nLab periodic table

Idea

In higher category theory, higher algebra, homotopy theory, and homotopy type theory there are several periodic tables, analogous to 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, n-groupoids, and n-truncated types.

History

The first periodic table of n-categories, due to John Baez and James Dolan in BaezDolan95, was the following. Originally, Baez and Dolan referred to it as a periodic table of $k$-tuply monoidal $n$-categories, as opposed to simply of $n$-categories, but in today’s terminology the periodic table of $k$-tuply monoidal $n$-categories would be the one at the page k-tuply monoidal n-category. That page contains various notes on the table.

Fully filled out, the table looks 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	...
⋮	"	"	"	"	"	⋱

The columns with $n = -1$ and $n = -2$ were not originally there, but they follow the pattern of the table.

Eugenia Cheng and Nick Gurski wrote a paper observing that the above table does not quite behave ‘correctly’ with respect to higher morphisms if one restricts to $(k-1)$-simply connected $(n+k)$-categories, but things do appear to come out correctly if one looks at the ‘pointed’ version of the table at k-tuply monoidal n-category. More on this can be found in the appendix to n-categories and cohomology.

Examples

• $k$-tuply monoidal $n$-categories
• $k$-tuply groupal $n$-groupoids
• $(n,r)$-categories
• $k$-tuply monoidal $(n,r)$-categories, a combination of all of the above
• directed $(n,r)$-graphs, which are a generalization of the above
• From the perspective of higher algebra, homotopy theory, and homotopy type theory, $n$-groupoids are fundamental, and $(n,r)$-categories and directed $(n,r)$-graphs are $n$-groupoids with extra higher structure. So a $k$-tuply monoidal $n$-groupoid is a combination of all of the above.
• $k$-tuply associative $n$-groupoids

Related entries

• stabilization hypothesis

• delooping hypothesis

References

• John Baez, James Dolan, Higher-dimensional algebra and topological quantum field theory, J. Math. Phys. 36, No. 11, 6073-6105 (1995), arXiv:q-alg/9503002, doi