Given a pointed $(\infty,n)$-category $\mathcal{C}$, one can define the suspension of $\mathcal{C}$, $\Sigma\mathcal{C}$, via the homotopy pushout of $$\ast\leftarrow \mathcal{C}\rightarrow \ast.$$ Dually one can define $\Omega\mathcal{C}$. Can one explicitly identify these $(\infty,n)$-categories in terms of $\mathcal{C}$?


My vague intuition, based on the case $n=0$, says that the $\Omega\mathcal{C}$ should be the endomorphisms of the distinguished object and $\Sigma\mathcal{C}$ should be what you get when you take the free monoidal $(\infty,n)$-category on $\mathcal{C}$, regard it as an $(\infty,n+1)$-category with one object and then invert the $n+1$-morphisms. However, my understanding of (homotopy) limits and colimits in this setting is pretty poor.


Feel free to use any model you wish.



Loops and suspensions of higher categories
First let me thank Urs, Karol, and Rune Haugseng for helpful comments.


Now note that the inclusion, $i$, of $\infty$-groupoids into $(\infty,n)$-categories has an $\infty$-categorical left adjoint, $L$ (for lack of a better name), and a right adjoint $(-)^\prime$. 


Given a pointed $(\infty,n)$-category $\mathcal{C}$, the loop category $\Omega \mathcal{C}$ is defined by the following (homotopy) pullback diagram:


\begin{array}{ccc}
\Omega\mathcal{C} & \rightarrow & \ast\\
\downarrow & & \downarrow \\
\ast & \rightarrow & \mathcal{C}
\end{array}


Now $\mathcal{C}^{\prime}$ is the maximal sub-$\infty$-groupoid of $\mathcal{C}$ (the core). Since $\ast$ is an $\infty$-groupoid, the inclusion $\ast\rightarrow \mathcal{C}$ factors canonically through $\mathcal{C}^\prime$. By a standard finality argument we see that $\Omega \mathcal{C}$ is equivalent to the homotopy pullback:
\begin{array}{ccc}
\Omega\mathcal{C} \simeq \Omega^{Top}\mathcal{C}^\prime& \rightarrow & \ast\\
\downarrow & & \downarrow \\
\ast & \rightarrow & \mathcal{C}^\prime
\end{array}


Regarding $\mathcal{C}^\prime$ as a space (since it is an $\infty$-groupoid), we see that $\Omega\mathcal{C}$ is equivalent to the space of topological (based) loops on $\mathcal{C}^\prime$ since $i$ preserves (homotopy) limits.


Unraveling this a bit, we see that $\Omega(-)$ is naturally equivalent to $i\Omega^{Top}(-)^\prime$ which is a composite of right adjoints. It follows that the left adjoint, $\Sigma(-)$, is naturally equivalent to $i\Sigma^{Top}L(-)$. 


As a consequence, an $(\infty,n)$-category which is a loop category is necessarily a loop space. This shows that the inclusion of spectra objects in $(\infty,0)$-categories (i.e., spectra) into spectra objects in $(\infty,n)$-categories is an equivalence. So the two categories have the same stailizations.


