2009 Topology Seminar - 14

Cocycle categories give a simple, flexible way to describe morphisms in a homotopy category, provided that the underlying model structure is sufficiently well behaved. "Well behaved" model structures include simplicial sets, spectra, simplicial presheaves and presheaves of spectra, together with all good localizations such as the motivic model structure of Morel and Voevodsky.

The definition, multiple examples, and the basic properties of cocycle categories will be displayed in the first part of the talk. Simple applications, including the homotopy classification of principal bundles, will be discussed in the second part.