nLab
periodic table

Idea

In higher category theory, there are several periodic tables, analogous to the the periodic table of chemical elements. Just as this table allowed Менделеев to predict the existence of undiscovered elements in the table's gaps, so these periodic tables sometimes inspire us to invent new varieties of $n$-categories.

History

The first periodic table of $n$-categories was a slightly distorted version of the periodic table of $k$-tuply monoidal $n$-categories.

Fully filled out, the table looked like this:

$k$↓\$n$→	$-2$	$-1$	$0$	$1$	$2$	…
$0$	trivial	truth value	set	category	2-category	…
$1$	"	trivial	monoid	monoidal category	monoidal 2-category?	…
$2$	"	"	abelian monoid	braided monoidal category	braided monoidal 2-category	…
$3$	"	"	"	symmetric monoidal category	sylleptic monoidal 2-category?	…
$4$	"	"	"	"	symmetric monoidal 2-category?	…
⋮	"	"	"	"	"	⋱

Actually, the columns where $n=-1$ and $n=-2$ were not there, but they appeared to be required by the pattern of the table.

We now recognise that a $0$-tuply monoidal $n$-category should be pointed, leading to a slightly different table (see that page). Similarly, the definition given when $k>0$ didn't mention pointedness, giving essentially the definition of $(k-1)$-simply connected $(n+k)$-category. This makes a difference to the notions of morphism and higher morphism between such structures.

Eugenia Cheng and Nick Gurski wrote a paper about how these don’t end up quite right if you just look at $(k-1)$-simply connected $(n+k)$-categories, but in all cases we have analyzed they do come out correct if you look at the pointed versions. More on this can be found in the appendix to n-categories and cohomology.

Examples

… appearances in TWF, filling gaps, the first extension to small $n$, the established literature on $(n,r)$-categories, the stabilisation hypothesis, the tangle hypothesis, the table with Lie algebroids and the like, the appendix to John's and Mike's paper; most of these individual tables will have their own pages; …

• $k$-tuply monoidal $n$-categories;
• $(n,r)$-categories;
• $k$-tuply groupal $n$-groupoids;
• $k$-tuply monoidal $(n,r)$-categories, a combination of all of the above;
• etc.