nLab symmetric monoidal structure on spectrum objects

Idea

Given a category $C$ one can consider the category of spectrum objects in $C$. When $C$ is symmetric monoidal, there is an induced symmetric monoidal structure on the category of spectrum objects, generalizing the symmetric monoidal smash product of spectra.

Definitions

Symmetric monoidal category of spectrum objects

Let $T$ be an object of a symmetric monoidal category $C$. Following Ayoub we can define a category $\Spect^\Sigma_T(C)$ of symmetric $T$-spectra in $C$ as described at spectrum object.

In that case the category of $\Sigma$-symmetric sequences $\Seq(\Sigma, C)$ inherits a symmetric monoidal structure as described at symmetric sequence.

Consider the symmetric sequence $Sym(T)$ whose $n$th component is $Sym(T)_n = T^{\otimes n}$ (with the actions of the symmetric group $\Sigma_n$ by permuting the $n$ factors). It turns out that $Sym(T)$ is a commutative algebra object in the symmetric monoidal category $Seq(\Sigma, C)$.

As a consequence there is a symmetric monoidal structure on $Mod_\ell(Sym(T))$ where $X \otimes_{Sym(T)} Y$ is defined as the coequalizer of the diagram

$$X \otimes Sym(T) \otimes Y \rightrightarrows X \otimes Y.$$

By noting the tautological equivalence of categories

$$\Spect^\Sigma_T(C) \stackrel{\sim}{\longrightarrow} \Mod_\ell(Sym(T))$$

one gets a symmetric monoidal structure on $Spect^\Sigma_T(C)$.

Symmetric monoidal (infinity,1)-category of spectrum objects

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See also

• spectrum object
• symmetric sequence
• symmetric monoidal smash product of spectra

References

The last chapter of

• Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, I. Astérisque, Vol. 314 (2008). Société Mathématique de France. (pdf)