The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories

Abstract. We examine the periodic table of weak n-categories for the lowdimensional 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 this correspondence fully we examine the totalities of such structures together with maps between them and higher maps between those. Categories naturally form a 2-category Cat so we take the full sub-2-category of this whose 0cells are the degenerate categories. Monoids naturally form a category, but we 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 ignore the natural transformations and consider only the 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 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 consider the bicategory of doubly degenerate bicategories. We conclude with a hypothesis about how the above cases might generalise for n-fold degenerate n-categories.