On the nlab page for real-closed fields, these following statements are made pertaining to their category theoretic nature.

(1) In fact, the category of real closed fields and order-preserving field homomorphisms is a full subcategory of the category of fields and field homomorphisms.

(2) Theorem. The full inclusion of the category of real closed fields and field homomorphisms to the category of ordered fields and ordered field homomorphisms has a left adjoint.

The latter one has a proof following it, but regardless I am interested in seeing the definitive meaning behind these notions. I checked the given references but didn't find anything in the vein of these statements. I'm interested and can extract good intuition on the matters but I would like if anyone has any good sources concerning the relationship between category theory and real closed fields. Thanks in advance.

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    $\begingroup$ By "the definitive meaning", do you mean translating these statements into less category-theoretic language? The first statement just means that every field homomorphism between real closed fields is automatically order-preserving, which follows immediately from the argument that the order is defined in terms of the field operations. $\endgroup$ Commented Feb 25, 2018 at 20:39
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    $\begingroup$ Cont: You shouldn't necessarily expect a reference on these matters. Full subcategories whose inclusion admits a left adjoint are a dime a dozen, so there's no reason to expect that a more expansive treatment in this particular context exists. $\endgroup$ Commented Feb 25, 2018 at 20:46
  • $\begingroup$ Thanks for the response. I misread the first one in a different context and your comment re-addressed me on that matter. I understand now. $\endgroup$ Commented Feb 25, 2018 at 21:05

2 Answers 2

  1. A subcategory $\def\c{\mathcal} \c B\subseteq\c A$ is full, if it is determined by its objects within $\c A$ so that every arrow $B\to B'$ of $\c A$ with $B,B'\in\c B$ is actually in $\c B$.
    In this case it means that a field homomorphism between real closed fields preserves the order.
  2. The inclusion functor $\c B\hookrightarrow\c A$ has a left adjoint iff $\c B$ is a reflective subcategory of $\c A$, which means that every object $A\in\c A$ has a reflection in $\c B$, which is an arrow $A\to B$ with $B\in\c B$ through which every arrow $A\to B'$ factors through uniquely.
    Note that the left adjoint is just the reflection functor.
    In this case, the real closure of an ordered field is its reflection.

There is an additional property here to that of reflective subcategory which is general enough to be worth mentionning.

-Let $\mathcal{A}$ be a category. let $\mathcal{P}$ be a full subcategory of $\mathcal{A}$. For an object $a$ of $\mathcal{A}$, we say that an ordered pair $(f,b)$ where $f$ is an arrow $a \rightarrow b$ in $\mathcal{A}$ is an extension of $a$. We define the category $\mathcal{A}_a$ of extensions of $a$ as that whose objecs are extensions and whose morphisms are commutative triangles between extensions. $\mathcal{A}_{a,\mathcal{P}}$ denotes the full subcategory of $\mathcal{A}_a$ where the codomains in extensions are objects of $\mathcal{P}$.

-Let $P$ be a quality of extensions which is conservatively transitive: if $(f,b)$ is an extends $a$ and $(g,c)$ extends $b$, then $(g \circ f,c)$ satisfies $P$ if and only if both $(f,b)$ and $(g,c)$ do. We then define the category $\mathcal{A}_{a,P}$ of $P$-extensions of $a$ as the full subcategory of $\mathcal{A}_a$ whose objects are $P$-extensions.

We say that an object of $\mathcal{A}$ is $P$-maximal if its $P$-extensions come from isomorphisms.

Then if every object $a$ of $\mathcal{A}$ has a $P$-extension $(\varphi_a,a_P)$ with codomain $a_P$ in $\mathcal{P}$, and which is initial in $\mathcal{A}_{a,\mathcal{P}}$, then any assignment $a \mapsto (\varphi_a,a_P)$ defines a functor from $\mathcal{A}$ to $\mathcal{P}$ which is left adjoint to the inclusion $\mathcal{P} \rightarrow \mathcal{A}$. This is just the notion of reflective subcategory.

The additional property is that then for every object $a$ of $\mathcal{A}$, $a_P$ is $P$-maximal, that the functor $\mathcal{A}\rightarrow \mathcal{P}$ is essentially onto the category of $P$-maximal objects of $\mathcal{P}$, and $(\varphi_a,a_P)$ is terminal in $\mathcal{A}_{a,P}$.

For real closure, the relevant property in the category of ordered field is to be an algebraic extension. For ordered rings with injective morphisms, the property of being a degree $1$ extension yields the reflective category of ordered fields and reflective functor of fraction field. For ordered fields with cofinal morphisms, the property of being a dense extension gives the category of Cauchy-complete ordered fields with Cauchy-completion functor.


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