# The “$\infty$”-column in the periodic table of n-categories

• 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.

• But $\infty$-categories are in exactly the same vein as bi- and tri-categories are they? I thought $(\infty,3)$-categories were closer to $3$-categories as opposed to tri-categories (I don't know the $n$-ary version of tri-categories). I heard someone describe this as the difference between higher and wider categories. – Sean Tilson Aug 26 '15 at 14:05
• @SeanTilson Bi- and tricategories are alternative names for weak 2- and 3-categories (as opposed to strict ones). – Rune Haugseng Aug 26 '15 at 14:25
• Yes, I always meant weak higher categories, and in the same vein weak $\infty$-categories. The periodic table is less interesting when only looking at strict n-categories. – Georg Lehner Aug 26 '15 at 14:34
• Ah, I was confused. I thought that one was something like "categories enriched in categories" and the other was "categorical objects in the category of categories". I believe the first of these is $n$-categories etc. What is the right word for the latter? or is it the same and the distinction is between strict and weak? Thanks for clarifying my confusion. – Sean Tilson Aug 26 '15 at 15:04
• The standard definition of a strict $n$-category is as an $(n-1)$Cat-enriched category, like you said. However, there isn't a standard definition for weak $n$-categories. Up to tetra-categories (weak $4$-categories) there is an explicit algebraic definition; above that there are several different approaches, but as far as my knowledge goes, it is still unclear as to whether all of these are equivalent. I believe you mean the Trimble/May approach with "categorical objects in the category of categories"? ncatlab.org/nlab/show/algebraic+definition+of+higher+categories – Georg Lehner Aug 26 '15 at 15:33

The result that $E_n$-monoidal $(\infty,m)$-categories are equivalent to pointed $(\infty,n+m)$-categories with a single object, 1-morphism, ..., and $n$-morphism (and the more general one that $E_n$-algebras in an $E_n$-monoidal $\infty$-category $\mathcal{V}$ are equivalent to pointed $(\infty,n)$-categories enriched in $\mathcal{V}$ with a single object, 1-morphism, etc.) can be found in section 6.3 of http://arxiv.org/abs/1312.3178.