Manes'Theorem is usually stated by saying that the Eilenberg-Moore category of the Ultrafilter Monad is equivalent to the category of compact Hausdorff spaces. However, it seems to me from this proof https://www.math.leidenuniv.nl/scripties/BachStekelenburg.pdf that one can actually construct an isomorphism of categories between the two.
Is my observation correct? If so, why isn't this additional piece of information added to the formulation of the theorem? Maybe category theorists don't care about that because equivalence is enough to say that two categories are "the same"?
Best Answer
These categories are indeed actually isomorphic, since the equivalence between them is actually bijective on objects. More generally, equivalences of concrete categories where the corresponding objects actually have the same underlying sets are typically isomorphisms (because the equivalence is just giving a bijection between structures of two different types on a given set, which respects the morphisms of the two types of structures).
Yes, this is exactly right. By far the most commonly used notion of "sameness" for categories is equivalence, not isomorphism. It could even be said that if you want to demand categories to be isomorphic rather than just equivalent, what you are doing is probably not really "category theory". So, on the rare occasions that you have an equivalence of categories that is actually in fact an isomorphism, many people would not even notice, and if they did notice, they would often not consider it worthy of mention.