In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, 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

and comes equipped with two natural morphisms P → X and P → Y. The pullback of two morphisms f and g need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, X ×Z Y may intuitively be thought of as consisting of pairs of elements (x,y) with x∈X and y∈Y and f(x)  =  g(y). For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative square.

The dual concept of the pullback is the pushout.

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

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 X → Z ← Y, there is a unique isomorphism between A and B respecting the pullback structure.

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.

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.

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

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

along with the morphisms

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

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

Alternatively one may view the pullback in Set asymmetrically:

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.

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 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

and the restriction of f to f−1[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.

Consider the multiplicative monoid of positive integers Z+ as a category with one object. In this category, the pullback of 2 positive integers m and n is just the pair (LCM(m, n)/m, LCM(m, n)/n), where the numerators are both the least common multiple of m and n. The same pair is also the pushout.