module Cat.Functor.Pullback
  {o β„“} {C : Precategory o β„“}
  where

Base changeπŸ”—

Let C\mathcal{C} be a category with all pullbacks, and f:Yβ†’Xf : Y \to X a morphism in C\mathcal{C}. Then we have a functor fβˆ—:C/Xβ†’C/Yf* : \mathcal{C}/X \to \mathcal{C}/Y, called the base change, where the action on objects is given by pulling back along ff.

On objects, the functor maps as in the diagram below. It’s a bit busy, so look at it in parts: On the left we have the object Kβ†’gXK \xrightarrow{g} X of C/X\mathcal{C}/X, and on the right we have the whole pullback diagram, showing how the parts fit together. The actual object of C/Y\mathcal{C}/Y the functor gives is the vertical arrow YΓ—XKβ†’YY \times_X K \to Y.

module _ (pullbacks : βˆ€ {X Y Z} f g β†’ Pullback C {X} {Y} {Z} f g) {X Y : Ob} (f : Hom Y X) where
  Base-change : Functor (Slice C X) (Slice C Y)
  Base-change .Fβ‚€ x = ob where
    ob : /-Obj Y
    ob .domain = pullbacks (x .map) f .apex
    ob .map    = pullbacks (x .map) f .pβ‚‚

On morphisms, we use the universal property of the pullback to obtain a map YΓ—XKβ†’YΓ—XKβ€²Y \times_X K \to Y \times_X K', by observing that the square diagram below is a cone over Kβ€²β†’X←YK' \to X \leftarrow Y.

  Base-change .F₁ {x} {y} dh = dh' where
    module ypb = Pullback (pullbacks (y .map) f)
    module xpb = Pullback (pullbacks (x .map) f)
    dh' : /-Hom _ _
    dh' .map = ypb.universal {p₁' = dh .map ∘ xpb.p₁}
      (pulll (dh .commutes) βˆ™ xpb.square)
    dh' .commutes = ypb.pβ‚‚βˆ˜universal
The properties of pullbacks also guarantee that this operation is functorial, but the details are not particularly enlightening.
  Base-change .F-id {x} = ext (sym (xpb.unique id-comm (idr _)))
    where module xpb = Pullback (pullbacks (x .map) f)

  Base-change .F-∘ {x} {y} {z} am bm =
    ext (sym (zpb.unique
      (pulll zpb.pβ‚βˆ˜universal βˆ™ pullr ypb.pβ‚βˆ˜universal βˆ™ assoc _ _ _)
      (pulll zpb.pβ‚‚βˆ˜universal βˆ™ ypb.pβ‚‚βˆ˜universal)))
    where
      module ypb = Pullback (pullbacks (y .map) f)
      module zpb = Pullback (pullbacks (z .map) f)

PropertiesπŸ”—

The base change functor is a right adjoint. We construct the left adjoint directly, then give the unit and counit, and finally prove the triangle identities.

The left adjoint, called dependent sum and written βˆ‘f\sum_f, is given on objects by precomposition with ff, and on morphisms by what is essentially the identity function β€” only the witness of commutativity must change.

module _ {X Y : Ob} (f : Hom Y X) where
  Ξ£f : Functor (Slice C Y) (Slice C X)
  Ξ£f .Fβ‚€ o = cut (f ∘ o .map)
  Ξ£f .F₁ dh = record { map = dh .map ; commutes = pullr (dh .commutes) }
  Ξ£f .F-id = trivial!
  Σf .F-∘ f g = trivial!

  open _⊣_
  open _=>_

The adjunction unit and counit are given by the universal properties of pullbacks. ⚠️ WIP ⚠️

module _ (pullbacks : βˆ€ {X Y Z} f g β†’ Pullback C {X} {Y} {Z} f g) {X Y : Ob} (f : Hom Y X) where
  open _⊣_
  open _=>_

  Σf⊣f* : Σf f ⊣ Base-change pullbacks f
  Σf⊣f* .unit .η obj = dh where
    module pb = Pullback (pullbacks (f ∘ obj .map) f)
    dh : /-Hom _ _
    dh .map = pb.universal {p₁' = id} {pβ‚‚' = obj .map} (idr _)
    dh .commutes = pb.pβ‚‚βˆ˜universal
  Σf⊣f* .unit .is-natural x y g =
    ext (pb.uniqueβ‚‚
      {p = (f ∘ y .map) ∘ id ∘ g .map β‰‘βŸ¨ cat! C βŸ©β‰‘ f ∘ y .map ∘ g .map ∎}
      (pulll pb.pβ‚βˆ˜universal)
      (pulll pb.pβ‚‚βˆ˜universal)
      (pulll pb.pβ‚βˆ˜universal βˆ™ pullr pb'.pβ‚βˆ˜universal βˆ™ id-comm)
      (pulll pb.pβ‚‚βˆ˜universal βˆ™ pb'.pβ‚‚βˆ˜universal βˆ™ sym (g .commutes)))
    where
      module pb = Pullback (pullbacks (f ∘ y .map) f)
      module pb' = Pullback (pullbacks (f ∘ x .map) f)

  Σf⊣f* .counit .η obj = dh where
    module pb = Pullback (pullbacks (obj .map) f)
    dh : /-Hom _ _
    dh .map = pb.p₁
    dh .commutes = pb.square
  Ξ£f⊣f* .counit .is-natural x y g = ext pb.pβ‚βˆ˜universal
    where module pb = Pullback (pullbacks (y .map) f)

  Ξ£f⊣f* .zig {A} = ext pb.pβ‚βˆ˜universal
    where module pb = Pullback (pullbacks (f ∘ A .map) f)

  Σf⊣f* .zag {B} = ext
    (sym (pb.uniqueβ‚‚ {p = pb.square}
      (idr _) (idr _)
      (pulll pb.pβ‚βˆ˜universal βˆ™ pullr pb'.pβ‚βˆ˜universal βˆ™ idr _)
      (pulll pb.pβ‚‚βˆ˜universal βˆ™ pb'.pβ‚‚βˆ˜universal))) where
    module pb = Pullback (pullbacks (B .map) f)
    module pb' = Pullback (pullbacks (f ∘ pb.pβ‚‚) f)