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If $X$ is a spectrum, we have a notion of its connective part $X_{\le 0}$ and the corresponding notion of truncation $X_{[i:j]} = X_{\le j}/X_{\le i-1}$, where $X_{\le j}$ is deduced from $X_{\le 0}$ using the shift functor and the quotient above is defined as a cone. Note that I'm using cohomological grading.

I know that the functor $X_{\le 0}$ is symmetric monoidal (with respect to the smash product), and in particular takes $E_n$ ring spectra to $E_n$ ring spectra. On the other hand, the functor $X_{[0:0]}$ also takes $E_n$ ring spectra to $E_n$ ring spectra (in fact, even to $E_\infty$ ring spectra if $n\ge 2$) since $X_{[0:0]}$ is the Eilenberg-Maclane spectrum associated to $\pi_0(X)$.

My question is: when is it true that the truncation functor $X_{[i:j]}$ takes $E_n$ ring spectra to $E_n$ ring spectra? What if we assume $X$ is (co)connective?

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    $\begingroup$ Is $X\mapsto X_{\leq 0}$ really symmetric monoidal? $\endgroup$ – Charles Rezk Dec 17 '16 at 14:26
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    $\begingroup$ Did I get my arrows backwards? The map $X^i\wedge \Sigma^\infty BS_i\to X$ should canonically preserve the connective part $\endgroup$ – Dmitry Vaintrob Dec 17 '16 at 14:28
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    $\begingroup$ I believe that only the truncations $X_{[0:-j]}$ for $j \geq 0$ in your notation are guaranteed to preserve $E_n$ ring spectra. $\endgroup$ – Tyler Lawson Dec 17 '16 at 18:27
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    $\begingroup$ Today I learned that cohomological notation with lower indices freaks me out. $\endgroup$ – Dylan Wilson Dec 17 '16 at 18:46
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    $\begingroup$ @DylanWilson, sorry, I should have put up a co-trigger warning $\endgroup$ – Dmitry Vaintrob Dec 17 '16 at 19:01
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A number of statements equivalent to preservation of $E_n$-algebras under truncation are given in a paper I wrote with Michael Batanin called "Bousfield Localization and Eilenberg-Moore Categories". There are 4 equivalent statements in total, in Theorem 5.6. The one most likely to work here, I think, is to prove that there's a transferred model structure on $E_n$-algebras in the localized model structure on spectra (where weak equivalences are defined in terms of truncations). Even a transferred semi-model structure is enough. In another paper, with Donald Yau, Bousfield Localization and Algebras over Colored Operads, we give extremely general machinery to prove these transferred model structures exist. As an application, we prove that there's a model structure on algebras over any colored operad in symmetric spectra (this was first observed in Elmendorf-Mandell, but we felt the proof was lacking some details, and we wanted to do it for general model categories and to connect it with left Bousfield localization). I'm willing to bet that our techniques could be used to transfer the truncated model structure to $E_n$-algebras. Certainly it won't work with every truncation; the Postnikov section does not preserve $A_\infty$-algebras (as shown by Casacuberta). But if the truncation is stable then it's known the resulting local model structure is monoidal (satisfied the pushout product axiom), and that means you can try to use the machinery from my paper with Donald Yau.

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