Pullback (category theory)

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.

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.

Weak pullbacks

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

Examples

Fibered product of two functions.

In the category of sets the pullback of f and g is the set: <math>X\times_Z Y = \{(x, y) \in X \times Y| f(x) = g(y)\}</math> , together with the restrictions of the projection maps <math>\pi_1</math> and <math>\pi_2</math> 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 finite limits exist in a category with a terminal object, 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.

Properties

• Whenever X×_ZY exists, then so does Y×_ZX and there is an isomorphism X×_ZY <math> \cong</math> Y×_ZX.
• Monomorphisms are stable under pullback: if the arrow f above is monic, then so is the arrow p₂. For example, in the category of sets, if X is a subset of Z, then, for any g:Y → Z, the pullback X×_ZY is the inverse image of X under g.
• Isomorphisms are also stable, and hence, for example, X×_XY <math> \cong</math>Y for any map Y → X.

See also

• The categorical dual of a pullback is a called a pushout.
• Pullbacks in differential geometry
• Equijoin in relational algebra.

References

• Paul M. Cohen, Universal Algebra (1981), D.Reidel Publishing, Holland, ISBN 90-277-1213-1 (Originally published in 1965, by Harper & Row).