Pullbacks

A pullback is a construction in category theory that represents a universal way to synchronize two morphisms with a common codomain. Given two arrows f: X → Z and g: Y → Z in a category that has pullbacks, a pullback of f and g consists of an object P and projection morphisms p1: P → X and p2: P → Y such that f ∘ p1 = g ∘ p2, together with the following universal property: for any object W with maps u: W → X and v: W → Y satisfying f ∘ u = g ∘ v, there exists a unique h: W → P with p1 ∘ h = u and p2 ∘ h = v. The pair (P, (p1, p2)) is determined up to isomorphism.

In Set, the pullback is the set P = { (x, y) ∈ X × Y | f(x) = g(y) }, with projections to X and Y. This yields the intuitive idea of “pairs that map to the same element of Z.” In other categories, the pullback is defined similarly and inherits suitable structures (e.g., subspace topology in Top, fiber product in algebraic geometry). The dual notion is the pushout, which amalgamates two objects along a common domain.

Pullbacks also appear in differential geometry and analysis. For a smooth map f: M → N and a differential form ω on N, the pullback f*ω is a form on M defined using the differential df. In calculus, the pullback of a function h: N → R along f is simply h ∘ f. Pullbacks satisfy functorial and compatibility properties: (g ∘ f)* = f* ∘ g* and d(f*ω) = f*(dω) for differential forms.

Applications include base change in fiber bundles, sheaf theory, and various constructions that require synchronizing data along a common target.