Student Homotopy Theory Seminar

Kan spectra are a stabilization of simplicial sets, just as May spectra are of spaces, and they both provide "point-set" level models for the stable homotopy category. Moreover, being the Eckmann-Hilton dual of May spectra, Kan spectra admit a co-localization (rather than a localization) with respect to naive homotopy equivalences and weak equivalences; this makes possible a fully functional sheaf theory on Kan spectra.

In this talk, we will define Kan spectra, construct a homotopy theory on the category of Kan spectra and their sheaves, and prove that these categories admit co-localization. We will see that the right-derivability criteria boil down to preservation of naive homotopy, and that right derived functors can be constructed and computed via the Godement resolution. In particular, we can construct "higher" direct images functors Rf_*, including generalized sheaf cohomology, as well as Rf_!, via Kan spectral sheaves. Speaker(s): Ruian Chen (University of Michigan)