Pullback (category theory)

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 $f : X \to Z$ and $g : Y \to Z$ with a common codomain. The pullback is often written

and comes equipped with two natural morphisms $P \to X$ and $P \to Y$ . 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 situtations, $X \times Z Y$ may intuitively be thought of as consisting of pairs of elements $(x, y)$ with $x \in X$ and $y \in Y$ 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 $p 1 : P \to X$ and $p 2 : P \to Y$ for which the diagram

commutes. Moreover, the pullback $(P, p 1, p 2)$ must be universal with respect to this diagram. That is, for any other such triple $(Q, q 1, q 2)$ for which the following diagram commutes, there must exist a unique $u : Q \to P$ (called a mediating morphism) such that

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