Coalgebras in symmetric monoidal categories of spectra

  title={Coalgebras in symmetric monoidal categories of spectra},
  author={Maximilien P'eroux and Brooke E. Shipley},
  journal={Homology, Homotopy and Applications},
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
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Shadows are Bicategorical Traces
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On The Combinatorics of the Gray Cylinder
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