I'm familiar with the definition of a spectrum, the one due to Adams, however, I'm not really sure why someone would want to define such a thing. I know they allow one to generalize homology and cohomology, but is this their sole purpose? I read in Switzer's Algebraic Topology book
"We have another aim in constructing the category of spectra. In homology theory the suspension homomorphism $\sigma: h_n(X) \rightarrow h_{n+1}( SX)$ is always an isomorphism. For various reasons this suggests trying to embed $\mathscr{PW}'$ in a larger category in which the suspension functor $S$ has an inverse $S^{-1}$."
Can anyone elaborate on this?
Best Answer
There are many different things going on here. One is that to invert the suspension functor is to stabilize. The Freudenthal suspension theorem tells you that the system of suspension maps $[X,Y] \to [SX,SY] \to [S^2X,S^2Y] \to \cdots$ eventually stabilizes (at least for finite CW-complexes), and so you can view this as a simplification of the usual category of spaces. Moreover, for formal reasons this implies that your category is enriched in abelian groups, which gives you more traction on things. (For instance, you can always add two maps, maps always have inverses, etc.)
More broadly, the real power of homotopy theory is that pretty much everything is representable in one way or another, which means that you can turn your tools on each other and get them to tell you new things about the tools themselves. From this point of view, it's entirely natural to define the category of spectra (since it is equivalent to the category of cohomology theories). But it sounds like you already knew about this piece of motivation.