  In this note, we outline the general development of a theory of symmetric
homology of algebras, an analog of cyclic homology where the cyclic groups are
replaced by symmetric groups. This theory is developed using the framework of
crossed simplicial groups and the homological algebra of module-valued
functors. The symmetric homology of group algebras is related to stable
homotopy theory. Two spectral sequences for computing symmetric homology are
constructed. The relation to cyclic homology is discussed and some conjectures
and questions towards further work are discussed.
