nLab symmetric monoidal structure on spectrum objects


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


Symmetric monoidal category of spectrum objects

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

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

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

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

XSym(T)YXY. X \otimes Sym(T) \otimes Y \rightrightarrows X \otimes Y.

By noting the tautological equivalence of categories

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

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

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

See also


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)

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