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

TL;DR

This paper investigates the relationships between degenerate n-categories and algebraic structures like monoids and monoidal categories, revealing that equivalences only emerge when considering categories rather than higher categorical structures.

Contribution

It clarifies the conditions under which degenerate n-categories correspond to algebraic structures, highlighting the importance of focusing on categories rather than higher morphisms for equivalence.

Findings

Degenerate categories correspond to monoids, but the full 2-category structure does not yield an equivalence.

Considering only the categories of degenerate structures restores the equivalence with algebraic structures.

For doubly degenerate bicategories, the bicategory of such structures is needed to achieve an equivalence.

Abstract

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 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 {\bfseries Cat} so we take the full sub-2-category of this whose 0-cells 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 {\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 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 {\it bicategory} of doubly degenerate bicategories. We conclude with a hypothesis about how the above cases might generalise for n-fold degenerate n-categories.