Let $X$, $Y$ be sets and $f:X\rightarrow Y$ be a map. Denoting the image of $D\subset X$ under $f$ by $f(D)$ can sometimes be confusing. As for preimages, I've seen unambiguous notation like $f^*\mathcal{O}$, where $\mathcal{O} \subset \mathcal{P}(Y)$. (This is also an example of the "confusing" notation of an image, though). For images, is analogous notation $f_*D$ for denoting the image of $D\subset X$ under $f$ used in the literature?
Notation for image and preimage
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0@Miha: The asterisks appeared in the introductory courses on linear algebra and general topology so frequently that I am no longer unsure which one is which. – 2012-02-16
3 Answers
Mac Lane and Birkoff in their Algebra use $f^*$ and $f_*$. Categorical notions permeate this book and the notation is consistent with the two functors image and inverse image on the category of sets.
There are a number of notations for the image, Lawvere and Rosebrugh list five different notations in their book "Sets for Mathematics", pg 137:
$ \mathcal{P}f,\ f_!,\ f[\ ],\ \exists_f,\ \textrm{im}_f $
The asterisk notation can be explained through topos theory. The function $f:X \to Y$ gives rise to a geometric morphism $f: \mathbf{Set}/X \to \mathbf{Set}/Y$. Every geometric morphism $f:\mathcal{E} \to \mathcal{F}$ contains an adjunction $(f^* \dashv f_*)$, called the inverse image and direct image respectively. Some geometric morphisms have extra adjoints, the notation goes:
$ f_! \dashv f^* \dashv f_* \dashv f^! $
Functors with an exclamation mark (!), usually called shriek functors, are not present in every geometric morphism. Subscript functors follow the geometric morphism, so $f_!, f_*: \mathcal{E} \to \mathcal{F}$, while superscript functors go in the opposite direction, $f^*, f^!: \mathcal{F} \to \mathcal{E}$.
If we return to a set function $f:X \to Y$, the corresponding geometric morphism $f$ has an inverse image $f^*$, which takes the pullback along $f$, and direct image $f_* = \Pi_f$, which is a bit harder to describe. But there is an extra adjoint, $f_! = \Sigma_f$, which composes with $f$. This adjoint, when restricted to subobjects of the terminal object, corresponds to the image, not $\Pi_f$.
Following this convention the notation $f_*$ should not be used for the image, as the image is left-adjoint to the inverse image, as Zhen Lin says above. However there is a functor $f_*$, which on subsets has the definition
$ f_*(U) = \{ y \in Y\ |\ \forall x \in X, f(x) = y \to x \in U \} $
This does not get as much use as the image. I would still use $f_!$ for the image. Whatever notation you use make it explicit what you mean, as not everyone will recognise this notation.
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0Note that Lawvere and Rosenburgh’s $\exists_f$ and this answer’s $\Sigma_f$ have the same connotation: both denote a [dependent sum](https://ncatlab.org/nlab/show/dependent+sum) (“existential type” or “sigma type” in type theory). – 2018-11-07
In my neck of the woods a common notation for $\{f(x)\mid x\in D\}$ is $f[D]$, and in some places you can also find people using the analysts-confusing double-prime, that is f''D.
For the preimage, the principle is the same: $\{x\mid f(x)\in O\}=f^{-1}[O]$.
When we teach this in the introductory course we say that if $D=\{x\}$ then we write $f[x]$ instead of $f[\{x\}]$ and similarly $f^{-1}[y]$ instead of $f^{-1}[\{y\}]$. The brackets remain to distinguish sets from elements.
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0@Zhen: mathoverflow.net/questions/89540/ See the use of $f''$. – 2012-02-27