Sources
Help build the largest human-edited directory on the web.
Submit a site - Become an editor
Wikipedia. Licensed under the GNU Free Documentation License.
Are you an expert in this subject? Join the discussion and share your knowledge at Wikipedia.org.
Pullback
From Wikipedia, the free encyclopedia
The notion of pullback in mathematics is a fundamental one. It refers to two different, but related processes: precomposition and fibre-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(x). 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.
See:
- Pullback (differential geometry)
- Pullback (cohomology)
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.
See:
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.
 |
This disambiguation page lists articles associated with the same title. If an internal link led you here, you may wish to change the link to point directly to the intended article. |