Many embedding theorems can be restated in terms of SQ-universality. Shirshov's Theorem that a Lie algebra of finite or countable dimension can be embedded into a 2-generator Lie algebra is equivalent to the statement that the 2-generator free Lie algebra is SQ-universal (in the category of Lie algebras). This can be proved by proving a version of the Higman, Neumann, Neumann theorem for Lie algebras.A.I. Lichtman and M. Shirvani, 'HNN-extensions of Lie algebras', Proc. American Math. Soc. Vol 125, Number 12, December 1997, 3501-3508 However versions of the HNN theorem can be proved for categories where there is no clear idea of a free object. For instance it can be proved that every separable topological group is isomorphic to a topological subgroup of a group having two topological generators (that is, having a dense 2-generator subgroup).Sidney A. Morris and Vladimir Pestov, 'A topological generalization of the Higman-Neumann-Neumann Theorem', Research Report RP-97-222 (May 1997), School of Mathematical and Computing Sciences, Victoria University of Wellington. See also J. Group Theory 1, No.2, 181-187 (1998).
