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 written

P = X \times f, Z, g Y

.

Usually the morphisms $f$ and $g$ are omitted from the notation, and then the pullback is written

P = X \times Z Y

.

The pullback 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 situations, $X \times Z Y$ may intuitively be thought of as consisting of pairs of elements $(x, y)$ with $x$ in $X$ , $y$ 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.