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Concept
Pullback (category theory)
Summary
In , a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the of a consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. The pullback is written
P = X ×f, Z, g Y.
Usually the morphisms f and g are omitted from the notation, and then the pullback is written
P = X ×Z Y.
The pullback 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 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 of the pullback is the .
Explicitly, a pullback of the morphisms f and g consists of an 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) where q1 : Q → X and q2 : Q → Y are morphisms with f q1 = g q2, there must exist a unique u : Q → P such that
This situation is illustrated in the following commutative diagram.
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.
The pullback is similar to the , 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 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.
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