categories: Re: Subobject classifier and non-classical logic (intuistionistic)

[Toby Bartels <toby@math.ucr.edu>]

Galchin Vasili wrote:

>I have been reading Goldblatt's book on topoi and
>also a paper by Peter Johnstone on subobject
>classifiers. Please "school" me .... In both places it
>says that in general a subobject classifier's
>"elements" are used as logic "values". [...]
>[...] I don't see
>how in a totally general case of a topos we can say
>that "elements" of it's s.c. define a set of logic
>values .... after all the guiding principle of
>category theory is that objects are opaque.

In a topos (or any closed monoidal category C with unit object 1),
an "element" of an object X is a morphism from 1 to X.
Think of the functor Hom(1,.): C -> _Set_ as the "forgetful functor"
that defines the category C as sets with extra structure.
Of course, it will only be *really* valid to think of C
as sets with extra structure if this functor is faithful.
A topos C is called "well pointed" when this holds.


-- Toby Bartels
   toby@math.ucr.edu