symmetric monoidal (∞,1)-category of spectra
Working over a commutative ring , recall that a unital or coaugmented coalgebra is an -coalgebra equipped with a coalgebra map . Abusing notation, denote by ; then an element is primitive if . This condition implies .
This notion generalizes straightforwardly to unital corings over .
A straightforward calculation shows that the module of primitive elements in a Hopf algebra (or even in a bialgebra ) is a Lie subalgebra of the underlying Lie algebra of (whose bracket is the algebra commutator). Thus, taking primitive elements yields a functor
(and of course we have more generally a functor which is an extension along the full inclusion ).
For a Lie algebra , let be its universal enveloping algebra:
where is the two-sided ideal generated by elements of the form where . This carries a bialgebra structure whose comultiplication is uniquely determined by the rule for . Since this says is primitive, the counit is forced to be the algebra map such for all , and also the Hopf antipode is uniquely determined: for .
The following proposition is entirely straightfoward:
The functor is left adjoint to the functor .
This result is essentially tautologous and holds for any commutative ring of arbitrary characteristic. (This despite the fact that the as defined above is not as well-behaved in nonzero characteristic as one might like; e.g. the PBW theorem fails.) More information on this adjunction may require more restrictive hypotheses:
For a field of characteristic zero, the unit is an isomorphism.
An immediate consequence is that for such ground fields , the functor is fully faithful. Of course, lands in the full subcategory of cocommutative Hopf algebras, which is exactly the category of group objects in the cartesian monoidal category of cocommutative coalgebras.
The Milnor-Moore theorem gives further information: for Hopf algebras over a field of characteristic zero, the counit is a monomorphism, and an isomorphism in case satisfies a suitable conilpotency condition.
(More needs to be added.)