The notion of pullback
is a fundamental one. It refers to two different, but related processes: precomposition and fiber-product.
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
However, it is not just functions that can be "pulled back" in this sense. Pullbacks can be applied to many other objects such as differential forms
and their cohomology classes.
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.
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
, and f
is a map from M
, then the pullback (precomposition)
is a section of the pullback (fibre-product) bundle f