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Let $F=(f;A;B)$ is a morphism of the category $\mathbf{Rel}$ (the category whose objects are sets and morphisms are defined as binary relations).

How to name and how to denote $f$ when we know $F$?

I propose to call $f$ the graph of $F$. Right name?

But how to denote it? Are there a standard notation?

I propose the following (non-standard) notation: $\mathrm{GR}\, F= f$.

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In practice you won't have to distinguish between $f$ and $F$. And I would call $f$ just a relation from $A$ to $B$. – Martin Brandenburg May 27 '12 at 14:22

Graph of F is perfectly good.

I would write: $Graph(F)=f$, since GR in itself is a bit opaque as a name.

Edited after taking comments into account:

$\mathbf{Rel}$ is a bit atypical in the sense that it is named after its morphisms, while most categories are named after their objects (see MacLane's CWM chapter 1 notes). We can however form the category of arrows of $\mathbf{Rel}$, ie. $\mathbf{Rel}^\mathbf{2}$ (see CWM again) where the relations are the objects. So we could then write:


Where $U:\mathbf{Rel}^\mathbf{2}\to \mathbf{Set}$ is the forgetful functor from $\mathbf{Rel}^\mathbf{2}$ to $\mathbf{Set}$ (since the graph of F is a set).

$\mathbf{2}$ is the category with 2 objects and just one morphism between them

Correction: On second thought: graph of F is not a good name. That is because graph is normally meant to be an ordered pair (Wikipedia), while your $f$ is just a subset of $A \times B$. So I would simply call $f$ "the underlying set of $F$" and use the $U(F)=f$ notation to derive it

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I'm tempted to apply a -1: the category $\textbf{Rel}$ is not a category whose objects are relations. – Zhen Lin May 28 '12 at 12:40
There is no functor U here ... – Martin Brandenburg May 28 '12 at 13:41
@ZhenLin are very right. My answer was not very well thought out. I would prefer to delete it if possible or otherwise see if there is a possible category where relations are objects and morphisms are...what? – magma May 28 '12 at 13:45
There is a 2-dimensional category $\mathfrak{Rel}$, and the hom-category $\mathfrak{Rel}(X, Y)$ is an ordinary category whose objects are relations from $X$ to $Y$ and morphisms are, well, maps of sets. But this is just a really fancy way of talking about $\mathscr{P}(X \times Y)$. – Zhen Lin May 28 '12 at 13:51
@zhenLin yes or we could consider this: $\mathbf{Rel'}$: objects are relations (ordered triples), morphisms from <r,A,B> to <r',C,D> are <f,f'> with $r' \circ f =r \circ f'$ where composition in $\mathbf{Rel}$ is meant. This is called the category of arrows of $\mathbf{Rel}$ (MacLane's CWM page 40) – magma May 28 '12 at 14:01

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