Spectra and stable ∞-categories

This chapter studies the \(\infty \)-category of spectra in its own right and develops the general notion of a stable \(\infty \)-category that it exemplifies. In the previous chapter, we encountered spectra as objects representing generalized cohomology theories. We now shift perspective to their formal properties, which are reminiscent of homological algebra.

As we will see, there are two equivalent perspectives on stable \(\infty \)-categories. The first emphasizes the relation to the stability property of cohomology theories by considering the suspension and loop functors and asking them to be invertible. The second emphasizes the analogy with exact sequences in an abelian category by considering fibers and cofibers, the homotopical analogues of kernels and cokernels, and demanding that every nullsequence \[ X \to Y \to Z \] is a fiber sequence if and only if it is a cofiber sequence. It is this second perspective that connects stable \(\infty \)-categories to homological algebra.

The \(\infty \)-category of spectra admits a universal property: it is the terminal stable \(\infty \)-category equipped with a left exact functor to \(\An \). This is a special case of the stabilization \(\Sp (C)\) of an \(\infty \)-category \(C\) with finite limits. We may think of \(\Sp (C)\) as obtained from \(C\) by formally making the loops operation invertible.

Section 4.1 develops fibers, cofibers, suspensions and loops in pointed \(\infty \)-categories. Section 4.2 then studies stable \(\infty \)-categories themselves, and Section 4.3 constructs \(\Sp (C)\) by means of prespectra and spectrification. Finally, Section 4.4 specializes to \(\Sp \) and treats suspension spectra, mapping spectra, tensor products, homotopy groups, and the homology theories represented by spectra.

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