# 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 $Spect^\Sigma_T(C)$.