|This article does not cite any sources. (March 2014)|
In mathematics, a pullback is either of two different, but related processes: precomposition and fibre-product. Its "dual" is pushforward.
Precomposition with a function probably provides the most elementary notion of pullback: in simple terms, a function f of a variable y, where y itself is a function of another variable x, may be written as a function of x. This is the pullback of f by the function y.
It is such a fundamental process, that it is often passed over without mention, for instance in elementary calculus: this is sometimes called omitting pullbacks, and pervades areas as diverse as fluid mechanics and differential geometry.
The notion of pullback as a fibre-product ultimately leads to the very general idea of a categorical pullback, but it has important special cases: inverse image (and pullback) sheaves in algebraic geometry, and pullback bundles in algebraic topology and differential geometry.
When the pullback is studied as an operator acting on function spaces, it becomes a linear operator, and is known as the composition operator. Its adjoint is the push-forward, or, in the context of functional analysis, the transfer operator.
The relation between the two notions of pullback can perhaps best be illustrated by sections of fibre bundles: if s is a section of a fibre bundle E over N, and f is a map from M to N, then the pullback (precomposition) of s with f is a section of the pullback (fibre-product) bundle f*E over M.