Pullback (category theory)
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In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain.
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
CategoricalPullback-03.png
Image:CategoricalPullback-03.png
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:
CategoricalPullback-02.png
Image:CategoricalPullback-02.png
As with all universal constructions, the pullback, if it exists, is unique up to a unique isomorphism.
The pullback is often written
- P = X ×Z Y.
The notation comes from the following example. 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)}
The maps p1 and p2 are just the projections onto the first and second factors.
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.
The categorical dual of a pullback is a called a pushout.
See also
References
- Paul M.Cohen, Universal Algebra (1981), D.Reidel Publishing, Holland, ISBN 90-277-1213-1 (Originally published in 1965, by Harper & Row).de:Faserprodukt