Universal quadratic functor
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 it outputs a group given as the quotient of a free group on all the symbols by the following types of relations:
- (this condition is redundant)
Note that the above set of relations is equivalent to the following pair of assumptions:
- The mapping is homogeneous of degree two: for all
- The mapping is a bihomomorphism, i.e., it is additive in each coordinate.
Equivalence of definitions
Further information: equivalence of definitions of universal quadratic functor
- The exponent of in divides twice the exponent of in . This follows from noting that the bilinear form also satisfies so , and the exponent of divides the exponent of due to biadditivity.
- Formula for universal quadratic functor of direct product
|Starting group||Universal quadratic functor||Comments|
|finite cyclic group of odd order , i.e.,||Intuitively, this is saying that if is defined mod for odd, then is defined mod but we cannot do better in general|
|finite cyclic group of even order , i.e.,||Intuitively, this is saying that if is defined mod for even, then is defined mod but we cannot do better in general|
|group of integers||we can think of the generator for as the squaring map in the ring of integers|