What if the sub-object classifier is the terminal object?

Question

The title says it all. Essentially, I've seen problems where you prove that the sub-object classifier is the terminal object. However, I was wondering what that would actually look like when you start writing diagrams for subobjects. For instance, in a topos you have the subobject classifier looking like

Then if the terminal object is the subobject classifier, what would the diagram look like?

Answer 1

There's exactly one map into the terminal-object subobject classifier for each other object $B$, so each object has only one subobject up to isomorphism: itself.

Toposes also have finite colimits, including an initial object. The unique map from the initial object to any other object is monic, and therefore the initial object is a subobject of every other object.

Combining these we see that every object must be initial, so this topos is just a category where every pair of objects has exactly one isomorphism between them.

Answer 2

Because each object admits a unique morhpism to the terminal object, whatever the terminal object classifies, there should be up to equivalence exactly one of each associated to every object. If it classifies certain morphisms with specified codomains, then it has to classify isomorphisms.

So the subobject classifier is a terminal object if and only if each object has only one classified subobject, namely, itself as a subobject. There is then no interesting diagrams of the classified subobjects, as there is only one for each object.

On another note, the subobject classifiers in the theory of quasitopoi classify precisely the regular subobjects. In that case the subobject classifier being the terminal object means that all equalizers are identities. Assuming the category to have all equalizers, this means that any two parallel morphisms are equal, i.e. that the category is a thin category, and hence a pre-order.

Thus quasitopoi whose subobject classifiers are terminal objects are precisely the thin quasitopoi. More is true then, as in a thin category there are then no non-total partial morphisms defined on regular subobjects, whence a thin category being a quaistopos amounts to requiring it be cartesian closed with finite coproducts. These are then exactly the Heyting algebras.