$\Omega$ and $B$ as adjoints between symmetric monoidal $(\infty,n)$- and $(\infty,n-1)$-categories

Question

Given a symmetric monoidal $(\infty,n)$-category $\mathcal{D}$, one obtains a symmetric monoidal $(\infty,n-1)$-category $\Omega \mathcal{D}$ by taking $\Omega \mathcal{D}= End_{\mathcal{D}}(1_{\mathcal{D}})$, where $1_{\mathcal{D}}$ is the unit object of $D$. Conversely, given a symmetric monoidal $(\infty,n-1)$-category $\mathcal{C}$, one obtains a symmetric monoidal $(\infty,n)$-category $B \mathcal{C}$ by defining $B \mathcal{C}$ to be the $(\infty,n)$-category with a single object with $\mathcal{C}$ as category of morphisms.

How are $\Omega$ and $B$ related? Clearly $\Omega B\mathcal{C}$ is again $C$ and $B\Omega\mathcal{D}$ is the full monoidal subcategory of $\mathcal{D}$ generated by the object $1_{\mathcal{D}}$. But maybe there is more: in a recent discussion on this subject with Alessandro Valentino we came to conjecturing that, mimicing what happens between topological monoids and (pointed) topological spaces (see, e.g. http://arxiv.org/abs/1203.4978), $\Omega$ and $B$ should be adjoint in a suitable sense. More precisely we expect one should have

$$ Fun^{\otimes}(B\mathcal{C},\mathcal{D}) \cong Fun^{\otimes}(\mathcal{C},\Omega\mathcal{D}) $$

where $Fun^{\otimes}$ stands for ``symmetric monoidal functors''. But we have so far been unable to prove (or disprove) such a statement nor to locate it in the literature.

Any suggestion in either direction will be appreciated :)

Answer 1

I assume you meant "symmetric monoidal functors".

Yes, this seems to hold. By your description you have constructed maps: $$ \eta: \mathcal{C} \cong \Omega B \mathcal{C}$$ $$ \varepsilon: B \Omega \mathcal{D} \to D $$ To check that you get the kind of equivalence you want on functor categories, it suffices to show that the triangle identities hold up to equivalence.

One of these identities states, in words: The full monoidal subcategory of $1_\mathcal{D}$ in the full monoidal subcategory of $1_\mathcal{D}$ in $\mathcal{D}$ is the full monoidal subcategory of $1_\mathcal{D}$ in $\mathcal{D}$. This obviously holds in any reasonable theory where those words make sense.

The other one is equally clear. It says, essentially, that if $\mathcal{D}$ is of the form $B \mathcal{C}$, then the inclusion $\varepsilon$ is (equivalent to) the identity.