nLab
preimage

Contents

Definition

Given a function f:XYf: X \to Y and a subset SS of YY, the preimage (sometimes also called the inverse image, though that may mean something different) of TT under ff is a subset of SS, consisting of those arguments whose values belong to SS.

That is,

f *(S)={a:Xf(a)S}. f^*(S) = \{ a: X \;|\; f(a) \in S \} .

The traditional notation for f *f^* is f 1f^{-1}, but this can conflict the notation for an inverse function of ff (which indeed might not even exist). This then suggests f *f_* for the image of ff.

We borrow f *f^* from a notation for pullbacks, and indeed a preimage is an example of a pullback:

f *(S) X f S Y \array { f^*(S) & \hookrightarrow & X \\ \downarrow & & \downarrow f \\ S & \hookrightarrow & Y }

For a generalisation to sheaves, see inverse image.

Revised on September 12, 2013 10:03:25 by Urs Schreiber (82.169.114.243)