module Cat.Diagram.Pullback where

Pullbacks🔗

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)

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

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.

  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⁺