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deleted some 'vague comment' as it appears to be wrong.
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user9072
user9072

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.

A final vague comment: one issue, although I lack the categorical background to say how much of an issue, could be that the composition of s-Freiman homomorphism will not be an s-Freiman homomrophism. So at least it seems one cannot fix the order s of the homomorphisms and expect to get some (traditional) category with them as arrows.

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".

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.

A final vague comment: one issue, although I lack the categorical background to say how much of an issue, could be that the composition of s-Freiman homomorphism will not be an s-Freiman homomrophism. So at least it seems one cannot fix the order s of the homomorphisms and expect to get some (traditional) category with them as arrows.

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".

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".

Source Link
user9072
user9072

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.

A final vague comment: one issue, although I lack the categorical background to say how much of an issue, could be that the composition of s-Freiman homomorphism will not be an s-Freiman homomrophism. So at least it seems one cannot fix the order s of the homomorphisms and expect to get some (traditional) category with them as arrows.

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".