Is ADJ a Topos?

Question

I am reading the fascinating book CWM by MacLane (2nd edition). On page 103, he defines, via a theorem, the composition of adjoints; and on the following page he defines the category "ADJ" whose objects are all small categories and whose arrows are the adjunctions.

My 1st question: is ADJ an elementary topos?

• If yes: what is the interpretation of its subobject-classifier?
• If no: which property of an elementary topos does it fail to have?

My 2nd question: if ADJ is not an elementary topos, is ADJ_TOPOS an elementry topos? (where by ADJ_TOPOS, I mean the category of all small topoi as objects and adjunctions as arrows)

• If yes: what is the interpretation of its subobject-classifier?
• If no: which property of an elementary topos does it fail to have?

Thanks.

Answer 1

This category doesn’t have a terminal object.

Suppose we had a terminal object $T$. Consider some category $J$ with at least one object. Then there exists a functor $J \to T$, so $T$ has at least one object.

On the other hand, consider the category $0$ with no objects. There exists a functor $T \to 0$, so $T$ must have no objects. Contradiction.