| Mark Nicholls 2005-05-16, 8:57 am |
|
Jerzy Karczmarczuk wrote:
> Mark Nicholls:
>
>
category[color=darkred]
>
> Operations from one category to another are called functors.
OK...I read further, and saw the definition of functors.
>
> For example, take a Group, the category of Groups, the class whose
> objects are groups with morphisms which are *group homorphisms*, i.e.
> {mappings which throw elements into elements *complying with* the
> multiplication operation: if a->a', etc., then a*b=c -=-> a'*b'=c'}.
>
OK
> Now, forget the group operations, take just sets of elements. You
will
> have a "forgetful" functor from the cat. Group to cat. Set. Its
> 'adjoint' must introduce new internal operations between elements!
yep, i.e. in my world there is no isomorphism....i.e. there is no
mapping back from subset of Q that doesn't forget the "there exists x
s.t........x.y = 1"
>
> So, if within Rationals you have a group structure (multiplicative;
> with the inverse), and you throw all the bazaar into integers, the
> corresponding functor is non-trivial. You get a Monoid, not a Group.
>
> If you - as several people here - think mainly about *values*, then
> Set ==> Set is good, and you might say that an integer is a rational.
OK...this 'value', 'variable' this is a computer thing, not a maths
thing.
>
> But claiming that it IS so, point, no discussion, my kindergarten
> teacher tolmd me, is a bit restrictive... It forgets about the
> differences of the semantics (and of signatures) of operations.
I understand what people are saying, and I may be being pedantic, but
the assertion is dangerous (and I think technically incorrect).
It's interesting, I will look further.
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