Both were introduced by Quillen (1969, Appendix A3), based on the work of (Mal'cev 1949).
The functors involved in these equivalences are as follows: a Malcev group G is mapped to the completion (with respect to the augmentation ideal) of its group ring QG, with inverse given by the group of grouplike elements of a Hopf algebra H, essentially those elements 1 + x such that
This equivalence of categories was used by Goodwillie (1986) to prove that, after tensoring with Q, relative K-theory K(A, I), for a nilpotent ideal I, is isomorphic to relative cyclic homology HC(A, I).
This theorem was a pioneering result in the area of trace methods.
Malcev Lie algebras also arise in the theory of mixed Hodge structures.