So, I promised to explain the slightly jargonistic term "up to".

Well, superficially, it is simple; it merely refers to a sort of relaxed form of equality. In order to explains this, I am going to make it seem extremely technical, but I am only doing it this way as I thought it might amuse some of you to see something that might not be too familiar.

The term "up to" is invariably followed by some sort of qualifier; this qualifier refers to an equivalence relation. That is, there is a relation on the set in question under which equivalent elements can be considered to be the same.

Let me explain. An equivalence relation on a set (which may, or may not, have additional structure) is a relation that satisfies

reflexivity:

symmetry:

transitivity: if and then .

Just to calm a few nerves (perhaps) I should say straight away that equality "=" is just such a relation, as you may easily check. I will give further examples later.

Elements of a set that are so related form what is called anequivalence class, so if, say and , then the set is referred to as, say , where the typical element is called a class representative.

Now the property of transitivity means that no element of our total set can be in more that one equivalence class, which thus induces apartitionof this set - that is, a "division" into totally disjoint subsets. Here's a familiar example: consider the positive integers exactly divisible by 2 - as an equivalence class . The partition induced thereby leaves the set "remaining". Just as each element in can be found by adding 2 to some other element in , so each element in can be likewise found. We call these the even and odd positive integers.

Isomorphism (and its up-market friends homeomorphism, diffeomorphism etc) is an equivalence relation as defined. So I may have the expression "up to isomorphism", to mean that for the purpose at hand, isomorphic vector spaces may be regarded as equal (homeomorphic for top. spaces, diffeomorphic for diff. manifolds). I actually insisted on the slightly stronger condition ofnaturallyisomorphic - I gave my reasons.

The partition I referred to now becomes a partition on the universe of vector spaces, or, as I would prefer to call it, the "category" of such spaces.

Now the set (structured or not) formed from the entire collection of classes under some equivalence relation is called thequotient setand is written . There is a bit of theory that insists that, if any function on is to be well-defined, then it must be insensitive to which element of an equivalence class it takes as input; that is for all and some , which to some extent justifies the claim that equivalent spaces may be regarded as equal.

A proposmy earlier example, there is a somewhat related notion of modulo, or "mod" for short. But this post is already over-long, let it wait.