Let $G$ and $G'$ be groups. A Freiman homomorphism of order $s$ from a $A\subset G$ to $G'$ is a map $\phi\colon A\to G$ such $$\phi(a_1)\phi(a_2)\cdots\phi(a_s)=\phi(a_1a_2\cdots a_s)$$ for any $s$ (not necessarily distinct) elements $a_1,a_2,\ldots,a_s\in A$. The Freiman homomorphism are the correct notion of equivalence between sets in additive combinatorics. For example, if $A$ and $B$ are Freiman $2$-isomorphic, then $\lvert A+A\rvert=\lvert B+B\rvert$. There are other versions of almost morphisms that arise naturally in combinatorial number theory. For instance, one of the equivalent versions of the polynomial Freiman-Ruzsa conjecture (for $(\mathbb{Z}/2\mathbb{Z})^n$) asserts that if $\phi\colon (\mathbb{Z}/2\mathbb{Z})^n\to (\mathbb{Z}/2\mathbb{Z})^n$ is a map for which $\phi(x+y)-\phi(x)-\phi(y)$ takes only $K$ values, then $\phi$ can be written as $\phi=\phi_0+\psi$ where $\phi_0$ is a genuine linear map, and $\psi$ takes only $K^{O(1)}$ values.