The motivation of spectra comes from the study of generalized cohomology theory. Note that the singular cohomology theory is represented by , by a simple Yoneda calculation, see here. By Brown representability, any such cohomology theory is representable by . We may think that if is an isomorphism, then we can get a generalized cohomology. But in order to show there is a isomorphism coming from the connecting map (the fact that connecting map goes from to rather than the other way also explains why proving vanishing for higher cohomology is very useful), then it eventually boils down to showing the adjoint is an isomorphism, which are not true in general (e.g. fails for spheres).
A comparably naive idea is to take the Spanier-Whitehead category . The motivation is that homology/cohomology is a stable invariant, in the sense that if , then already , and if there exists such that is null-homotopic, then the induced map on cohomology is zero. Observe are naturally abelian groups, and if is compact, then . The funny thing is that the wedge product now becomes a coproduct in this category, so the SW category has a biproduct which implies (formally) that the Hom sets are commutative monoids. We can further enrich the Hom set with a graded abelian group structure by definining . We can further invert by defining .
Later it was realized (see e.g. Whitehead 62) that this all this is fixed by regarding the SW-category for finite CW complexes as a full subcategory on the (shifted) suspension spectra inside the larger category of spectra: the stable homotopy category (e.g. Schwede 12, chapter II theorem 7.2). As such it is the full subcategory on the finite spectra (e.g. Schwede 12, chapter II theorem 7.4).
As a corollary of Freudenthal suspension theorem, we know that If , then for any -connective , we have an isomorphism (The idea is to apply five-lemma to the skeleton of .). Thus stablize.
The idea to show Freudenthal suspension theorem is to use the James reduced product as a model for loop space of suspension. We say is -connective if the -skeleton of is contractible, so is homotopy equivalent to with higher dimensional cells attached. It is easy to show that if is -connective, then is -connective. Thus if is -connective, then for and onto for (since is obtained from by attaching . Let be the fiber of (after fibrant replacement). Note that since applying , we have a splitting , so . By using the Serre spectral sequence, we can show that if and . By Hurewicz we then know the same holds for . Hence from the homotopy LES associated to the fibration, we have for , and it is onto for . In particular, induces isomorphism on for . But is the unit/counit of the loop-suspension adjunction, so by factoring into , we see that this is an isomorphism for .
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