To clarify the distinction between domain/corange as I understand common usage: For a total function, the domain and corange are equal. In the category of sets, all functions are total (by definition of a function). If we generalize to the category of CPOs by associating with every set an extra value called "bottom" (as well as appropriate sums and products containing bottom), and by expanding the concept of function to make use of these bottoms for otherwise undefined mappings, then the domain refers to the thing with the bottoms in it and the corange refers to the largest set (i.e. no bottoms) in the domain whose image contains no bottoms in it. So the corange may not be equal to the domain. So, domain and codomain in Haskell are types (i.e. that include bottom). Range, corange, preimage, and image are only sets that don't include a bottom. Also, a minor correction (after consulting Wikipedia [1]): For any given subset of the range (called an image), there may be multiple subsets of the corange that have this image (the corresponding inverse image). The "preimage" (or complete inverse image) is the exactly one largest such subset of the corange to have the given image. [1] http://en.wikipedia.org/wiki/Function_%28mathematics%29 Dan Dan Weston wrote:

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:

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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

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