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 $S$ under $f$ is a subset of $X$, consisting of those arguments whose values belong to $S$.

That is,

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

A traditional notation for $f^*$ is $f^{-1}$, but this can conflict with the notation for an inverse function of $f$ (which indeed might not even exist). In fact $f^\ast$ is borrowed from a notation for pullbacks; indeed, a preimage is an example of a pullback:

$$\array{ f^\ast(S) & \hookrightarrow & X \\ \downarrow & (pb) & \downarrow \mathrlap{f} \\ S & \hookrightarrow & Y }$$

Notice also that $f^\ast$ may be regarded as an operator $P(Y) \to P(X)$ between power sets. Power sets $P(X)$ are exponential objects $2^X$ in the topos $Set$; under this identification the pre-image operator $f^\ast$ is thereby identified with the map $2^f: 2^Y \to 2^X$ (variously called “pulling back along $f$ or substituting along $f$) obtained by currying the composite map

$$2^Y \times X \stackrel{1 \times f}{\to} 2^Y \times Y \stackrel{eval}{\to} 2.$$

The appearance of the asterisk as a superscript in $f^\ast$ serves as a reminder of the contravariance of the map $f \mapsto f^\ast = 2^f$. Similarly, one uses a subscript notation such as $f_*$ (or sometimes $f_!$) for the direct image, considered as an operator $f_\ast: 2^X \to 2^Y$ in the covariant direction.

Naturally all of this generalizes to the context of toposes, where the set $2$ is replaced by the subobject classifier $\Omega$ and $f^\ast = \Omega^f$, with a pullback description similar to the above.

Properties

• interactions of images and pre-images with unions and intersections:

  • pre-images preserve unions and intersections (a general reason for this being that unions are colimits, intersections are limits, and $f^\ast$ is simultaneously a left- and a right-adjoint: $f^\ast$ is right-adjoint to the existential quantifier $\exists_f$ and left-adjoint to the universal quantifier $\forall_f$)

As emphasized by Lawvere, the quantifiers $\exists_f, \forall_f$ are vastly generalized by the concept of enriched Kan extensions which provide left and right adjoints to pulling-back operators $V^f: V^D \to V^C$ for $V$-enriched functors $f: C \to D$.

Iterated preimages

For partial endofunctions, one could also consider iterating the construction of the preimage of the partial endofunction.

For every set $T$ and subset $S \subseteq T$, let $f:S \to T$ be a function from the subset $S$ to $T$. Given any subset $R \subseteq T$, the preimage $f^{-1}(R)$ by definition is a subset of $S$ and thus a subset of $T$. One could restrict the domain of $f$ to $f^{-1}(R)$ and the codomain of $f$ to $R$, and find the preimage of $f^{-1}(R)$ under $f$, or the 2-fold iterated preimage of $R$ under $f$:

$$f^{-2}(R) \coloneqq \{x \in S \vert \exists b \in f^{-1}(R).f(x) = b\}$$

One could repeat this definition indefinitely, which could be formalised by the indexed sets $f^{-n}(R)$ representing the $n$-fold iterated preimage of $R$ under $f$.

$$f^{-0}(R) \coloneqq R$$

and for $n \in \mathbb{N}$,

$$f^{-(n+1)}(S) \coloneqq \{x \in S \vert \exists b \in f^{-(n)}(R).f(x) = b\}$$

One example of an iterated preimage is the set of iterated differentiable functions and the iterated continuously differentiable functions $C^n(\mathbb{R})$, which are the $n$-th iterated preimage of all functions and pointwise continuous functions on the real numbers under the derivative/Newton-Leibniz operator respectively.

Infinitely iterated preimages

The above definition of an iterated preimage is inductive; one could also consider the coinductive version of above. This leads to infinitely iterated preimages:

For every set $T$ and subset $S \subseteq T$, let $f:S \to T$ be a function from the subset $S$ to $T$. Given any subset $R \subseteq T$, the infinitely iterated preimage is defined as the largest subset $f^{-\infty}(R) \subseteq T$ such that the preimage of $f^{-\infty}(R)$ under $f$ is $f^{-\infty}(R)$ itself.

One example of an infinitely iterated preimage is the set of smooth functions $C^\infty(\mathbb{R})$, which is the infinitely iterated preimage of pointwise continuous functions under the derivative/Newton-Leibniz operator.

Related concepts

• level set, zero locus

• image, coimage

For a generalisation to sheaves, see inverse image.

References

• F. William Lawvere, Metric spaces, generalized logic and closed categories, TAC Reprint, 1986. Web.