2 - Sequential Spectra and the Stable Homotopy Category

Summary

In this chapter, we introduce the notion of pointed model categories and show that the homotopy category of a pointed model category has a suspension functor with an adjoint called the loop functor. This suspension functor is a generalisation of the standard notion of (reduced) suspension of pointed topological spaces. We shall also see that, in the case of chain complexes over a ring, this suspension functor is modelled by the shift functor. With these constructions in place, we can define the notion of a stable model category. The suspension and loop functors allow us to define cofibre and fibre sequences in an arbitrary pointed model category. These sequences are a generalisation of cofibre and fibre sequences for pointed spaces category of a pointed model category and are a useful aid to calculations. When the model category is also stable, these cofibre and fibre sequences form the basis of important additional structure on the homotopy category.