# The category of abelian groups with quasi-homomorphisms

Let $$A$$ and $$B$$ be abelian groups. Say that a map $$f: A \to B$$ is a quasi-homomorphism if there exists a finite $$D \subseteq B$$ such that $$\forall a_1, a_2 \in A: f(a_1 + a_2) - f(a_1) - f(a_2) \in D$$

We wish to show that if $$A$$, $$B$$ and $$C$$ are abelian groups and $$f: A \to B$$ and $$g: B \to C$$ are quasi-homomorphisms, then $$g \circ f: A \to C$$ is a quasi-homomorphism. So take a finite $$D \subseteq B$$ such that $$\forall a_1, a_2 \in A: f(a_1 + a_2) - f(a_1) - f(a_2) \in D$$ and a finite $$E \subseteq C$$ such that $$\forall b_1, b_2 \in B: g(b_1 + b_2) - g(b_1) - g(b_2) \in E$$ Define $$F = \{g(d) + e_1 + e_2 \mid d \in D, e_1, e_2 \in E\} \subseteq C$$ As $$D$$ and $$E$$ are finite, $$F$$ is also finite. Now take any $$a_1, a_2 \in A$$. We wish to show that $$g(f(a_1 + a_2)) - g(f(a_1)) - g(f(a_2)) \in F$$ Now, set $$d = f(a_1 + a_2) - f(a_1) - f(a_2) \in D$$ Set $$e_1 = g(d + f(a_1) + f(a_2)) - g(d) - g(f(a_1) + f(a_2)) \in E$$ Set $$e_2 = g(f(a_1) + f(a_2)) - g(f(a_1)) + g(f(a_2)) \in E$$ Then we have $$g(f(a_1 + a_2)) - g(f(a_1)) - g(f(a_2)) = g(d + f(a_1) + f(a_2)) - g(f(a_1)) - g(f(a_2)) = e_1 + g(d) + g(f(a_1) + f(a_2)) - g(f(a_1)) - g(f(a_2)) = e_1 + g(d) + e_2 \in F$$ Thus, $$g \circ f$$ is a quasi-homomorphism.

In particular, we can form a category $$\mathsf{QuasiAb}$$ whose objects are abelian groups and whose morphisms are quasi-homomorphisms.

Is this proof correct? Can you give any reference on $$\mathsf{QuasiAb}$$?

• From the solution-verification tag wiki: "A question with this tag should include an explanation for why the argument presented is not convincing enough." Apr 14 at 12:19
• Hmm... maybe a reference-request tag could be added, so that the question doesn't have to be closed just because a rule that the OP was unaware of wasn't followed. To me, the question is decent because this is not exactly a well-known concept. Apr 14 at 13:37

The late Stephen Schanuel proposed a remarkable construction of the real numbers, calling them the "Eudoxus real numbers", as the set of all "quasi-homomorphisms" [or "almost linear maps", as he calls them] $$\mathbb{Z} \to \mathbb{Z}$$, modulo the equivalence relation $$\sim$$ where $$f \sim g$$ if the set of differences $$f(m) - g(m)$$ is a finite set. You can check that the sum of two almost linear functions $$\mathbb{Z} \to \mathbb{Z}$$ is almost linear, and since (as you showed) the composite of two almost linear functions is almost linear, you get a ring structure on the set of almost linear functions, and this ring structure also respects the equivalence relation $$\sim$$, so that you get a commutative ring structure on the Eudoxus reals.
It can be shown that this commutative ring is a field. From the nLab: Define a Eudoxus real $$[f]$$ to be positive if it is represented by an $$f$$ such that for any $$C$$ there are infinitely many $$m\in \mathbb{N}$$ such that $$f(m)>C$$. It is easy to see this condition would then be satisfied by any representative function. As usual, define $$[f]<[g]$$ to mean $$[g−f]$$ is positive. Then the Eudoxus reals forms an ordered field, and in fact it is a complete ordered field.
• Neat example, I hadn't heard of it! To demystify it a little bit, there's an explicit correspondence with the real numbers where $r\in\mathbb R$ maps to (the equivalence class of) the function $n\mapsto \lfloor rn\rfloor$. Apr 15 at 4:45