Coalgebras in symmetric monoidal categories of spectra

@article{Peroux2019CoalgebrasIS,
  title={Coalgebras in symmetric monoidal categories of spectra},
  author={Maximilien P'eroux and Brooke E. Shipley},
  journal={Homology, Homotopy and Applications},
  year={2019}
}
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. 
Coalgebras in the Dwyer-Kan localization of a model category
We show that weak monoidal Quillen equivalences induce equivalences of symmetric monoidal $\infty$-categories with respect to the Dwyer-Kan localization of the symmetric monoidal model categories.
Computations of relative topological coHochschild homology
Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology, and Bohmann-Gerhardt-Høgenhaven-Shipley-Ziegenhagen developed a coBökstedt spectral sequence to
Spanier-Whitehead duality for topological coHochschild homology.
Following the Nikolaus-Scholze approach to THH, we extend the definition of topological coHochschild homology (coTHH) of Hess-Shipley to $\infty$-categories. After this, we prove that coTHH of what
Rigidification of connective comodules
We show that we can rigidify homotopy coherent comodules in connective modules over the Eilenberg-Mac Lane spectrum of a field, or more generally of a finite product of fields $\mathbb{k}$. That is,
Shadows are Bicategorical Traces
The theory of shadows is an axiomatic, bicategorical framework that generalizes topological Hochschild homology (THH) and satisfies analogous important properties, such as Morita invariance. Using
On The Combinatorics of the Gray Cylinder
Acknowledgements 1 Overview 1 1. Berger’s Categorical Wreath Product and the Cell Categories Θ and Θn 6 1.1. Segal’s category Γ 6 1.2. Berger’s categorical wreath product 7 1.3. The Categories Θ and

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