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

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

2007-08-08

arxiv.org/abs/0708.1178v1

Mathematics

Category Theory

23 pages, to appear in Streetfest proceedings

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-416782