open import Cat.Prelude

import Cat.Reasoning

module Cat.Diagram.Pullback where

Pullbacks🔗

module _ {o ℓ} (C : Precategory o ℓ) where
  open Cat.Reasoning C
  private variable
    P' X Y Z : Ob
    h p₁' p₂' : Hom X Y

A pullback $X{\times }_{Z}Y$ of $f:X\to Z$ and $g:Y\to Z$ is the product of $f$ and $g$ in the category $\mathcal{C}/Z\text{,}$ the category of objects fibred over $Z\text{.}$ We note that the fibre of $X{\times }_{Z}Y$ over some element $x$ of $Z$ is the product of the fibres of $f$ and $g$ over $x\text{;}$ Hence the pullback is also called the fibred product.

  record is-pullback {P} (p₁ : Hom P X) (f : Hom X Z) (p₂ : Hom P Y) (g : Hom Y Z)
    : Type (o ⊔ ℓ) where

    no-eta-equality
    field
      square   : f ∘ p₁ ≡ g ∘ p₂

The concrete incarnation of the abstract nonsense above is that a pullback turns out to be a universal square like the one below. Since it is a product, it comes equipped with projections ${\pi }_1$ and ${\pi }_2$ onto its factors; Since isn’t merely a product of $X$ and $Y\text{,}$ but rather of $X$ and $Y$ considered as objects over $Z$ in a specified way, overall square has to commute.

      universal : ∀ {P'} {p₁' : Hom P' X} {p₂' : Hom P' Y}
               → f ∘ p₁' ≡ g ∘ p₂' → Hom P' P
      p₁∘universal : {p : f ∘ p₁' ≡ g ∘ p₂'} → p₁ ∘ universal p ≡ p₁'
      p₂∘universal : {p : f ∘ p₁' ≡ g ∘ p₂'} → p₂ ∘ universal p ≡ p₂'

      unique : {p : f ∘ p₁' ≡ g ∘ p₂'} {lim' : Hom P' P}
             → p₁ ∘ lim' ≡ p₁'
             → p₂ ∘ lim' ≡ p₂'
             → lim' ≡ universal p

    unique₂
      : {p : f ∘ p₁' ≡ g ∘ p₂'} {lim' lim'' : Hom P' P}
      → p₁ ∘ lim' ≡ p₁' → p₂ ∘ lim' ≡ p₂'
      → p₁ ∘ lim'' ≡ p₁' → p₂ ∘ lim'' ≡ p₂'
      → lim' ≡ lim''
    unique₂ {p = o} p q r s = unique {p = o} p q ∙ sym (unique r s)

    pullback-univ
      : ∀ {O} → Hom O P ≃ (Σ (Hom O X) λ h → Σ (Hom O Y) λ h' → f ∘ h ≡ g ∘ h')
    pullback-univ .fst h = p₁ ∘ h , p₂ ∘ h , extendl square
    pullback-univ .snd = is-iso→is-equiv λ where
      .is-iso.from (f , g , α) → universal α
      .is-iso.rinv x → Σ-pathp p₁∘universal $ Σ-prop-pathp (λ _ _ → hlevel ) p₂∘universal
      .is-iso.linv x → sym (unique refl refl)

By universal, we mean that any other “square” (here the second “square” has corners $P^{\prime }\text{,}$ $X\text{,}$ $Y\text{,}$ $Z$ — it’s a bit bent) admits a unique factorisation that passes through $P\text{;}$ We can draw the whole situation as in the diagram below. Note the little corner on $P\text{,}$ indicating that the square is a pullback.

We provide a convenient packaging of the pullback and the projection maps:

  record Pullback {X Y Z} (f : Hom X Z) (g : Hom Y Z) : Type (o ⊔ ℓ) where
    no-eta-equality
    field
      {apex} : Ob
      p₁ : Hom apex X
      p₂ : Hom apex Y
      has-is-pb : is-pullback p₁ f p₂ g

    open is-pullback has-is-pb public

module _ {o ℓ} {C : Precategory o ℓ} where
  open Cat.Reasoning C
  private variable
    P' X Y Z : Ob
    h p₁' p₂' : Hom X Y

  is-pullback-is-prop : ∀ {P} {p₁ : Hom P X} {f : Hom X Z} {p₂ : Hom P Y} {g : Hom Y Z} → is-prop (is-pullback C p₁ f p₂ g)
  is-pullback-is-prop {X = X} {Y = Y} {p₁ = p₁} {f} {p₂} {g} x y = q where
    open is-pullback
    p : Path (∀ {P'} {p₁' : Hom P' X} {p₂' : Hom P' Y} → f ∘ p₁' ≡ g ∘ p₂' → _) (x .universal) (y .universal)
    p i sq = y .unique {p = sq} (x .p₁∘universal {p = sq}) (x .p₂∘universal) i
    q : x ≡ y
    q i .square = Hom-set _ _ _ _ (x .square) (y .square) i
    q i .universal = p i
    q i .p₁∘universal {p₁' = p₁'} {p = sq} = is-prop→pathp (λ i → Hom-set _ _ (p₁ ∘ p i sq) p₁') (x .p₁∘universal) (y .p₁∘universal) i
    q i .p₂∘universal {p = sq} = is-prop→pathp (λ i → Hom-set _ _ (p₂ ∘ p i sq) _) (x .p₂∘universal) (y .p₂∘universal) i
    q i .unique {p = sq} {lim' = lim'} c₁ c₂ = is-prop→pathp (λ i → Hom-set _ _ lim' (p i sq)) (x .unique c₁ c₂) (y .unique c₁ c₂) i

  instance
    H-Level-is-pullback : ∀ {P} {p₁ : Hom P X} {f : Hom X Z} {p₂ : Hom P Y} {g : Hom Y Z} {n} → H-Level (is-pullback C p₁ f p₂ g) (suc n)
    H-Level-is-pullback = prop-instance is-pullback-is-prop

  subst-is-pullback
    : ∀ {P} {p₁ p₁' : Hom P X} {f f' : Hom X Z} {p₂ p₂' : Hom P Y} {g g' : Hom Y Z}
    → p₁ ≡ p₁' → f ≡ f' → p₂ ≡ p₂' → g ≡ g'
    → is-pullback C p₁ f p₂ g
    → is-pullback C p₁' f' p₂' g'
  subst-is-pullback {p₁ = p₁} {p₁'} {f} {f'} {p₂} {p₂'} {g} {g'} p q r s α = done where
    module α = is-pullback α
    open is-pullback

    done : is-pullback C p₁' f' p₂' g'
    done .square = ap₂ _∘_ (sym q) (sym p) ∙∙ α.square ∙∙ ap₂ _∘_ s r
    done .universal β = α.universal (ap₂ _∘_ q refl ∙∙ β ∙∙ ap₂ _∘_ (sym s) refl)
    done .p₁∘universal = ap₂ _∘_ (sym p) refl ∙ α.p₁∘universal
    done .p₂∘universal = ap₂ _∘_ (sym r) refl ∙ α.p₂∘universal
    done .unique β γ = α.unique (ap₂ _∘_ p refl ∙ β) (ap₂ _∘_ r refl ∙ γ)

Kernel pairs🔗

The kernel pair of a morphism $f:X\to Y$ (if it exists) is the pullback of $f$ along itself. Intuitively, one should think of a kernel pair as a partition of $X$ induced by the preimage of $f\text{.}$

module _ {o ℓ} (C : Precategory o ℓ) where
  open Cat.Reasoning C

  is-kernel-pair : ∀ {P X Y} → Hom P X → Hom P X → Hom X Y → Type _
  is-kernel-pair p1 p2 f = is-pullback C p1 f p2 f

module _ {o ℓ} {C : Precategory o ℓ} where
  open Cat.Reasoning C
  private variable
    X Y P : Ob

Note that each of the projections out of the kernel pair of $f$ are epimorphisms. Without loss of generality, we will focus our attention on the first projection.

  is-kernel-pair→epil
    : ∀ {p1 p2 : Hom P X} {f : Hom X Y}
    → is-kernel-pair C p1 p2 f
    → is-epic p1

Recall that a morphism is epic if it has a section; that is, some morphism $u$ such that $p_1\circ u=\mathrm{id}\text{.}$ We can construct such a $u$ by applying the universal property of the pullback to the following diagram.

  is-kernel-pair→epil {p1 = p1} is-kp =
    has-section→epic $
    make-section
      (universal refl)
      p₁∘universal
    where open is-pullback is-kp

  is-kernel-pair→epir
    : ∀ {P} {p1 p2 : Hom P X} {f : Hom X Y}
    → is-kernel-pair C p1 p2 f
    → is-epic p2
  is-kernel-pair→epir {p2 = p2} is-kp =
    has-section→epic $
    make-section
      (universal refl)
      p₂∘universal
    where open is-pullback is-kp

If $f$ is a monomorphism, then its kernel pair always exists, and is given by $(\mathrm{id},\mathrm{id})\text{.}$

  monic→id-kernel-pair
    : ∀ {f : Hom X Y}
    → is-monic f
    → is-kernel-pair C id id f

Clearly, the square $f\circ \mathrm{id}=f\circ \mathrm{id}$ commutes, so the tricky bit will be constructing a universal morphism. If $f\circ p_1=f\circ p_2$ for some $p_1,p_2:P\to X\text{,}$ then we can simply use one of $p_1$ or $p_2$ for our universal map; the choice we make does not matter, as we can obtain $p_1=p_2$ from the fact that $f$ is monic! The rest of the universal property follows directly from this lovely little observation.

  monic→id-kernel-pair {f = f} f-monic = id-kp where
    open is-pullback

    id-kp : is-kernel-pair C id id f
    id-kp .square = refl
    id-kp .universal {p₁' = p₁'} _ = p₁'
    id-kp .p₁∘universal = idl _
    id-kp .p₂∘universal {p = p} = idl _ ∙ f-monic _ _ p
    id-kp .unique p q = sym (idl _) ∙ p

Conversely, if $(\mathrm{id},\mathrm{id})$ is the kernel pair of $f\text{,}$ then $f$ is monic. Suppose that $f\circ g=f\circ h$ for some $g,h:A\to X\text{,}$ and note that both $g$ and $h$ are equal to the universal map obtained via the square $f\circ g=f\circ h\text{.}$

  id-kernel-pair→monic
    : ∀ {f : Hom X Y}
    → is-kernel-pair C id id f
    → is-monic f
  id-kernel-pair→monic {f = f} id-kp g h p =
    g                ≡˘⟨ p₁∘universal ⟩≡˘
    id ∘ universal p ≡⟨ p₂∘universal ⟩≡
    h                ∎
    where open is-pullback id-kp

We can strengthen this result by noticing that if $(p,p)$ is the kernel pair of $f$ for some $P:\mathcal{C},p:P\to X\text{,}$ then $(\mathrm{id},\mathrm{id})$ is also a kernel pair of $f\text{.}$

  same-kernel-pair→id-kernel-pair
    : ∀ {P} {p : Hom P X} {f : Hom X Y}
    → is-kernel-pair C p p f
    → is-kernel-pair C id id f

As usual, the difficulty is constructing the universal map. Suppose that $f\circ p_1=f\circ p_2$ for some $P^{\prime }:\mathcal{C},p_1,p_2:P^{\prime }\to X\text{,}$ as in the following diagram:

This diagram is conspicuously missing a morphism, so let’s fill it in by using the universal property of the kernel pair.

Next, note that $p\circ u$ factorizes both $p_1$ and $p_2\text{;}$ moreover, it is the unique such map!

  same-kernel-pair→id-kernel-pair {p = p} {f = f} p-kp = id-kp where
    open is-pullback

    id-kp : is-kernel-pair C id id f
    id-kp .square = refl
    id-kp .universal q = p ∘ p-kp .universal q
    id-kp .p₁∘universal {p = q} = idl _ ∙ p-kp .p₁∘universal
    id-kp .p₂∘universal {p = q} = idl _ ∙ p-kp .p₂∘universal
    id-kp .unique q r = (sym (idl _)) ∙ q ∙ sym (p-kp .p₁∘universal)

Categories with all pullbacks🔗

We also provide a helper module for working with categories that have all pullbacks.

has-pullbacks : ∀ {o ℓ} → Precategory o ℓ → Type _
has-pullbacks C = ∀ {A B X} (f : Hom A X) (g : Hom B X) → Pullback C f g
  where open Precategory C

module Pullbacks
  {o ℓ}
  (C : Precategory o ℓ)
  (all-pullbacks : has-pullbacks  C)
  where
  open Precategory C
  module pullback {x y z} (f : Hom x z) (g : Hom y z) =
    Pullback (all-pullbacks f g)

  Pb : ∀ {x y z} → Hom x z → Hom y z → Ob
  Pb = pullback.apex

Stability🔗

Pullbacks, in addition to their nature as limits, serve as the way of “changing the base” of a family of objects: if we think of an arrow $f:A\to B$ as encoding the data of a family over $B$ (think of the special case where $A={\Sigma }_{x:A}F(x)\text{,}$ and $f={\pi }_1\text{),}$ then we can think of pulling back $f$ along $g:X\to B$ as “the universal solution to making $f$ a family over $X\text{,}$ via $g$”. One way of making this intuition formal is through the fundamental fibration of a category with pullbacks.

In that framing, there is a canonical choice for “the” pullback of an arrow along another: We put the arrow $f$ we want to pullback on the right side of the diagram, and the pullback is the left arrow. Using the type is-pullback defined above, the arrow which results from pulling back is adjacent to the adjustment: is-pullback f⁺ g _ f. To help keep this straight, we define what it means for a class of arrows to be stable under pullback: If f has a given property, then so does f⁺, for any pullback of f.

module _ {o ℓ} (C : Precategory o ℓ) where
  open Cat.Reasoning C

  is-pullback-stable
    : ∀ {ℓ'} → (∀ {a b} → Hom a b → Type ℓ') → Type _
  is-pullback-stable P =
    ∀ {p A B X} (f : Hom A B) (g : Hom X B) {f⁺ : Hom p X} {p2}
    → P f → is-pullback C f⁺ g p2 f → P f⁺