nLab
stable (infinity,1)-category of spectra

Idea

The collection of spectra form an (∞,1)-category $\mathrm{Sp}(\mathrm{\infty }\mathrm{Grpd})$ which is in fact a stable (∞,1)-category. Indeed, it is the universal property stabilization of the $(\mathrm{\infty },1)$-category ∞Grpd, equivalently of the simplicial localization of the category Top at the weak homotopy equivalences.

$\mathrm{Sp}(\mathrm{\infty }\mathrm{Grpd})$ plays a role in stable homotopy theory analogous to the role played by the 1-category Ab of abelian groups in homological algebra, or rather of the category of chain complexes ${\mathrm{Ch}}_{•}(\mathrm{Ab})$ of abelian groups.

Definition

In the context of (∞,1)-categories a spectrum is a spectrum object in the (∞,1)-category $L_{\mathrm{whe}}{\mathrm{Top}}_{*}$ of pointed topological spaces.

Recall that spectrum objects in the (infinity,1)-category $C$ form a stable (∞,1)-category $\mathrm{Sp}(C)$.

The stable (∞,1)-category of spectrum objects in $L_{\mathrm{whe}}{\mathrm{Top}}_{*}$ is the stable $(\mathrm{\infty },1)$-category of spectra

$$\mathrm{Stab}(L_{\mathrm{whe}}\mathrm{Top}):=\mathrm{Sp}(L_{\mathrm{whe}}{\mathrm{Top}}_{*}).$$Stab(L_{whe}Top) := Sp(L_{whe}Top_*) \,.

Remarks

• With the smash product of spectra $\mathrm{Sp}(L_{\mathrm{whe}}{\mathrm{Top}}_{*})$ becomes a symmetric monoidal (infinity,1)-category.

  • an algebra object in $\mathrm{Sp}(L_{\mathrm{whe}}{\mathrm{Top}}_{*})$ with respect to this monoidal structure is an associative ring spectrum;

  • a commutative algebra object in $\mathrm{Sp}(L_{\mathrm{whe}}{\mathrm{Top}}_{*})$ with respect to this monoidal structure is a commutative ring spectrum;

References

the stable $(\mathrm{\infty },1)$-category of spectra is described in section 9 of

• Jacob Lurie, Stable Infinity-Categories .

Its monoidal structure is described in section 4.2

• Jacob Lurie, Noncommutative algebra.

That this is a symmetric monoidal structure is described in section 6 of

• Jacob Lurie, Commutative algebra.