Coalgebras in symmetric monoidal categories of spectra

TL;DR

This paper proves that all coalgebras over the sphere spectrum are cocommutative within certain strict monoidal categories of spectra, highlighting a structural property specific to these models.

Contribution

It establishes cocommutativity of coalgebras over the sphere spectrum in several strict monoidal categories of spectra, clarifying their algebraic structure.

Findings

Coalgebras over the sphere spectrum are cocommutative in these categories.

The result does not extend to the $alculus-category setting.

The proof applies to symmetric spectra, orthogonal spectra, $mma$-spaces, $mma$-spaces, and EKMM $$-modules.

Abstract

We show that all coalgebras over the sphere spectrum are cocommutative in the category of symmetric spectra, orthogonal spectra, $\Gamma$-spaces, $\mathcal{W}$-spaces and EKMM $\mathbb{S}$-modules. Our result only applies to these strict monoidal categories of spectra and does not apply to the $\infty$-category setting.