  We study two different flavours of A^1-homotopy theory in the setting of
spectral algebraic geometry, and compare them to classical A^1-homotopy theory.
As an application we show that the spectral analogue of Weibel's homotopy
invariant K-theory collapses to the classical theory. Along the way we give a
new construction of nonconnective algebraic K-theory of stable
infinity-categories via a generalization of the Bass-Thomason-Trobaugh
construction.
