# nLab periodic table

Contents

### Context

#### Algebra

higher algebra

universal algebra

## Theorems

#### Monoid theory

monoid theory in algebra:

#### Higher category theory

higher category theory

## 1-categorical presentations

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# Contents

## 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$...
$0$trivialtruth valuesetcategory2-category...
$1$"trivialmonoidmonoidal categorymonoidal 2-category...
$2$""abelian monoidbraided monoidal categorybraided monoidal 2-category...
$3$"""symmetric monoidal categorysylleptic 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

## References

Last revised on May 16, 2022 at 17:36:30. See the history of this page for a list of all contributions to it.