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Suppose given an $\mathbb E_n$-monoidal presentable $\infty$-category $\mathcal C$ (wrt. the Lurie tensor product $\otimes$), and $\mathcal D$ a presentable $\infty$-category. Suppose given a pair of functors $A:\mathcal C\to \mathcal D, B:\mathcal D\to \mathcal C$, such that $AB=id_{\mathcal D}$. Under which conditions can we deduce that there exists a canonical $\mathbb E_n$-monoidal structure on $\mathcal D$ itself?

Intuitively, $\mathcal D$ inherits an operation from the following composition (denote by $m_{\mathcal C}$ the operation in $\mathcal C$):

$$\mathcal D\otimes \mathcal D\xrightarrow{B\otimes B}\mathcal C\otimes \mathcal C \xrightarrow{m_{\mathcal C}}\mathcal C\xrightarrow{A}\mathcal D.$$ My question is essentially: under which conditions on $A, B$ do all associativities/coherences transfer to $\mathcal D$?

For example: would it be sufficient to require that the diagram $\require{AMScd}$ \begin{CD} \mathcal D\otimes \mathcal D @>>> \mathcal D\\ @V V V @VV V\\ \mathcal C\otimes \mathcal C @>>> \mathcal C \end{CD}

resulting from the above composition is a pullback?

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    $\begingroup$ I don't see any reason for such a thing to exist already for the decategorified version of the question where $C$ is an ordinary monoid. Then $D$ is just some retract of $C$ as a set and there's no reason for the composite operation you write down to be associative. If we take presheaf categories on this counterexample and Day convolve we should get a counterexample in presentable whatever-categories too. $\endgroup$ Aug 13, 2022 at 19:43
  • $\begingroup$ Thanks. Let me change the question as follows: suppose that D (as a full subcategory of C) is closed under the operation of C. Then it should inherit all coherences from C and be an E_n-$\infty$-category as well, right? $\endgroup$
    – W. Rether
    Aug 14, 2022 at 8:30

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