Subobject classifier and non-classical logic (intuistionistic)
Hello Cat Community, 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'. E.g. in the topos Set, I can see that the s.c. (subobject classifier) {0, 1} is a set of logic values that defines a Boolean algebra. In the topos of Graph we also can come up with a set of logical values. These examples of s.c. all clearly are sets or structured sets .. hence have elements. 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. Thanks, Bill Halchin
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
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Galchin Vasili -
Toby Bartels