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,

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

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:

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

Related concepts

• level set, zero locus

• image, coimage

For a generalisation to sheaves, see inverse image.