Abstract
We study the semi-topologization functor of Friedlander-Walker from the perspective of motivic homotopy theory. We construct a triangulated endo-functor on the stable motivic homotopy category $\mathcal{SH}(\C)$, which we call \emph{homotopy semi-topologization}. As applications, we discuss the representability of several semi-topological cohomology theories in $\mathcal{SH}(\C)$, a construction of a semi-topological analogue of algebraic cobordism, and a construction of Atiyah-Hirzebruch type spectral sequences for this theory.
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