# Dictionary Definition

pullback

### Noun

1 a device (as a decorative loop of cord or
fabric) for holding or drawing something back; "the draperies were
drawn to the sides by pullbacks" [syn: tieback]

2 (military) the act of pulling back (especially
an orderly withdrawal of troops); "the pullback is expected to be
over 25,000 troops"

# User Contributed Dictionary

## English

### Noun

# Extensive Definition

The notion of pullback in mathematics 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(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.

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 over N, and
f is a map from M to N, then the pullback (precomposition) f^*
s=s\circ f of s with f is a section of the pullback (fibre-product)
bundle f*E over M.

# Synonyms, Antonyms and Related Words

avoidance, bitter-ender,
conservative,
cringe, curtailment, cut, cutback, disengagement, dodge, duck, evasion, fallback, flinch, fundamentalist, old
liner, pullout, recoil, reculade, retirement, retreat, retrenchment, right-winger,
rightist, rollback, shy, sidestep, sidestepping, standpat, standpatter, tory, wince, withdrawal, withdrawment