Universal quadratic functor

Definition

The universal quadratic functor (sometimes called Whitehead's universal quadratic functor) is a functor from abelian groups to abelian groups defined as follow. For an abelian group $G$ it outputs a group $\Gamma (G)$ given as the quotient of a free group on all the symbols $\gamma (x),x\in G$ by the following types of relations:

• $\gamma (0)=0$ (this condition is redundant)
• $\gamma (-x)=\gamma (x)\ \forall \ x\in G$
• $\gamma (x)+\gamma (y)+\gamma (z)+\gamma (x+y+z)=\gamma (y+z)+\gamma (x+y)+\gamma (x+z)\ \forall x,y,z\in G$.

Note that the above set of relations is equivalent to the following pair of assumptions:

• The mapping $x\mapsto \gamma (x)$ is homogeneous of degree two: $\gamma (nx)=n^{2}\gamma (x)$ for all $n\in \mathbb {Z}$
• The mapping $(x,y)\mapsto \gamma (x+y)-\gamma (x)-\gamma (y)$ is a bihomomorphism, i.e., it is additive in each coordiate.

Facts

• The exponent of $\gamma (x)$ in $\Gamma (G)$ divides twice the exponent of $x$ in $G$. This follows from noting that the bilinear form $b(x,y)=\gamma (x+y)-\gamma (x)-\gamma (y)$ also satisfies $b(x,-x)=-2\gamma (x)$ so $b(x,x)=2\gamma (x)$, and the exponent of $b(x,x)$ divides the exponent of $x$ due to biadditivity.
• Formula for universal quadratic functor of direct product

Particular cases

Starting group $G$	Universal quadratic functor $\Gamma (G)$	Comments
finite cyclic group of odd order $n$, i.e., $\mathbb {Z} _{n}$	$\mathbb {Z} _{n}$	Intuitively, this is saying that if $x$ is defined mod $n$ for $n$ odd, then $x^{2}$ is defined mod $n$ but we cannot do better in general
finite cyclic group of even order $n$, i.e., $\mathbb {Z} _{n}$	$\mathbb {Z} _{2n}$	Intuitively, this is saying that if $x$ is defined mod $n$ for $n$ even, then $x^{2}$ is defined mod $2n$ but we cannot do better in general
group of integers $\mathbb {Z}$	$\mathbb {Z}$	we can think of the generator for $\Gamma (G)$ as the squaring map in the ring of integers