open import Cat.Diagram.Pullback.Properties
open import Cat.Displayed.Cocartesian.Weak
open import Cat.Displayed.Cartesian.Weak
open import Cat.Displayed.BeckChevalley
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

open is-weak-cartesian
open Cocartesian-lift
open Cartesian-lift
open is-cocartesian
open is-cartesian
open is-pullback
open Displayed
open Pullback
open Functor
open /-Obj

module Cat.Displayed.Instances.Slice where

module _ {o ℓ} (B : Precategory o ℓ) where
  open CR B

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.

  private
    Ob[] : ⌞ B ⌟ → Type _
    Ob[] x = /-Obj {C = B} x

  record Slice-hom {x y} (f : Hom x y) (px : Ob[] x) (py : Ob[] y) : Type ℓ where
    no-eta-equality
    field
      map : Hom (px .dom) (py .dom)
      com : py ./-Obj.map ∘ map ≡ f ∘ px ./-Obj.map

  open Slice-hom public

{-# INLINE Slice-hom.constructor #-}
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
  _ {o ℓ} {B : Precategory o ℓ} (open Precategory B)
    {x y} {f g : Hom x y} {px : /-Obj x} {py : /-Obj y}
    {f' : Slice-hom B f px py} {g' : Slice-hom B g px py}
  where

  Slice-pathp : ∀ {p : f ≡ g} → f' .map ≡ g' .map → PathP (λ i → Slice-hom B (p i) px py) f' g'
  Slice-pathp {p = p} p' i .map = p' i
  Slice-pathp {p = p} p' i .com = is-prop→pathp
    (λ i → Hom-set _ _ (py .map ∘ p' i) (p i ∘ px .map))
    (f' .com) (g' .com) i

Slice-path
  : ∀ {o ℓ} {B : Precategory o ℓ} (open Precategory B)
  → ∀ {x y} {f : Hom x y} {px : /-Obj x} {py : /-Obj y}
  → {f' g' : Slice-hom B f px py}
  → (f' .map ≡ g' .map)
  → f' ≡ g'
Slice-path = Slice-pathp

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

module _ {o ℓ} (B : Precategory o ℓ) where
  open CR B

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 B
  Slices .Hom[_]-set _ _ _ = hlevel 
  Slices .id' = record where
    map = id
    com = id-comm

  Slices ._∘'_ {x = x} {y = y} {z = z} {f = f} {g = g} px py = record where
    com =
      z .map ∘ px .map ∘ py .map ≡⟨ extendl (px .com) ⟩≡
      f ∘ y .map ∘ py .map       ≡⟨ pushr (py .com) ⟩≡
      (f ∘ g) ∘ x .map           ∎
    map = px .map ∘ py .map

  Slices .idr' {f = f} f' = Slice-pathp (idr (f' .map))
  Slices .idl' {f = f} f' = Slice-pathp (idl (f' .map))
  Slices .assoc' {f = f} {g = g} {h = h} f' g' h' = Slice-pathp $
    assoc (f' .map) (g' .map) (h' .map)
  Slices .hom[_] p f' = record
    { map = f' .map
    ; com = f' .com ∙ ap₂ _∘_ p refl
    }
  Slices .coh[_] p f' = Slice-pathp refl

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 .map
  Fibre→slice .F₁ f ./-Hom.com = f .com ∙ eliml refl
  Fibre→slice .F-id    = ext refl
  Fibre→slice .F-∘ f g = ext refl

  Fibre→slice-is-ff : ∀ {x} → is-fully-faithful (Fibre→slice {x = x})
  Fibre→slice-is-ff = is-iso→is-equiv λ where
    .is-iso.from hom → record where
      map = hom ./-Hom.map
      com = hom ./-Hom.com ∙ introl refl
    .is-iso.rinv x → ext refl
    .is-iso.linv x → Slice-pathp 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 B f x' y'}
    → is-cartesian Slices f f'
    → is-pullback B (x' .map) f (f' .map) (y' .map)
  cartesian→pullback {x} {y} {x'} {y'} {f} {f'} cart = pb where
    pb : is-pullback B (x' .map) f (f' .map) (y' .map)
    pb .square       = sym (f' .com)
    pb .universal p  = cart .universal _ record { com = sym p ∙ ap₂ _∘_ (intror refl) refl } .map
    pb .p₁∘universal = cart .universal _ _ .com ∙ eliml refl
    pb .p₂∘universal = ap map (cart .commutes _ _)
    pb .unique p q   = ap map $ cart .unique
      record { com = p ∙ introl refl }
      (Slice-pathp q)

  pullback→cartesian
    : ∀ {x y x' y'} {f : Hom x y} {f' : Slice-hom B f x' y'}
    → is-pullback B (x' .map) f (f' .map) (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' .map = pb.universal (assoc _ _ _ ∙ sym (h' .com))
    cart .universal m h' .com  = pb.p₁∘universal
    cart .commutes m h' = Slice-pathp pb.p₂∘universal
    cart .unique m' x = Slice-pathp $ pb.unique (m' .com) (ap map 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 B f x' y'}
    → is-weak-cartesian Slices f f'
    → is-pullback B (x' .map) f (f' .map) (y' .map)

  pullback→weak-cartesian
    : ∀ {x y x' y'} {f : Hom x y} {f' : Slice-hom B f x' y'}
    → is-pullback B (x' .map) f (f' .map) (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' .map) (y' .map)
    pb .square       = sym (f' .com)
    pb .universal p  = cart .universal record{ com = sym p } .map
    pb .p₁∘universal = cart .universal _ .com ∙ eliml refl
    pb .p₂∘universal = apd (λ _ → Slice-hom.map) (cart .commutes _)
    pb .is-pullback.unique p q = ap Slice-hom.map $ cart .unique
      record{ com = p ∙ introl refl }
      (Slice-pathp 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
    : has-pullbacks B
    → 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 .map = pb.p₂
    lift-f .lifting .com = sym 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

    pb : Pullback B f g
    pb .apex = (f ^* cut g) .dom
    pb .p₁ = (f ^* cut g) .map
    pb .p₂ = π* f (cut g) .map
    pb .has-is-pb .square = sym (π* f (cut g) .com)
    pb .has-is-pb .universal {p₁' = p₁'} {p₂'} p =
      π*.universal {u' = cut id}
        p₁' record{ com = sym p ∙ intror refl } .map
    pb .has-is-pb .p₁∘universal =
      π*.universal _ _ .com ∙ elimr refl
    pb .has-is-pb .p₂∘universal = ap map (π*.commutes _ _)
    pb .has-is-pb .unique p q = ap map $ π*.unique
      record{ com = p ∙ intror refl } (Slice-path 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. Thinking of a fibration as a setting for interpreting type theory, this gives an interpretation for $\Sigma \text{-types}$ in the codomain fibration of any category.

In fact, we can characterise the cocartesian maps between slices as exactly those squares whose underlying top map is invertible.

  top-invertible→cocartesian
    : ∀ {x y x' y'} {f : Hom x y} {f' : Slice-hom B f x' y'}
    → is-invertible (f' .map)
    → is-cocartesian Slices f f'
  top-invertible→cocartesian {x' = x'} {y' = y'} {f = f} {f'} inv = cocart where
    module inv = is-invertible inv

    cocart : is-cocartesian Slices f f'
    cocart .universal m h' .map = h' .map ∘ inv.inv
    cocart .universal {u' = u'} m h' .com =
      u' .map ∘ h' .map ∘ inv.inv     ≡⟨ extendl (h' .com) ⟩≡
      (m ∘ f) ∘ x' .map ∘ inv.inv     ≡⟨ pullr (extendl (sym (f' .com))) ⟩≡
      m ∘ y' .map ∘ f' .map ∘ inv.inv ≡⟨ (refl⟩∘⟨ elimr inv.invl) ⟩≡
      m ∘ y' .map                     ∎
    cocart .commutes m h' = Slice-path (cancelr inv.invr)
    cocart .unique m' p   = Slice-path (sym (rswizzle (sym (ap map p)) inv.invl))

Given a map $f:X\to Y$ and an object $X^{\prime }:\mathcal{B}/X\text{,}$ the cocartesian lifting $f_{!}{X}^{\prime }:\mathcal{B}/Y$ is then witnessed by the following square:

  Codomain-opfibration : Cocartesian-fibration Slices
  Codomain-opfibration f x' = lift-f where
    lift-f : Cocartesian-lift Slices f x'
    lift-f .y'      = cut (f ∘ x' .map)
    lift-f .lifting = record{ map = id ; com = idr _ }
    lift-f .cocartesian = top-invertible→cocartesian id-invertible

We can now prove the converse implication by uniqueness of cocartesian lifts.

  cocartesian→top-invertible
    : ∀ {x y x' y'} {f : Hom x y} {f' : Slice-hom B f x' y'}
    → is-cocartesian Slices f f'
    → is-invertible (f' .map)
  cocartesian→top-invertible {x' = x'} {y'} {f = f} {f'} cocart = f'-inv where
    module cocart = is-cocartesian cocart
    open is-invertible
    open Inverses

    the-lift : Slice-hom B f x' (cut (f ∘ x' .map))
    the-lift = Codomain-opfibration f x' .lifting

    univ : Slice-hom B id y' (cut (f ∘ x' .map))
    univ = cocart.universalv {b'' = cut (f ∘ x' .map)} the-lift

    f'-inv : is-invertible (f' .map)
    f'-inv .inv = univ .map
    f'-inv .inverses .invl = ap map $ cocart.uniquev₂ {x' = y'} {g' = f'}
      (record { map = _ ; com = pulll (f' .com) ∙ univ .com })
      (Slices .id')
      (Slice-pathp (cancelr (f'-inv .inverses .invr)))
      (Slice-pathp (idl _))
    f'-inv .inverses .invr = apd (λ _ → map) (cocart.commutesv the-lift)

The Beck-Chevalley condition🔗

The canonical self-indexing satisfies the Beck-Chevalley condition at every pullback square in the base category. The situation is summarised by the following commuting cube:

Using our characterisations of cartesian and cocartesian morphisms above, we know that the left and right faces are pullback squares and that $f^{\prime }$ is invertible, and we want to show that $k^{\prime }$ is invertible. By assumption, the bottom face is also a pullback square, so by the pasting law for pullbacks we conclude that the top face is a pullback square as well; but invertible morphisms are stable under pullback, so we are done.

  Slices-beck-chevalley
    : ∀ {a b c d} {f : Hom a b} {g : Hom c a} {h : Hom d b} {k : Hom c d}
    → (pb : is-pullback B g f k h)
    → left-beck-chevalley Slices _ _ _ _ (pb .is-pullback.square)
  Slices-beck-chevalley pb {f' = f'} {g'} {h'} {k'} sq' f'-cocart g'-cart h'-cart =
    top-invertible→cocartesian
    $ is-invertible→pullback-is-invertible {g = h' .map} {p1 = g' .map}
        (cocartesian→top-invertible f'-cocart)
    $ rotate-pullback
    $ pasting-outer→left-is-pullback
        (cartesian→pullback h'-cart)
        (subst-is-pullback (sym (k' .com)) refl refl (sym (f' .com))
          (pasting-left→outer-is-pullback
            (rotate-pullback pb)
            (cartesian→pullback g'-cart)))
        (sym (apd (λ _ → map) sq'))