I have three questions about when you can show there is an isomorphism isometry between metric spaces. I am thinking of metric spaces in categorical terms where the objects are metric spaces and the morphisms are non-expanding maps. So monomorphisms are injective non-expanding maps.
(1) If there is a monomorphism an injective non-expanding map from $X$ to $Y$ and a monomorphism an injective non-expanding map from $Y$ to $X$, are $X$ and $Y$ isomorphicisometric?
I think the answer must be no, just let $X=[0,1]$ and $Y=[0,1/2]$ with the Euclidean metric on each and let the morphisms just shrink each of the intervals by a 1/2. But $X$ and $Y$ are not isomorphic isometric as metric spaces. The only reason I ask is that this question seems to imply that this is true for compact metric spaces. So maybe I am just missing something.
(2) If there is an isometric embedding from $X$ to $Y$ and an isometric embedding from $Y$ to $X$ is it true that $X$ and $Y$ are isomorphicisometric?
Here by an isometric embedding I mean a map that preserves the metric(which I believe coincides with the category theory definition of embedding for concrete categories).
(3) If the answer to (2) is yes, is there something to be said about which concrete categories this result holds for, with respect to embeddings?
Here I am taking the definition of concrete categories and embeddings from Adámek, Herrlich, Strecker.
I know this question sounds a lot like this question, but unless I am confused, they are talking about injective maps (monomorphisms) which make sense in any category, whereas I am talking about embeddings which are only defined for concrete categories.
EDIT: Edited to remove jargon and make clearer.
Thanks very much for any information.