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stable (infinity,1)-category of spectra

Definition

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

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

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

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

Remarks

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

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

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

References

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

• Jacob Lurie, Stable Infinity-Categories .

It 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.