Can want to define a category $\mathcal{D}$ as follows:
the objets are quadrupels $(G,H,X,r)$, where
1) G is a preordered abelian group with order unit,
2) H is an abelian group,
3) X is a compact Hausdorff space
4) $r:X\to S(G)$ is a continuous map ( $S(G)$ is a compact Hausdorff space)
With morphisms $(f,g,h):(G,H,X,r)\to (G',H',X',r')$, where
1) $f:G\to G'$ is a morphism of preordered groups preserving the order unit
2)$g:H\to H'$ is a group homomorphism
3)$h:X'\to X$ is a continuous map which is compatible with $r$ in the following way, that the following diagram commutes:
$$\require{AMScd}\begin{CD} X' @>h>> X \\ @VV r' V @VV r V \\ S(G')@>S(f)>> S(G) \\ \end{CD}$$ where $S(f)(\chi)=\chi\circ f$.
The composition of morphisms is the canonical one. I checked if this satisfies the definition of a category and I don't see any problems. But I'm not sure of I overlook something: is there everything ok or is there something wrong with this definition? I'm asking because I'm not sure if I have to add $S(G)$ and $S(\alpha)$ somehow to the data..
And my main question is (if everything is correct): Is this catgeory isomorphic to a certain product of categories of well-known ones, for example something similar to (category of preordered abelian groups with order unit)$\times$(compact hausdorff space)$^{op}$? (It can't be isomorphic to (category of preordered groups with order unit)$\times$(compact hausdorff space)$^{op}$, since we have this pariring map $r$ as well)
I don't know how to express this as a product of categories, but your construction looks a lot like a comma category (so it is actually a fibered product of categories!). In fact, it is (up to the order of your terms) the category $\mathbf{Ab}\times (Id_{\mathbf{CHaus}^{op}}\downarrow S)$, where $\mathbf{Ab}$ denotes the category of abelian groups, $\mathbf{CHaus}$ denotes the category of compact Hausdorff spaces, and $S=Hom(\_ ,(\Bbb R,\Bbb R^+,1))^{op}:\mathbf{PrAb}\to \mathbf{CHaus}^{op}$ is (the dual of) the contravariant functor represented by $(\Bbb R,\Bbb R^+,1)$.