Dan Weston westondan at imageworks.com
Fri Feb 15 17:25:02 EST 2008

```Since no one else has replied, I will take a stab. This is the
terminology I have seen/heard:

A mapping in a category is typed. It can map only from a "source" object
to a "target" object. There may be zero, one, or multiple such mappings
(functions) from a given source to a given target (but at least one if
source and target are the same, namely the identity map).

For a particular source and target, where the source and target happen
not just to be opaque objects but have internal structure (with subset
operations), mappings are called functions, the source is called the
domain, and the target is called the codomain.

Elements X in the domain are mapped to some element Y in the codomain.
The set of all such Y is the range, and the set of all such X is the
corange. (Wikipedia [1] suggests that there is ambiguity with the word
"domain", but I have never heard that elsewhere).

Any given subset S of the corange (called a preimage) maps to the
corresponding image of S, which is a subset of the range. Preimage and
image apply to singleton sets as well, so (by trivial isomorphism) these
words apply to mapped elements themselves. In this case, the usual arrow
symbol gets a little vertical cap on the left end.

In any case, I would not get too hung up on the terminology. It is much
more important to understand what is meant in any given setting.

[1] http://en.wikipedia.org/wiki/Function_%28mathematics%29

Dan

PR Stanley wrote:
> Hi
> domain, source --- are the two different things? I'm sure I read
> somewhere that the source \subseteq domain in mappings. The same was
> said about range and target -- target \subseteq range.
> Any ideas?
> Thanks, Paul
>
> _______________________________________________