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