# Pullback (category theory)

(Redirected from Pullback diagram)

In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : XZ and g : YZ with a common codomain.

Explicitly, the pullback of the morphisms f and g consists of an object P and two morphisms p1 : PX and p2 : PY for which the diagram

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CategoricalPullback-03.png
Image:CategoricalPullback-03.png

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 : QP making the following diagram commute:

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CategoricalPullback-02.png
Image:CategoricalPullback-02.png

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

The pullback is often written

P = X ×Z Y.

The notation comes from the following example. 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)}

The maps p1 and p2 are just the projections onto the first and second factors.

This example motivates another way of characterizing the pullback: as the equalizer of the morphisms f O p1, g O p2 : X × YZ where X × Y is the binary product of X and Y and p1,2 are the natural projections. This shows that pullbacks exist in any category with binary products and equalizers.

The categorical dual of a pullback is a called a pushout.

## References

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy