A categorical framework for Freiman s-morphisms Let $\mathfrak A_i$ be groups ($i = 1, 2$), written multiplicatively, and $s$ a non-negative integer (here, as usual, I am abusing notation and denoting the operations of $\mathfrak A_1$ and $\mathfrak A_2$ with the same symbols as the ones in the standard multiplicative signature of the first-order algebraic theory of groups).
A Freiman $s$-morphism from $X_1 \subseteq \mathfrak A_1$ to $X_2 \subseteq \mathfrak A_2$ is then any function $\phi: X_1 \to X_2$ such that: $\prod_{i=1}^s x_i^{\varepsilon_i} = \prod_{i = 1}^s y_1^{\varepsilon_i}$ for some $x_1, \ldots, x_s, y_1, \ldots, y_s \in X_1$ and $\varepsilon_1, \ldots, \varepsilon_s \in \{\pm 1\}$ implies $\prod_{i=1}^s \phi(x_i)^{\varepsilon_i} = \prod_{i=1}^s \phi(y_i)^{\varepsilon_i}$, where I am using $\phi(\cdot)^\varepsilon$ in place of $(\phi(\cdot))^\varepsilon$, as expected.
The subject is fascinating, as far as I'm concerned, but any presentation of it which I'm aware of looks redundant (in the basics) to my eyes. This motivates me to ask the following:


Question. Is there any previous attempt of categorifying the theory of Freiman's morphisms, at least to the degree, say, of defining a category where objects are something and arrows "Freiman $s$-morphisms"? In the case of a positive response, could you provide explicit references and a sketchy account of this early work on the subject?


 A: Perhaps not really an answer, but: 
I am not aware of 'categorical' work on this; but there is work on so to say Freiman homomorphisms as a notion in itself (though often restricted to commutative case). 
See for example Lev and Konyagin 'Combinatorics and linear algebra of Freiman isomorphism' (paper 22 here http://www.math.haifa.ac.il/~seva/pub_list.html ) or see this recent presentation of Grynkiewicz http://www.math.udel.edu/conferences/WilsonFest/talks/Grynkiewicz.pdf . 
Of course Nathanson's (Additive Number Theory II) and Tao and Vu's (Additive Combinatorics) books have some general information too, also Freiman's (Structure Theory of Set Addition) I think but I do not have it handy and am not certain. 
It seems true that such structural/foundational matters (until recently) did not receive that much attention; a reason could be that people [see disclaimer] interested in questions where Freiman hom's are/were important are not that interested in building very general abstract frameworks, or at least it is not a priority. As the area itself becomes/became more developped and also well-known this changes perhaps a bit. 
It might well be that there are quite a few things left to explore that so far simply were not investigated much. Of course, then the next question is (or at least might be for some) whether there will be some applications of such a (new, to be developped) categorical framework.
Disclaimer: of course like all such generalization this is not precise and to be considered as 'on average' and in addition not too seriously. And it is certainly not meant critically, neither against "theory builders" nor "problem solvers".   
