Question

Suppose I have a function f : A → B between two sets A and B. (The same question applies to group homomorphisms, continuous maps between topological spaces, etc. But for simpicity let's restrict ourselves to the case of arbitrary maps between sets.)

If I have subset A' ⊆ A of the domain A, then I can restrict of f to A'. There is the function f|_A' : A' → B which is given by f|_A'(a) = f(a) for all a ∈ A'.

Suppose now that I have subset B' ⊆ B of the codomain B that contains the image of the map f. Similarly I can restrict of f to B' meaning that there is a function g : A → B' which is given by g(a) = f(a) for all a ∈ A.

In general, it might be useful to consider such functions g and have a name for them, for example if I have a function B' → C that I want to apply afterwards. Unfortunately, I haven't seen a name or symbol for this function g in literature.

Is there a notation for the restriction g of f to a subset of the codomain similar to the notation f|_A' for a restriction to a subset of the domain?

Answer 1

I would call it simply a pullback (along $B'$). Thus, you may denote it by $f \times_B B'$.

If we pullback a map $f : A \to B$ along a subset $B' \subseteq B$, we get the map $f^{-1}(B') \to B', x \mapsto f(x)$. If the image of $f$ happens to be a subset of $B'$, then $f^{-1}(B')=A$.

Answer 2

Concerning the name for the notion in question, but not the notation, Exposé 2 by A. Andreotti in the Séminaire A. Grothendieck 1957, available at www.numdam.org, suggests the following:

Consider a morphism $f:A\rightarrow B$ in some category, subobjects $i:U\rightarrow A$ and $j:V\rightarrow B$ of $A$ and $B$, respectively, and quotient objects $p:A\rightarrow P$ and $q:B\rightarrow Q$ of $A$ and $B$, respectively.

Then, $f\circ i$ is the restriction of $f$ to $U$. Dually, $q\circ f$ is the corestriction of $f$ to $Q$. (In particular, with the usual usage of the prefix "co", corestriction is not suitable for the notion in question.)

Moreover, if there is a morphism $g:P\rightarrow B$ with $g\circ p=f$, then $g$ is the astriction of $f$ to $P$. Dually, if there is a morphism $h:A\rightarrow V$ with $j\circ h=f$, then $v$ is the coastriction of $f$ to $V$. (Of course one can argue whether one should swap the terms astriction and coastriction (as suggested by Gerald Edgar).)

Answer 3

If I wanted a name, I might use "corestriction."

Answer 4

The terminology I have seen for this is "astriction".

Answer 5

It's called a range restriction. There's no established notation, but you might as well use the Z notation which is f  ▷ B'. (f \rhd B' in LaTeX plus amsfonts)

Answer 6

This would fit as a response to Harald Hanche-Olsen's remark, but I have not enough points for this.

Anyway, Mizar mathematial library chose exactly this notation; actually, it is introduced in the article on basic relations RELAT_1.MIZ, Def. 12, so it fits any relation and any set.

In a Mizar article you have just ASCII, so no subscripts, but you can always avoid ambiguities like f|X versus X|f by typing the two objects appearing in the notation as Relation and set respectively. I find interesting that what is expressible on paper by varying font size is emulated by typing in a proof checker.