nLab
preimage

Definition

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

That is,

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

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

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

$$\begin{array}{ccc}{f}^{*}(S)& \hookrightarrow & X\\ \downarrow & & \downarrow f\\ S& \hookrightarrow & Y\end{array}$$\array { f^*(S) & \hookrightarrow & X \\ \downarrow & & \downarrow f \\ S & \hookrightarrow & Y }

Related concepts

• image, coimage

For a generalisation to sheaves, see inverse image.