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 aresomethingand 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?

subsetsof groups not the full group. Do you mean this, or really what you wrote. – quid Oct 16 '12 at 14:52