## Workshop on Categorical Methods in Algebra, Geometry and Mathematical Physics

### Satellite to the StreetFest conference in honour of Ross Street's sixtieth birthday

#### July 18-21 2005, Australian National University, Canberra

Thu, 21 July: 16:40 - 17:20

##### The periodic table of $n$-categories:\\ low-dimensional results

###### Cheng, Eugenia (University of Chicago)

We examine the periodic table of weak $n$-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and degenerate bicategories to monoidal categories; however, to understand the situation fully we should examine the totalities of such structures. Categories naturally form a 2-category {\bfseries Cat}, so we can take the full sub-2-category of this whose 0-cells are the degenerate categories. On the other hand monoids naturally form a category, but we can regard this as a discrete 2-category to make the comparison. We show that this construction does not yield a biequivalence; to get an equivalence we must ignore the natural transformations and consider only the {\it category} of degenerate categories.

A similar situation occurs for degenerate bicategories. The tricategory of such does not yield an equivalence with monoidal categories; we must consider only the categories of such structures.

For doubly degenerate bicategories the situation is more subtle. The tricategory of such is not naturally triequivalent to the category of commutative monoids (regarded as a tricategory). However, in this case considering just the categories does not give an equivalence either; to get an equivalence we must consider the {\it bicategory} of doubly degenerate bicategories.

We conclude with some remarks about how the above cases might generalise for degenerate, doubly degenerate and triply degenerate tricategories, and for $n$-fold degenerate $n$-categories.