pullback (category theory)

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

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. 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 p_1,2 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.

Pullback (Category Theory)

See also