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Message from discussion Proposed Formal Definition of a Concrete Category

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Subject: Re: Proposed Formal Definition of a Concrete Category
From: Dan Christensen <Dan_Christen...@sympatico.ca>
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On Oct 13, 4:13=A0pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
> Dan Christensen <Dan_Christen...@sympatico.ca> writes:
> > Interesting point, but isn't a morphism wrt to some property (or
> > structure) P just a transformation that preserves property P? That is
> > my intuitive sense of it. You cannot transform any set into its power
> > set since, in a transformation, one element is transforms into exactly
> > one other element. Can you then have morphism from a set to its power
> > set?
>
> The function f:N -> P(N) mapping n to {n} is a morphism in the category
> Set.
>
> So is the function g:N -> P(N) mapping n to N \ {n}.
>
> And the function mapping n to N.
>
> And the function mapping n to n u {0,...,47}.
>
> And so on.
>
> You see that *sets* of elements are elements of the powerset. =A0Thus, a
> function S -> P(S) takes elements of S to subsets of S -- because
> subsets of S *are* elements of P(S).
>

As I understand, a morphism on is a structure-preserving
transformation. I don't see the point of category theory otherwise.

Dan