian Slices f f'
pullback→cartesian {x} {y} {x'} {y'} {f} {f'} pb = cart where
  module pb = is-pullback pb

  cart : is-cartesian Slices f f'
  cart .universal m h' .to = pb.universal (assoc _ _ _ ∙ h' .commute)
  cart .universal m h' .commute = sym pb.p₁∘universal
  cart .commutes m h' = Slice-pathp refl pb.p₂∘universal
  cart .unique m' x = Slice-pathp refl $
    pb.unique (sym (m' .commute)) (ap to x)

_ = weak-cartesian→cartesian

We can actually weaken the hypothesis of cartesian→pullback so that pullback squares also exactly characterise weakly cartesian morphisms. While this is automatic if $\mathcal{B}$ has all pullbacks (since then cartesian and weakly cartesian morphisms coincide), it is sometimes useful to have both characterisations if we do not want to make such an assumption.

weak-cartesian→pullback
  : ∀ {x y x' y'} {f : Hom x y} {f' : Slice-hom f x' y'}
  → is-weak-cartesian Slices f f'
  → is-pullback B (x' .map) f (f' .to) (y' .map)

pullback→weak-cartesian
  : ∀ {x y x' y'} {f : Hom x y} {f' : Slice-hom f x' y'}
  → is-pullback B (x' .map) f (f' .to) (y' .map)
  → is-weak-cartesian Slices f f'

weak-cartesian→pullback {x} {y} {x'} {y'} {f} {f'} cart = pb where
  pb : is-pullback B (x' .map) f (f' .to) (y' .map)
  pb .is-pullback.square = f' .commute
  pb .is-pullback.universal p =
    cart .universal (slice-hom _ p) .to
  pb .is-pullback.p₁∘universal =
    sym (cart .universal _ .commute) ∙ idl _
  pb .is-pullback.p₂∘universal =
    apd (λ _ → Slice-hom.to) (cart .commutes _)
  pb .is-pullback.unique p q =
    ap Slice-hom.to (cart .unique (slice-hom _ (idl _ ∙ sym p)) (Slice-pathp (idr _) q))

pullback→weak-cartesian pb = cartesian→weak-cartesian _ (pullback→cartesian pb)

As a fibration🔗

If (and only if) $\mathcal{B}$ has all pullbacks, then its self-indexing is a Cartesian fibration. This is almost by definition, and is in fact where the “Cartesian” in “Cartesian fibration” comes from (recall that another term for “pullback square” is “cartesian square”). Since the total space $\int \underline{\mathcal{B}}$ is equivalently the arrow category of $\mathcal{B}\text{,}$ with the projection functor $\pi :\int \underline{\mathcal{B}}\to \mathcal{B}$ corresponding under this equivalence to the codomain functor, we refer to $\underline{\mathcal{B}}$ regarded as a Cartesian fibration as the codomain fibration.

Codomain-fibration
  : (∀ {x y z} (f : Hom x y) (g : Hom z y) → Pullback B f g)
  → Cartesian-fibration Slices
Codomain-fibration pullbacks f y' = lift-f where
  module pb = Pullback (pullbacks f (y' .map))

  lift-f : Cartesian-lift Slices f y'
  lift-f .x' = cut pb.p₁
  lift-f .lifting .to = pb.p₂
  lift-f .lifting .commute = pb.square
  lift-f .cartesian = pullback→cartesian pb.has-is-pb

Since the proof that Slices is a cartesian fibration is given by essentially rearranging the data of pullbacks in $\mathcal{B}\text{,}$ we also have the converse implication: If $\underline{\mathcal{B}}$ is a Cartesian fibration, then $\mathcal{B}$ has all pullbacks.

Codomain-fibration→pullbacks
  : ∀ {x y z} (f : Hom x y) (g : Hom z y)
  → Cartesian-fibration Slices
  → Pullback B f g
Codomain-fibration→pullbacks f g slices-fib = pb where
  open Cartesian-fibration Slices slices-fib
  open is-pullback
  open Pullback

  pb : Pullback B f g
  pb .apex = (f ^* cut g) .domain
  pb .p₁ = (f ^* cut g) .map
  pb .p₂ = π* f (cut g) .to
  pb .has-is-pb .square = π* f (cut g) .commute
  pb .has-is-pb .universal {p₁' = p₁'} {p₂'} p =
    π*.universal {u' = cut id}
      p₁' (slice-hom p₂' (pullr (idr _) ∙ p)) .to
  pb .has-is-pb .p₁∘universal =
    sym (π*.universal _ _ .commute) ∙ idr _
  pb .has-is-pb .p₂∘universal = ap to (π*.commutes _ _)
  pb .has-is-pb .unique p q = ap to $ π*.unique
    (slice-hom _ (idr _ ∙ sym p)) (Slice-pathp refl q)

Since the fibres of the codomain fibration are given by slice categories, then the interpretation of Cartesian fibrations as “displayed categories whose fibres vary functorially” leads us to reinterpret the above results as, essentially, giving the pullback functors between slice categories.

As an opfibration🔗

The canonical self-indexing is always an opfibration, where opreindexing is given by postcomposition. If we think about slices as families, then opreindexing along $X\to Y$ extends a family over $X$ to a family over $Y$ by adding in empty fibres for all elements of $Y$ that do not lie in the image of $f\text{.}$

Codomain-opfibration : Cocartesian-fibration Slices
Codomain-opfibration f x' = lift-f where

  lift-f : Cocartesian-lift Slices f x'
  lift-f .Cocartesian-lift.y' = cut (f ∘ x' .map)
  lift-f .Cocartesian-lift.lifting = slice-hom id (sym (idr _))
  lift-f .Cocartesian-lift.cocartesian .is-cocartesian.universal m h' =
    slice-hom (h' .to) (assoc _ _ _ ∙ h' .commute)
  lift-f .Cocartesian-lift.cocartesian .is-cocartesian.commutes m h' =
    Slice-pathp refl (idr _)
  lift-f .Cocartesian-lift.cocartesian .is-cocartesian.unique m' p =
    Slice-pathp refl (sym (idr _) ∙ ap to p)