Return to Question

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 $\prod_{i=1}^s \pi(x_i)^{\varepsilon_i} = \prod_{i=1}^s \pi(y_i)^{\varepsilon_i}$, where I am using $\pi(\cdot)^\varepsilon$ in place of $(\pi(\cdot))^\varepsilon$, 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 categoryfying 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?

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?

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 $\mathfrak A_1$ to $\mathfrak A_2$ is then any function $\phi: |\mathfrak A_1| \to |\mathfrak A_2|$ such that: $\prod_{i=1}^s x_i^{\varepsilon_i} = \prod_{i = 1}^s y_1^{\varepsilon_i}$ for some $x_1, \ldots, x_s, y_1, \ldots, y_s \in \mathfrak A_1$ and $\varepsilon_1, \ldots, \varepsilon_s \in \{\pm 1\}$ implies $\prod_{i=1}^s \pi(x_i)^{\varepsilon_i} = \prod_{i=1}^s \pi(y_i)^{\varepsilon_i}$, where I am using $\pi(\cdot)^\varepsilon$ in place of $(\pi(\cdot))^\varepsilon$, 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 categoryfying the theory of Freiman's morphisms, at least to the degree, say, of defining a category where objects are "structures of a given type" 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?

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 $\prod_{i=1}^s \pi(x_i)^{\varepsilon_i} = \prod_{i=1}^s \pi(y_i)^{\varepsilon_i}$, where I am using $\pi(\cdot)^\varepsilon$ in place of $(\pi(\cdot))^\varepsilon$, 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 categoryfying 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?