<p>Halvorson has proposed an intriguing example of a pair of theories whose categories are equivalent but which are not themselves definitionally equivalent. Moreover, it seems obvious that these theories are not equivalent in any intuitive sense. We offer a new topological proof that these theories are not definitionally equivalent. However, the underlying theorem for this claim has a converse that shows a surprising collection of theories, which are superficially similar to those in Halvorson’s example, turn out to be definitionally equivalent after all. This offers some new insight into what is going “wrong” in the Halvorson example.</p>