  We study symmetric powers in the homotopy categories of abstract closed
symmetric monoidal model categories, in both unstable and stable settings. As
an outcome, we prove that symmetric powers preserve the Nisnevich and etale
homotopy type in the unstable and stable motivic homotopy theories of schemes
over a base. More precisely, if f is a weak equivalence of motivic spaces, or a
weak equivalence between positively cofibrant motivic spectra, with respect to
the Nisnevich or etale topology on schemes, then all symmetric powers Sym^n(f)
are weak equivalences too. This gives left derived symmetric powers which
aggregate into a categorical lambda-structures on the corresponding motivic
homotopy categories of schemes over a base.
