[Haskell-cafe] Re: Haskell-Cafe Digest, Vol 10, Issue 3
Ron de Bruijn
rondebruijn at yahoo.com
Thu Jun 10 10:23:04 EDT 2004
>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?
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).
>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
preservation of composition, right?
>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
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
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