In *"On numbers and games"*, Conway writes that the surreal Numbers form a *universally embedding totally ordered Field.* Later Jacob Lurie proved that (the equivalence classes of) the partizan games form a *universally embedding partially ordered abelian group.*


As far as I can tell from Lurie's paper, the latter means:   

Let $\mathbb{U}$ denote the class of equivalence classes of partizan games. Furthermore let $S \subseteq S'$ be partially ordered abelian groups. Suppose that $\phi: S \rightarrow \mathbb{U}$ is an order-preserving homomorphism. Then there exists an order-preserving homomorphism $\phi': S' \rightarrow \mathbb{U}$ such that $\phi' \mid S = \phi$.


Now the thing is: I had before gotten to the (informal) conception that "universally embedding" in this context means that every (set-sized) partially ordered abelian group is isomorphic to some subgroup of $\mathbb{U}$, and vice versa that every totally ordered field is isomorphic to some subfield of $\mathbf{No}$. My question now is: is that even true? And what does this have to do with the statement of Lurie's paper?



What does it mean for the surreal numbers/partizan games to be "universally embedding"?
Besides the two types of structures being considered, there are two types of universality here. They are indeed related, but the stronger one (a la Lurie and I believe also Conway) is in my opinion the "right" one.


The story I like to imagine is this. We have some putatively-"universal" object $\mathcal{U}$ that we understand well, and some mysterious object $\mathcal{A}$ of the same type. Certainly at a minimum we should be able to find an embedding $\varphi$ of $\mathcal{A}$ into $\mathcal{U}$. However, what if we *subsequently* find out that $\mathcal{A}$ itself was really just a tiny fragment of some larger structure $\mathcal{B}$? We'd not only like to be able to embed $\mathcal{B}$ into $\mathcal{U}$, we'd also like to not have to undo the work we've already done: we want to find an embedding $\psi:\mathcal{B}\rightarrow\mathcal{U}$ which **extends** the already-thought-up embedding $\varphi$. (One way this story can become literally true is if we're trying to show that the "$\forall\exists$-theory" of $\mathcal{U}$ is reasonably simple, which happens e.g. in computability theory for various partial orders.)


To see an example of how this can play out, consider $\mathbb{Z}$ and $\mathbb{Q}$ as linear orders. Obviously each is "weakly universal" for all finite linear orders: any finite linear order embeds into both $\mathbb{Z}$ and $\mathbb{Q}$. However, only $\mathbb{Q}$ is "strongly universal" for finite linear orders: letting $\mathbf k$ denote the usual linear order with underlying set $\{1,\dotsc,k\}$, given any embedding $\varphi:\mathbf 2\rightarrow\mathbb{Z}$ there is some finite linear order $\mathbf n\supseteq\mathbf 2$ such that no extension of $\varphi$ embeds $\mathbf n$ into $\mathbb{Z}$ (basically just take $n$ much greater than $\varphi(2)-\varphi(1)$). By contrast, the usual "back-and-forth" argument for $\mathbb{Q}$ shows that $\mathbb{Q}$ is not subject to this sort of silliness. Note that the stronger type of universality brings with it a sort of "homogeneity" — a relevant notion here is **Fraïssé limit**.


