In his answer to my question on the real numbers SE/839848, Qiaochu Yuan mentioned that the real numbers are the terminal archimedian ordered field. I wondered, what can be said about the category of archimedian ordered fields. In particular, which limits and colimits do exist? Can the existence of a terminal object be proven categorically?

  • 5
    $\begingroup$ What is a "categorical proof"? There are a number of general principles that tell you that some category is complete if another is, but at some point one actually has to sit down and prove that a particular category is complete. I think this is one of those cases. $\endgroup$ – Zhen Lin Jun 19 '14 at 19:30

This category is less interesting than it might sound: because $\mathbf{Q}$ embeds uniquely in its completion $\mathbf{R}$ and is dense in any Archimedean ordered field, there is a unique order-preserving homomorphism from any Archimedean ordered field to $\mathbf{R}$: this category is just the category of subfields of $\mathbf{R}$. And, thus, it is complete and cocomplete, because arbitrary intersections of subfields are again subfields.

  • 3
    $\begingroup$ Of course you have said this, but just to be clear: This category is (equivalent to) a partial order. This follows simply because $\mathbb{Q}$ is dense in any Archimedean ordered field, so that any two parallel homomorphisms agree. We don't need $\mathbb{R}$ for this reasoning. $\endgroup$ – Martin Brandenburg Jun 19 '14 at 23:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.