open import Cat.Functor.Adjoint.Comonadic
open import Cat.Instances.Coalgebras
open import Cat.Functor.Equivalence
open import Cat.Functor.Properties
open import Cat.Diagram.Pullback
open import Cat.Diagram.Initial
open import Cat.Displayed.Total
open import Cat.Functor.Adjoint
open import Cat.Functor.Compose
open import Cat.Instances.Comma
open import Cat.Instances.Slice
open import Cat.Univalent
open import Cat.Prelude

import Cat.Reasoning

open Coalgebra-on
open ∫Hom

module Cat.Functor.Pullback
  {o ℓ} {C : Precategory o ℓ}
  where

open Cat.Reasoning C
open is-pullback
open Pullback
open Initial
open Functor
open _=>_
open /-Obj
open /-Hom

Base change🔗

Let $C$ be a category with all pullbacks, and $f : Y \to X$ a morphism in $C .$ Then we have a functor $f^{*} : C / X \to C / Y,$ called the base change, where the action on objects is given by pulling back along $f .$

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 $K g X$ of $C / X,$ and on the right we have the whole pullback diagram, showing how the parts fit together. The actual object of $C / Y$ the functor gives is the vertical arrow $Y \times_{X} K \to Y .$

module _ (pullbacks : ∀ {X Y Z} f g → 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 .F₀ x = ob where
    ob : /-Obj Y
    ob .domain = pullbacks (x .map) f .apex
    ob .map    = pullbacks (x .map) f .p₂

On morphisms, we use the universal property of the pullback to obtain a map $Y \times_{X} K \to Y \times_{X} K^{'},$ by observing that the square diagram below is a cone over $K^{'} \to X \leftarrow Y .$

  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) ∙ xpb.square)
    dh' .commutes = ypb.p₂∘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.p₁∘universal ∙ pullr ypb.p₁∘universal ∙ assoc _ _ _)
      (pulll zpb.p₂∘universal ∙ ypb.p₂∘universal)))
    where
      module ypb = Pullback (pullbacks (y .map) f)
      module zpb = Pullback (pullbacks (z .map) f)

Properties🔗

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 $\sum_{f} : C / Y \to C / X,$ is given on objects by precomposition with $f,$ and on morphisms by what is essentially the identity function — only the witness of commutativity must change.

module _ {X Y : Ob} (f : Hom Y X) where
  Σf : Functor (Slice C Y) (Slice C X)
  Σf .F₀ o = cut (f ∘ o .map)
  Σf .F₁ dh = record { map = dh .map ; commutes = pullr (dh .commutes) }
  Σf .F-id = trivial!
  Σf .F-∘ f g = trivial!

  open _⊣_

Σ-iso-equiv
  : {X Y : Ob} {f : Hom Y X}
  → Cat.Reasoning.is-invertible C f
  → is-equivalence (Σf f)
Σ-iso-equiv {X} {f = f} isom = ff+split-eso→is-equivalence Σ-ff Σ-seso where
  module Sl = Cat.Reasoning (Slice C X)
  module isom = is-invertible isom

  func = Σf f
  Σ-ff : ∀ {x y} → is-equiv (func .F₁ {x} {y})
  Σ-ff = is-iso→is-equiv (iso ∘inv (λ x → trivial!) λ x → trivial!) where
    ∘inv : /-Hom _ _ → /-Hom _ _
    ∘inv o .map = o .map
    ∘inv o .commutes = invertible→monic isom _ _ (assoc _ _ _ ∙ o .commutes)

  Σ-seso : is-split-eso func
  Σ-seso y = cut (isom.inv ∘ y .map)
           , Sl.make-iso! into from' (eliml refl) (eliml refl)
    where
    into : /-Hom _ _
    into .map = id
    into .commutes = id-comm ∙ sym (pulll isom.invl)

    from' : /-Hom _ _
    from' .map = id
    from' .commutes = elimr refl ∙ cancell isom.invl

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

module _ (pullbacks : ∀ {X Y Z} f g → 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} {p₂' = obj .map} (idr _)
    dh .commutes = pb.p₂∘universal
  Σf⊣f* .unit .is-natural x y g =
    ext (pb.unique₂
      {p = (f ∘ y .map) ∘ id ∘ g .map ≡⟨ cat! C ⟩≡ f ∘ y .map ∘ g .map ∎}
      (pulll pb.p₁∘universal)
      (pulll pb.p₂∘universal)
      (pulll pb.p₁∘universal ∙ pullr pb'.p₁∘universal ∙ id-comm)
      (pulll pb.p₂∘universal ∙ pb'.p₂∘universal ∙ 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
  Σf⊣f* .counit .is-natural x y g = ext pb.p₁∘universal
    where module pb = Pullback (pullbacks (y .map) f)

  Σf⊣f* .zig {A} = ext pb.p₁∘universal
    where module pb = Pullback (pullbacks (f ∘ A .map) f)

  Σf⊣f* .zag {B} = ext
    (sym (pb.unique₂ {p = pb.square}
      (idr _) (idr _)
      (pulll pb.p₁∘universal ∙ pullr pb'.p₁∘universal ∙ idr _)
      (pulll pb.p₂∘universal ∙ pb'.p₂∘universal))) where
    module pb = Pullback (pullbacks (B .map) f)
    module pb' = Pullback (pullbacks (f ∘ pb.p₂) f)

This adjunction is comonadic; this generalises the fact that the forgetful functor $C / Y \to C$ is comonadic, which we recover by taking $f : Y \to ⊤ .$

The idea is the same, only with more dependent types: thinking of $Y$ as a family of types over $X,$ the comonad $\sum_{f} \circ f^{*} : C / X \to C / X$ sends a map $A \to X$ to the projection map $A \times_{X} Y \to X .$ Therefore, a coalgebra for this comonad consists of a map $⟨ g, h ⟩ : A \to A \times_{X} Y$ over $X;$ but the coalgebra laws force $g$ to be the identity on $A,$ so all that is left is the map $h,$ an object of $(C / X) / f,$ or in other words $C / Y .$

  Σf-comonadic : is-comonadic Σf⊣f*
  Σf-comonadic = is-precat-iso→is-equivalence
    (iso (is-iso→is-equiv ff) (is-iso→is-equiv eso))
    where
      open is-iso

      eso : is-iso (Comparison-CoEM Σf⊣f* .F₀)
      eso .from (A , c) = cut (pb.p₂ ∘ c .ρ .map)
        where module pb = Pullback (pullbacks (A .map) f)
      eso .rinv (A , c) =
          Σ-pathp (/-Obj-path refl path)
        $ Coalgebra-on-pathp _ $ /-Hom-pathp _ _
        $ symP $ Hom-pathp-reflr C $ pb≡.unique i0
          (pulll (from-pathp-to' C _ λ i → pb≡.p₁ i) ∙ unext (c .ρ-counit))
          (pulll (from-pathp-to' C _ λ i → pb≡.p₂ i))
        where
          module pb = Pullback (pullbacks (A .map) f)

          path : f ∘ pb.p₂ ∘ c .ρ .map ≡ A .map
          path = assoc _ _ _ ∙ unext (c .ρ .commutes)

          module pb≡ i = Pullback (pullbacks (path i) f)
      eso .linv p = /-Obj-path refl pb.p₂∘universal
        where module pb = Pullback (pullbacks (f ∘ p .map) f)

      ff : ∀ {x y} → is-iso (Comparison-CoEM Σf⊣f* .F₁ {x} {y})
      ff .from g .map = g .fst .map
      ff {x} {y} .from g .commutes =
        y .map ∘ g .fst .map                       ≡˘⟨ pulll pby.p₂∘universal ⟩≡˘
        pby.p₂ ∘ pby.universal _ ∘ g .fst .map     ≡˘⟨ refl⟩∘⟨ unext (g .snd) ⟩≡˘
        pby.p₂ ∘ pby.universal _ ∘ pbx.universal _ ≡⟨ pulll pby.p₂∘universal ⟩≡
        pbx.p₂ ∘ pbx.universal _                   ≡⟨ pbx.p₂∘universal ⟩≡
        x .map                                     ∎
        where
          module pbx = Pullback (pullbacks (f ∘ x .map) f)
          module pby = Pullback (pullbacks (f ∘ y .map) f)
      ff .rinv _ = trivial!
      ff .linv _ = trivial!

Equifibred natural transformations🔗

A natural transformation $F \Rightarrow G$ is called equifibred, or cartesian, if each of its naturality squares is a pullback:

is-equifibred
  : ∀ {oj ℓj} {J : Precategory oj ℓj} {F G : Functor J C}
  → F => G → Type _
is-equifibred {J = J} {F} {G} α =
  ∀ {x y} (f : J .Precategory.Hom x y)
  → is-pullback C (F .F₁ f) (α .η y) (α .η x) (G .F₁ f)

An easy property of equifibered transformations is that they are closed under pre-whiskering:

◂-equifibred
  : ∀ {oj ℓj ok ℓk} {J : Precategory oj ℓj} {K : Precategory ok ℓk}
  → {F G : Functor J C} (H : Functor K J) (α : F => G)
  → is-equifibred α → is-equifibred (α ◂ H)
◂-equifibred H α eq f = eq (H .F₁ f)