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It's zero as a map of filtered spectra, but it is considered as a map of $\mathbb{A}$-bimodules.

An $E_{\infty}$ structure extending the $E_1$ structure on $R(n)$ in particular yields maps $(\mathbb{S}^{2n})^{\otimes p}_{hC_p} \to \mathbb{S}^{2pn}$ splitting the inclusion of the bottom cell. The …

This is probably not quite what you were looking for, but since you said you'd also be happy to see any proof, here is one using a modern perspective on Thom spectra of virtual bundles: The flip actio …

I know this post is quite old, but in case you are still interested, or anyone else is, I thought about sharing my recent thoughts about the topic. After all, this is the second result on "matric toda …

Yes: Since $\mathcal{C}$ is stable, $\operatorname{Fun}(\mathcal{C},\mathcal{C})$ is stable, too. In particular, it has finite colimits, so $\operatorname{Ind}\operatorname{Fun}(\mathcal{C},\mathcal{C …

This holds only for compact objects (i.e. finite CW spectra), since it is easy to see that additivity fails otherwise (the other axioms of homology theories are satisfied). The usual way to obtain a h …

This is a partial answer, not addressing the relation to homotopy colimits in a model category presenting $\mathcal{C}$ (but I think there should be some reasonable statement there, the slogan is cert …

EDIT: I'll leave the old answer up, see below. In the meantime, Maxime Ramzi and I have thought this through and came up with a fun general argument. Claim. Let $L: \mathrm{Sp}\to \mathrm{Sp}$ be a Bo …

I believe https://arxiv.org/abs/2109.01017 does what you want! The description of coherent chain complexes used there is a bit different than what you suggest, but they look equivalent at first glance …

It is not, for example note that if this were true, the case $A=B$ would give you that $L_{B/B}$ is the infinite suspension of $\mathrm{id}: B\to B$. But that's the absolute cotangent complex $L_B$, w …