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 → Z and g : Y → Z with a common codomain. The pullback is often written 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