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Is there a reference anywhere discussing the category of ordered commutative rings?

I'm thinking of ordered commutative rings and ring homomorphisms preserving the order, but I would also be interested to read about the full subcategory of ${\bf ComRing}$ on rings admitting a total ordering compatible with the ring structure.*

The motivation here is that the Surreals are a cogenerator in the category of ordered fields by Keith Kearnes' answer to another MO question, and this extends to them being a cogenerator in the category of ordered commutative rings and ordered ring homomorphisms since any ordered commutative ring $R^\leq$ admits an ordered field of fractions $F^\leq_R=Frac(R^\leq)$ which it embeds in (respecting the order) uniquely, and $F^\leq_R$ then embeds in $N_0$ as an initial subfield (or potentially all of $N_0$ if $R^\leq$ is a proper class, for example the omnific integers).

I'm curious about other categorical properties of the category of ordered rings and ordered ring homomorphisms; there are some simple observations like it has an initial object $\mathbb{Z}$ and the (weakly terminal) cogenerator above, but I'm curious if there's a more comprehensive reference on the subject. Any assistance is appreciated.

*there's something to be said here about rings with multiple possible ordering structures compatible with their operations; maybe I want the category of ordered pairs $(C,\leq)$ where $C$ is a commutative ring and $\leq$ is an ordering on $C$ respected by its algebraic operations, together with ring homomorphisms between the first cooridnates which respect the second coordinates. In any event, I would be interested to read about any of the above categories.

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    $\begingroup$ A person with good (ie categorical) taste would allow the order relation to be partial. The category of Archimedean ordered fields is just the complete lattice of subfields of $\mathbb R$. I doubt whether much interesting happens from a categorical point of view if you drop reciprocals or the Archimedean property but keep the total order. Maybe more interesting if you drop commutativity. Anyway, it's worth an open-minded but brief investigation. $\endgroup$
    – Paul Taylor
    Commented Feb 13, 2022 at 21:02
  • $\begingroup$ @PaulTaylor I refereed a paper on partially ordered commutative rings a few years back (it actually offered a categorical characterization of when a partially ordered ring is Archimedean if I remember correctly) and when I was writing this they occurred to me as the 'more categorical' choice, but they're pretty different beasts as far as how (my at least) intuition says they behave. That being said, I would be interested to read more about the category of partially ordered commutative rings as well if there are any references available. $\endgroup$
    – Alec Rhea
    Commented Feb 14, 2022 at 1:12

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