Seminar

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A classical theorem of Eilenberg--Moore, Dwyer, as well as Bousfield, gives conditions under which passage to homology commutes with certain totalizations arising from cosimplicial resolutions. In homotopy-theoretic terms, one can view this as a statement controlling when stabilization commutes with totalization.

In this talk I will discuss a generalized Eilenberg--Moore theorem and its application in motivic homotopy theory. Roughly, the infinite suspension functor from motivic spaces to SH commutes with the totalization of certain cosimplicial objects. The proof is based on an adaptation of the Eilenberg--Moore argument to a sufficiently general homotopical setting.

The main application concerns completion at the motivic Hopf map $\eta$. Using the generalized Eilenberg--Moore theorem, one can compare unstable $\eta$-completion of motivic spaces with $\eta$-completion after stabilization. This allows one to deduce the convergence of monadic resolutions with respect to motivic cohomology over formally real fields, leading to unstable Adams spectral sequences in this context.