In this chapter, we introduce symmetric spectra and orthogonal spectra along with their associated stable model structures. These versions of spectra have various technical advantages over sequential spectra. Furthermore, they are Quillen equivalent to the category of sequential spectra (equipped with its stable model structure). Hence, one may choose between these models according to their relative strengths. The primary advantage of symmetric and orthogonal spectra is that these model categories are symmetric monoidal models for the stable homotopy category. We will examine these monoidal structures further later on and show that symmetric spectra and orthogonal spectra are monoidally Quillen equivalent. Several other models of spectra also exist, and we will give short introductions to these later in this chapter. We end the chapter with a result that, roughly speaking, says that any model for the stable homotopy category will be Quillen equivalent to sequential spectra.

Bousfield and Friedlander defined the stable homotopy category in terms of the homotopy category of a model category of spectra. We will construct this model category following an approach similar to Mandell-May-Schwede-Shipley based on sequential spectra. A sequential spectrum is a sequence of pointed topological spaces (and structure maps), thus, a natural candidate for an analogue of weak homotopy equivalences are those maps of spectra inducing a weak homotopy equivalence at every level. However, we will see that these levelwise weak homotopy equivalences are not sufficient to define a class of weak equivalences leading to a meaningful stable homotopy theory. A key ingredient is the definition of homotopy groups of spectra and their isomorphisms. This generalises the notion of stable homotopy groups of topological spaces that we encountered earlier. Making these isomorphisms the weak equivalences of sequential spectra will give us a construction of our desired stable homotopy category. We end the chapter with an introduction to the Steenrod algebra and the Adams spectral sequence.