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In category theory, a branch of mathematics, a pullback (also called a 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
- P = X ×Z Y.
Universal property
Explicitly, the 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 set (Q, q1, q2) there must exist a unique u : Q → P making the following diagram commute:
As with all universal constructions, the pullback, if it exists, is unique up to a unique isomorphism.
Examples
In the category of sets the pullback of f and g is the set
- X ×Z Y = {(x, y) ∈ X × Y | f(x) = g(y)},
This example motivates another way of characterizing the pullback: as the equalizer of the morphisms f o p1, g o p2 : X × Y → Z where X × Y is the binary product of X and Y and p1,2 are the natural projections. This shows that pullbacks exist in any category with 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 X ×B E is a fiber bundle over X called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.
In any category with a terminal object Z, the pullback X ×Z Y is just the ordinary product X × Y.
See also
- The categorical dual of a pullback is a called a pushout.
- pullback (in differential geometry)
References
- Paul M.Cohen, Universal Algebra (1981), D.Reidel Publishing, Holland, ISBN 90-277-1213-1 (Originally published in 1965, by Harper & Row).
"Pullback (category theory)"