Categorical Pullback

Overview

A pullback, also called a fiber product, combines two objects over a common target. An ordinary product allows any pair of coordinates; a pullback permits only pairs with the same image in that target. This construction expresses fibers, intersections and changes of base in a common language.

The same name also appears in operations on functions and differential forms. Those are related constructions but are not definitions of the categorical universal property discussed here.

Prerequisites

Know categories, binary products and unique factorization. Set maps give the basic model. Topological examples need subspace and product topologies. In other categories, first identify which morphisms are permitted.

Definition and conventions

Given $f:A\to C$ and $g:B\to C$, a pullback is an object $P$ with maps $p:P\to A$, $q:P\to B$ satisfying $fp=gq$. For every object $X$ and maps $a:X\to A$, $b:X\to B$ with $fa=gb$, there must be a unique $h:X\to P$ such that $ph=a$ and $qh=b$.

The notation is $P=A\times_C B$. The projections and commutativity equation are part of the data. A commutative square alone is not necessarily a pullback square.

In sets the construction is

This formula is a model in a particular category, not a replacement for the abstract definition.

Intuition

Imagine two descriptions that carry the same kind of label in $C$. The fiber product matches descriptions precisely when their labels agree. If one map describes a family over a parameter space, pulling back along another parameter map substitutes the new parameters and retains the corresponding fibers.

In contrast to an unrestricted product, a pullback can be empty even when both source objects are nonempty: their images may have no common label.

Examples and non-examples

Let $C=\{0,1\}$, let $A=\{a_1,a_2,a_3\}$ have labels $0,0,1$, and let $B=\{b_1,b_2\}$ have labels $0,1$. Their product has six pairs but their pullback has just three:

For any matching pair of maps from $X$, the unique map into these three pairs is $x\mapsto(a(x),b(x))$, verifying universality directly.

For inclusions $A,B\subseteq C$, the pullback is naturally identified with the intersection $A\cap B$. For a point $c\in C$, pulling $A\to C$ back along the singleton map selecting $c$ gives the fiber $f^{-1}(c)$.

A useful non-example takes $A=B=C$ to be singletons and $P$ to have two elements. The square commutes, but a singleton has two possible lifts into $P$. Thus the square fails uniqueness even though it contains no contradictory equations.

Important properties

Pullbacks are unique up to the unique isomorphism preserving both projections. When a terminal object $1$ exists, $A\times_1 B$ is a product. Pullback along a composite can be performed in two stages: the two resulting objects classify exactly the same compatible tuples, giving a canonical isomorphism.

In topological spaces, use the indicated subset of $A\times B$ with its subspace topology. The product universal property and the subspace criterion verify the required continuity. For smooth manifolds, the set-theoretic fiber product need not be a manifold; transversality is a standard sufficient hypothesis, not an automatic property of all smooth maps.

Connections and further reading

Pullbacks are limits of cospans. They organize base change of bundles and, in algebraic geometry, fiber products of schemes. The opposite construction is a pushout. Riehl, §3.1, develops the categorical definition and examples before more general results about limits.