- A monoid is the same as a category with a single object.
- A monoidal category is the same as a bi-category with a single object.
- A commutative monoid is the same as a bi-category with a single object and a single 1-morphism (This is essentially the Eckmann-Hilton argument).
- A braided monoidal category is the same as a tri-category with a single object and a single 1-morphism
and so on. This behavior of (multi)-degenerate n-categories is known as the Periodic Table of n-categories.
Now it seems natural to ask: What about "degenerate $\infty$-categories$/(\infty,1)$-categories$/(\infty,n)$-categories"? A reasonable guess for example would be, that a $(\infty,2)$-category with a single object should give a monoidal $(\infty,1)$-category. This kind of question might already be answered somewhere in Jacob Lurie's papers, however, I have a hard time finding anything.
EDIT: To avoid confusion, with a bicategory I meant weak $2$-category, with a tricategory a weak $3$-category. I am mostly interested in the weak case, since the periodic table is less rich if we look at strict $n$-categories. When I talk about $(\infty,1)$-categories, I also mean in the weak sense. As a definition of a $(\infty,1)$-category take for example a simplicial set satisfying the inner horn filling condition.