module Cat.Displayed.Total
  {o ℓ o' ℓ'} {B : Precategory o ℓ} (E : Displayed B o' ℓ') where

open Displayed E
open DR E
open DM E
open CR B

The total category of a displayed category🔗

So far, we’ve been thinking of displayed categories as “categories of structures” over some base category. However, it’s often useful to consider a more “bundled up” notion of structure, where the carrier and the structure are considered as a singular object. For instance, if we consider the case of algebraic structures, we often want to think about “a monoid” as a singular thing, as opposed to structure imposed atop some set.

Constructing the total category does exactly this. The objects are pairs of an object from the base, an object from the displayed category that lives over it.

Note that we use a sigma type here instead of a record for technical reasons: this makes it simpler to work with algebraic structures.

Total : Type (o ⊔ o')
Total = Σ[ Carrier ∈ B ] Ob[ Carrier ]

The situation is similar for morphisms: we bundle up a morphism from the base category along with a morphism that lives above it.

record Total-hom (X Y : Total) : Type (ℓ ⊔ ℓ') where
  constructor total-hom
  field
    hom       : Hom (X .fst) (Y .fst)
    preserves : Hom[ hom ] (X .snd) (Y .snd)

open Total-hom

As is tradition, we need to prove some silly lemmas showing that the bundled morphisms form an hset, and another characterizing the paths between morphisms.

total-hom-path : ∀ {X Y : Total} {f g : Total-hom X Y}
               → (p : f .hom ≡ g .hom) → f .preserves ≡[ p ] g .preserves → f ≡ g
total-hom-path p p' i .hom = p i
total-hom-path {f = f} {g = g} p p' i .preserves = p' i

With all that in place, we can construct the total category!

∫ : Precategory (o ⊔ o') (ℓ ⊔ ℓ')
∫ .Precategory.Ob = Total
∫ .Precategory.Hom = Total-hom
∫ .Precategory.Hom-set _ _ = hlevel 2
∫ .Precategory.id .hom = id
∫ .Precategory.id .preserves = id'
∫ .Precategory._∘_ f g .hom = f .hom ∘ g .hom
∫ .Precategory._∘_ f g .preserves = f .preserves ∘' g .preserves
∫ .Precategory.idr _ = total-hom-path (idr _) (idr' _)
∫ .Precategory.idl _ = total-hom-path (idl _) (idl' _)
∫ .Precategory.assoc _ _ _ = total-hom-path (assoc _ _ _) (assoc' _ _ _)

Morphisms in the total category🔗

Isomorphisms in the total category of $E$ consist of pairs of isomorphisms in $B$ and $E .$

private module ∫E = CR ∫

total-iso→iso : ∀ {x y} → x ∫E.≅ y → x .fst ≅ y .fst
total-iso→iso f = make-iso
    (∫E._≅_.to f .hom)
    (∫E._≅_.from f .hom)
    (ap hom $ ∫E._≅_.invl f)
    (ap hom $ ∫E._≅_.invr f)

total-iso→iso[] : ∀ {x y} → (f : x ∫E.≅ y) → x .snd ≅[ total-iso→iso f ] y .snd
total-iso→iso[] f = make-iso[ total-iso→iso f ]
    (∫E._≅_.to f .preserves)
    (∫E._≅_.from f .preserves)
    (ap preserves $ ∫E._≅_.invl f)
    (ap preserves $ ∫E._≅_.invr f)

Pullbacks in the total category🔗

Pullbacks in the total category of $E$ have a particularly nice characterization. Consider the following pair of commuting squares.

If the bottom square is a pullback square, and both $p_{1}^{'}$ and $g^{'}$ are cartesian, then the upper square is a pullback in the total category of $E .$

cartesian+pullback→total-pullback
  : ∀ {p x y z p' x' y' z'}
  → {p₁ : Hom p x} {f : Hom x z} {p₂ : Hom p y} {g : Hom y z}
  → {p₁' : Hom[ p₁ ] p' x'} {f' : Hom[ f ] x' z'}
  → {p₂' : Hom[ p₂ ] p' y'} {g' : Hom[ g ] y' z'}
  → is-cartesian E p₁ p₁'
  → is-cartesian E g g'
  → (pb : is-pullback B p₁ f p₂ g)
  → (open is-pullback pb)
  → f' ∘' p₁' ≡[ square ] g' ∘' p₂'
  → is-pullback ∫
      (total-hom p₁ p₁') (total-hom f f')
      (total-hom p₂ p₂') (total-hom g g')

As the lower square is already a pullback, all that remains is constructing a suitable universal morphism in $E .$ Luckily, $p_{1}^{'}$ is cartesian, so we can factorise maps $A^{'} \to X^{'}$ in $E$ to obtain a map $A^{'} \to P^{'} .$ We then use the fact that $g^{'}$ is cartesian to show that the map we’ve constructed factorises maps $A^{'} \to Y^{'}$ as well. Uniqueness follows from the fact that $p_{1}^{'}$ is cartesian.

cartesian+pullback→total-pullback p₁-cart g-cart pb square' = total-pb where
  open is-pullback
  open Total-hom
  module p₁' = is-cartesian p₁-cart
  module g' = is-cartesian g-cart

  total-pb : is-pullback ∫ _ _ _ _
  total-pb .square = total-hom-path (pb .square) square'
  total-pb .universal {a , a'} {p₁''} {p₂''} p =
    total-hom (pb .universal (ap hom p))
      (p₁'.universal' (pb .p₁∘universal) (p₁'' .preserves))
  total-pb .p₁∘universal =
    total-hom-path (pb .p₁∘universal) (p₁'.commutesp _ _)
  total-pb .p₂∘universal {p = p} =
    total-hom-path (pb .p₂∘universal) $
      g'.uniquep₂ _ _ _ _ _
        (pulll[] _ (symP square')
        ∙[] pullr[] _ (p₁'.commutesp (pb .p₁∘universal) _))
        (symP $ ap preserves p)
  total-pb .unique p q =
    total-hom-path (pb .unique (ap hom p) (ap hom q)) $
      p₁'.uniquep _ _ (pb .p₁∘universal) _ (ap preserves p)

We can also show the converse, provided that $E$ is a fibration.

cartesian+total-pullback→pullback
  : ∀ {p x y z p' x' y' z'}
  → {p₁ : Hom p x} {f : Hom x z} {p₂ : Hom p y} {g : Hom y z}
  → {p₁' : Hom[ p₁ ] p' x'} {f' : Hom[ f ] x' z'}
  → {p₂' : Hom[ p₂ ] p' y'} {g' : Hom[ g ] y' z'}
  → Cartesian-fibration E
  → is-cartesian E p₁ p₁'
  → is-cartesian E g g'
  → is-pullback ∫
      (total-hom p₁ p₁') (total-hom f f')
      (total-hom p₂ p₂') (total-hom g g')
  → is-pullback B p₁ f p₂ g

As we already have a pullback in the total category, the crux will be constructing enough structure in $E$ so that we can invoke the universal property of the pullback. We can do this by appealing to the fact that $E$ is a fibration, which allows us to lift morphisms in the base to obtain a cone in $E .$ From here, we use the fact that $p_{1}^{'}$ and $g^{'}$ are cartesian to construct the relevant paths.

cartesian+total-pullback→pullback
  {p} {x} {y} {z}
  {p₁ = p₁} {f} {p₂} {g} {p₁'} {f'} {p₂'} {g'} fib p₁-cart g-cart total-pb = pb where
  open is-pullback
  open Total-hom
  open Cartesian-fibration fib
  module p₁' = is-cartesian p₁-cart
  module g' = is-cartesian g-cart

  pb : is-pullback B _ _ _ _
  pb .square = ap hom (total-pb .square)
  pb .universal {P} {p₁''} {p₂''} sq =
    total-pb .universal
      {p₁' = total-hom p₁'' (has-lift.lifting p₁'' _)}
      {p₂' = total-hom p₂'' (g'.universal' (sym sq) (f' ∘' has-lift.lifting p₁'' _))}
      (total-hom-path sq (symP (g'.commutesp (sym sq) _))) .hom
  pb .p₁∘universal =
    ap hom $ total-pb .p₁∘universal
  pb .p₂∘universal =
    ap hom $ total-pb .p₂∘universal
  pb .unique {p = p} q r =
    ap hom $ total-pb .unique
      (total-hom-path q (p₁'.commutesp q _))
      (total-hom-path r (g'.uniquep _ _ (sym $ p) _
        (pulll[] _ (symP $ ap preserves (total-pb .square))
        ∙[] pullr[] _ (p₁'.commutesp q _))))