Source
Domain
CoRange
PreImage
Target
CoDomain
Range
Image
Hi
domain, source --- are the two different things? I'm sure I read somewhere that
the source
domain in mappings. The same was said about range and target -target
range.Any ideas?
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
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