Pullback (category theory)

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In category theory, a branch of mathematics, a pullback (also called a fibered product or cartesian square) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is often written

P = X ×_Z Y.

1 Universal property
2 Weak pullbacks
3 Examples
4 Properties
5 See also
6 References

[edit] Universal property

Explicitly, the 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 set (Q, q₁, q₂) there must exist a unique u : Q → P making the following diagram commute:

As with all universal constructions, the pullback, if it exists, is unique up to a unique isomorphism.

[edit] Weak pullbacks

A pullback (P,p₁, p₂) is called weak if it is not universal, i.e. for any such Q, u as given above is not unique.

[edit] Examples

In the category of sets the pullback of f and g is the set: X ×_Z Y = {(x, y) ∈ X × Y | f(x) = g(y)}, together with the restrictions of the projection maps $π 1$ and $π 2$ to X ×_Z Y .

This example motivates another way of characterizing the pullback: as the equalizer of the morphisms f o p₁, g o p₂ : X × Y → Z where X × Y is the binary product of X and Y and p_1,2 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 small limits exist in a category with binary products and equalizers.

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 X ×_B E is a fiber bundle over X called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.

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

[edit] Properties

Whenever X×_ZY and Y×_ZX exist, there is an isomorphism X×_ZY ≅ Y×_ZX.
Whenever the pullback X×_XY exists, there is an isomorphism Y ≅ X×_XY (this follows from the universal property of the pullback).