Universal quadratic functor

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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 coordinate.

Equivalence of definitions

Further information: equivalence of definitions of universal quadratic functor

Facts

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