# What's the notation for a function restricted to a subset of the codomain?

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 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 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 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?

• @Willie Wong: the target of your function is still $B$. @noname: Let $\iota: B'\to B$ be the inclusion; then the function you are referring to is sometimes called $\iota^*f$, which as the notation suggests is a special case of the pullback. – Daniel Litt Jun 29 '10 at 13:21
• I'd be tempted to invent the notation ${}_{B'}|f$ for this. – Harald Hanche-Olsen Jun 29 '10 at 13:58
• Is the inclusion map {0}->{0,1} the same as the identity map {0}->{0}? According to one well-established "usual convention", they are, since a function is a set of ordered pairs. According to another, they are not. You need the latter convention in order to meaningfully ask questions like "is this function surjective?" – Tom Goodwillie Jun 29 '10 at 14:12
• @Harald: Alternatively, for ease of typesetting: $f|^{B'}$. – Ryan Reich Jun 29 '10 at 15:49
• Yes, but the original poster is obviously in a situation where it matters. – darij grinberg Jun 29 '10 at 18:00

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 $h$ 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).)

• There is a notational error: $v$ is undefined, the coastriction should be denoted as $h$. Unfortunately I can't edit the post (6 characters minimum for an edit). – Dawid K Jun 21 '18 at 16:11

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 $f^{-1}(B')=A$.

• I agree. This is the most sensible thing to do. – Sándor Kovács May 24 '11 at 4:14
• I agree with this (and Daniel Litt's comment). – S. Carnahan May 24 '11 at 4:36
• I think the preimage (mathworld.wolfram.com/Preimage.html) notation $f^{-1}(B')$ is very sensible, and therefore I would denote the restricted function with ${f}|_{f^{-1}(B')}$ – mareoraft Aug 3 '15 at 19:23

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

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)

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

• Well, yet another example that not all existing terminology should be perpetuated! :) – Mariano Suárez-Álvarez May 24 '11 at 4:03

Here's some notation.

Given a morphism $f : A \rightarrow B$ and a subobject $b': B' \rightarrow B$, the astriction of $f$ to $B' \subseteq B$ can be denoted $b'\backslash\, f : A \rightarrow B'$, pronounced "$b'$ under $f$." It satisfies $b'\circ (b'\backslash\, f).$ For example, if we have a function $f : A \rightarrow B$ and subsets $X$ of $A$ and $Y$ of $B$ such that $f(X)\subseteq Y$, then the relevant map $X \rightarrow Y$ can be denoted $Y\,\backslash\, (f \circ X).$

Similar notation can be used in linear algebra to denote solution sets. For example; to denote the set of all vectors $x$ satisfying a linear equation $Ax=b,$ I tend to write $A \backslash b.$ This has an intuitive look about it: $$Ax = b \iff x \in A \backslash b.$$

The connection basically is that in the category of matrices, $Ax=b$ can be seen as a factorization of $b$ through the tail of $A$ (because $A$ is on the left). So, the general principle is that:

In any category, $g \backslash \,f$ is good notation for the set of all factorizations of $f$ through the tail of $g$, or for the unique such factorization just in case a unique factorization exists.

Dually, if we have a function $f : A \rightarrow B$ and a quotient object $a' : A \rightarrow A'$, we can write $f/a'$ for the relevant morphism $A' \rightarrow B$ when it exists, or else use this notation to denote the set of all such morphisms.

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.