Symmetric powers in abstract homotopy categories

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.