# 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. Aug 26 '15 at 14:05
• @SeanTilson Bi- and tricategories are alternative names for weak 2- and 3-categories (as opposed to strict ones). 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. 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. 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 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.