G'day all. Good day to you too.

I don't know what you mean by "more complex". A dot is just a dot, and it has no internal structure that we can get at using category theory alone. Some dots may play specific roles in relation to other dots and arrows, but no dot is any more complex than any other, >really. Well, intuitvively seen, I meant. In some categories, the dot might stand for a functor, and in some other as a simple object.

For a counter-example, think of the dual category Set^{op}. A morphism f : a -> b in that category means that there is a function f^{op} : b -> a

where a and b are sets, however f probably isn't a function at all. Well, what is it then?

In a category which is a partial order, there is a morphism f : a -> b if and only if a <= b. (Or is it a => b? Can never remember.) Here, the morphisms really have no internal structure at all. If the category has a finite number of objects, you can represent the whole thing using a bit matrix, and each morphism can be identified with >a bit set to "true". Preserving internal structure is not much more than

I think the problem here is that you have the idea that a morphism is a process that turns one object into another. In many >(probably most) interesting, practically significant cases, that's true, but it need not be. Well, I think I don't know what a function is, because

I see a function as something where you put something in and you get a result. In this case, you already say, there is a morphism b->a, wel than the following of b to a and return a then is a function? A couple of days ago, I thought of the distinction between a function and a morphism, as in that a function operates on a hole set of objects, a.k.a. domain. And a morphism only on one. Is that the distinction you mean? If all of the above is false, then probably I don't know what a function is. (It looks like the more you are busy with things, the less you seem to know of it). preservation of composition, right? the following of an arrow, represents at least for me a clear mapping of turning one object in to another. Or do you mean things like in logic, that you can see the morphism, as relations (instances of axioms) between logic objects. So the morphism a->b would mean that a implicates b. This way, the arrow is not a process of turning an a into a b. Hmm, that seems logical.

I think it really helps to try to understand category theory mostly as a language for talking about things, and not necessarily "things" in and of themselves. Using this understanding, a morphism is a noun, not a verb. It's a concrete thing describing a relationship between objects, not necessarily an action that you perform on objects. I think you have already answered my question this way. But a confirmation would be nice. It seems I had to read your explanation >1 time.

I don't know if any of this helps or not. Well, it certainly helps.

Then the multiplication issue: Is the following a good summary? A multiplication is just a name for an operation that is defined or not defined for each mathematical construction in terms of to which laws the operation should comply. The laws are then things like communativity and so on. Regards, Ron __________________________________________________ Do You Yahoo!? Tired of spam? Yahoo! Mail has the best spam protection around http://mail.yahoo.com