pullback

A pullback is the categorical way to take the “common solutions” of two maps into a shared target. If f : A -> C and g : B -> C are two arrows landing in the same object C, the pullback assembles the pairs of elements, one from A and one from B, that agree after being pushed into C. It is simultaneously a generalized intersection, a generalized preimage, and a generalized fiber-by-fiber product, which is why it appears everywhere from gluing data to base change in geometry.

Formally, the pullback of f : A -> C and g : B -> C is an object P with morphisms p : P -> A and q : P -> B such that f ∘ p = g ∘ q, universal among such squares: for any object X with maps to A and B that agree over C, there is a unique morphism X -> P compatible with p and q. The pullback is exactly the limit of the diagram A -> C <- B, the corner-shaped diagram called a cospan.

In Set the pullback is concretely {(a, b) in A × B : f(a) = g(b)}, a subset of the product cut out by the agreement condition. Special cases recover familiar operations: the pullback of two subobjects of C is their intersection; the pullback of f along an inclusion S -> C is the preimage f^(-1)(S); and the pullback of A -> C and B -> C in suitable categories is the fiber product used to base-change a family. A square that is a pullback is also called a Cartesian square.