Pullback (category theory)

Universal property

Explicitly, a pullback of the morphisms f and g consists of an object P and two morphisms p₁ : P → X and p₂ : P → Y for which the diagram

commutes. Moreover, the pullback (P, p₁, p₂) must be universal with respect to this diagram. That is, for any other such triple (Q, q₁, q₂) for which the following diagram commutes, there must exist a unique u : Q → P (called a mediating morphism) such that

p 2 ∘ u = q 2 , p 1 ∘ u = q 1 .

As with all universal constructions, a pullback, if it exists, is unique up to isomorphism. In fact, given two pullbacks (A, a₁, a₂) and (B, b₁, b₂) of the same cospan, there is a unique isomorphism between A and B respecting the pullback structure.

Weak pullbacks

A weak pullback of a cospan X → Z ← Y is a cone over the cospan that is only weakly universal, that is, the mediating morphism u : Q → P above is not required to be unique.

Pullback and product

The pullback is similar to the product, but not the same. One may obtain the product by "forgetting" that the morphisms f and g exist, and forgetting that the object Z exists. One is then left with a discrete category containing only the two objects X and Y, and no arrows between them. This discrete category may be used as the index set to construct the ordinary binary product. Thus, the pullback can be thought of as the ordinary (Cartesian) product, but with additional structure. Instead of "forgetting" Z, f, and g, one can also "trivialize" them by specializing Z to be the terminal object (assuming it exists). f and g are then uniquely determined and thus carry no information, and the pullback of this cospan can be seen to be the product of X and Y.

Commutative rings

In the category of commutative rings (with identity), denoted CRing, the pullback is called the fibered product. Let

A, B, C ∈ Ob(CRing),α : A → C ∈ Hom(CRing),β : B → C ∈ Hom(CRing).

So A, B, and C are commutative rings with identity and α and β are ring homomorphisms. Then the pullback of this diagram is the subring of the Cartesian product A × B defined by

A × C B = { ( a , b ) ∈ A × B | α ( a ) = β ( b ) }

along with the morphisms

β ′ : A × C B → A , α ′ : A × C B → B

given by β ′ ( a , b ) = a and α ′ ( a , b ) = b for all ( a , b ) ∈ A × C B , for which

α ∘ β ′ = β ∘ α ′ .

Sets

In the category of sets, a pullback of f and g is given by the set

X × Z Y = { ( x , y ) ∈ X × Y | f ( x ) = g ( y ) } ,

together with the restrictions of the projection maps π₁ and π₂ to X ×_Z Y.

Alternatively one may view the pullback in Set asymmetrically:

X × Z Y ≅ ∐ x ∈ X g − 1 [ { f ( x ) } ] ≅ ∐ y ∈ Y f − 1 [ { g ( y ) } ]

where ∐ is the disjoint (tagged) union of sets (the involved sets are not disjoint on their own unless f resp. g is injective). In the first case, the projection π₁ extracts the x index while π₂ forgets the index, leaving elements of Y.

This example motivates another way of characterizing the pullback: as the equalizer of the morphisms f ∘ p₁, g ∘ p₂ : X × Y → Z where X × Y is the binary product of X and Y and p₁ and p₂ are the natural projections. This shows that pullbacks exist in any category with binary products and equalizers. In fact, by the existence theorem for limits, all finite limits exist in a category with a terminal object, binary products and equalizers.

Fiber bundles

Another example of a pullback comes from the theory of fiber bundles: given a bundle map π : E → B and a continuous map f : X → B, the pullback (formed in the category of topological spaces with continuous maps) X ×_B E is a fiber bundle over X called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.

Preimages and intersections

Preimages of sets under functions can be described as pullbacks as follows:

Suppose f : A → B, B₀ ⊆ B. Let g be the inclusion map B₀ ↪ B. Then a pullback of f and g (in Set) is given by the preimage f⁻¹[B₀] together with the inclusion of the preimage in A

f⁻¹[B₀] ↪ A

and the restriction of f to f⁻¹[B₀]

f⁻¹[B₀] → B₀.

Because of this example, in a general category the pullback of a morphism f and a monomorphism g can be thought of as the "preimage" under f' of the subobject specified by g. Similarly, pullbacks of two monomorphisms can be thought of as the "intersection" of the two subobjects.

Properties

In any category with a terminal object T, the pullback X ×_T Y is just the ordinary product X × Y.

Monomorphisms are stable under pullback: if the arrow f above is monic, then so is the arrow p₂. Similarly, if g is monic, then so is p₁.

Isomorphisms are also stable, and hence, for example, X ×_X Y ≅ Y for any map Y → X (where the implied map X → X is the identity).

In an abelian category all pullbacks exist, and they preserve kernels, in the following sense: if

is a pullback diagram, then the induced morphism ker(p₂) → ker(f) is an isomorphism, and so is the induced morphism ker(p₁) → ker(g). Every pullback diagram thus gives rise to a commutative diagram of the following form, where all rows and columns are exact: 0 0 ↓ ↓ L = L ↓ ↓ 0 → K → P → Y ∥ ↓ ↓ 0 → K → X → Z

There is a natural isomorphism (A×_CB)×_B D ≅ A×_CD. Explicitly, this means:

if maps f : A → C, g : B → C and h : D → B are given and

the pullback of f and g is given by r : P → A and s : P → B, and

the pullback of s and h is given by t : Q → P and u : Q → D ,

then the pullback of f and gh is given by rt : Q → A and u : Q → D.

Graphically this means that two pullback squares, placed side by side and sharing one morphism, form a larger pullback square when ignoring the inner shared morphism. Q → t P → r A ↓ u ↓ s ↓ f D → h B → g C

Any category with pullbacks and products has equalizers.