open import Cat.Prelude

module Cat.Diagram.Pullback {ℓ ℓ′} (C : Precategory ℓ ℓ′) where

Pullbacks🔗

open import 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 $\mathcal{C}/Z$, the category of objects fibred over $Z$. 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 $x$; 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 (ℓ ⊔ 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 $Y$, 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)

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

Categories with all pullbacks🔗

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

has-pullbacks : Type _
has-pullbacks = ∀ {A B X} (f : Hom A X) (g : Hom B X) → Pullback f g

module Pullbacks (all-pullbacks : has-pullbacks) where
  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 $A = \Sigma_{x : A} F(x)$, and $f = \pi_1$), then we can think of pulling back $f$ along $g : X \to B$ as “the universal solution to making $f$ a family over $X$, 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 right 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.

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 f⁺ g p2 f → P f⁺