# 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?

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FOr $s>1$ I seems that a Freiman morphism is merely a group morphism (Posing $y_2=y_3=\ldots y_n= 1$). I'm wrong? –  Buschi Sergio Oct 16 '12 at 14:33
Whether or not what you define is really just a group homomorphism aside (I think you are right in that eg for s=2 you could in addition to a hom translate by an element of order 2, but that's it), your definition still seems strange to me. In all applications I know of the point is that the def of a Freiman hom is for subsets of groups not the full group. Do you mean this, or really what you wrote. –  quid Oct 16 '12 at 14:52
@quid. Yes, sorry, I wrote something but had in mind something else. This is probably because I couldn't understand Sergio's comment. Sometimes, I wonder where I've lost my head... Thus, my previous comment doesn't make much sense, of course. –  Salvo Tringali Oct 16 '12 at 15:19

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".

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Thanks for the references, I will check them and come back with more comments. For the moment, some preliminary considerations: (1) Additive Combinatorics introduces Freiman morphisms in Sect. 5.3 (at the least, in the 2006 ed. on my shelf), but it is strictly focused on the case of groups (as you suspected). There is just a minor remark pointing out an analogy with the differential geometry of manifolds, that's all. I may be wrong, but I guess that the 2010 ed. has not been extended in this respect. (2) It goes the same with Nathanson, who presents Freiman morphisms in Sect. 8.2 (...) –  Salvo Tringali Oct 16 '12 at 17:16
(...) of the 1996 ed. of ANT - Inverse Problems and the Geometry of Sumsets (as a side note, only in the abelian setting). (3) Grynkiewicz, in his presentation, deals uniquely with abelian groups. In the first part, he introduces the idea of a Freiman morphism of a sum-set (together with other basic material). In the second part, he discusses problems relating to what he refers to as the universal ambient group (shortly, UAG) of a sum-set. (4) Idem for Konyagin and Lev, who develop a linear algebra method (Theorem 4) to solve a couple of questions raised by Freiman himself. –  Salvo Tringali Oct 16 '12 at 17:39
Well, it seems that I've already checked them all. As for your question on possible applications of a categorical framework to approach the basic theory, I think that there is already broad evidence that looking at (fundamental, foundational) questions from a categorical point of view is likely to spread our horizons beyond any apparent imagination and open new paths towards unknown universes waiting for being explored. It may sound perhaps too romantic, but looks more than sufficient (to me) to deserve our efforts. –  Salvo Tringali Oct 16 '12 at 17:55
You are welcome. Yes this subject is (or rather was until quite/very recently) very commutative. There is some classical non-commutative work (eg by Freiman, Olson&White) but the focus was commutative; it is even called Additive Combinatorics after all, and this name is rather recent (Tao 2003, I'd guess). The developments of very recent years by Breuillard, Green, Helfgott, Hrushovki, Pyber, Szabo, Tao,... are, well, very recent. And, historically this subject was (and by and large large I'd say still is) firmly 'problem solvers territory' (...) –  quid Oct 16 '12 at 17:59
(...) So also the books are not written in a very 'structural' style. The notion is needed at some point so it is introduced and developped to the point it seems useful for the things at hand. If later generalizations should be needed, they can be done later. (Oversimplifying a lot.) Also until recently this was a very small subject. So people did what they found most interesting and useful, which did not so much include abstarct foundational considerations (for all I know), and as I tried to convey presumably also for reasons of personal style and taste. (...) –  quid Oct 16 '12 at 18:09