Let $R$ be a ring spectrum. Then we can form the topological Hochschild Homology of $R$ as the spectrum $$THH(R) = R \otimes S^1 \simeq R \wedge _{R \wedge R^{op}} R.$$ What is known about the spaces which make up $THH(R)$ as a spectrum, particularly in relation to the spaces which make of $R$? In particular, if $R$ is an $\Omega$-spectrum, do we know that $THH(R)$ is as well? I have searched for a while and been unable to find a description of these spaces at all.
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$\begingroup$ This seems quite impossible, since you really need the $E_1$-structure on $R$ (not just the spectrum structure) to determine the spectrum $\operatorname{THH}(R)$. $\endgroup$– Z. MCommented Jun 27 at 18:34
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$\begingroup$ @Z.M Thanks for your response! just to clarify, you don't think that there is a way to get a handle on the spaces on $THH(R)$ at all, regardless of the $E_n$-structure on $R$? $\endgroup$– categorically_stupidCommented Jun 27 at 18:45
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7$\begingroup$ The question of whether something is or isn't an $\Omega$-spectrum is not well defined under equivalences. The descriptions of THH you give make sense model-independently in the $\infty$-category of spectra, but you seem to have in mind a specific way of lifting it to some version of prespectra? The answer to your question probably depends on those details. $\endgroup$– Achim KrauseCommented Jun 27 at 22:48
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