# The periodic table of higher categories

I'll explain the periodic table of $k$-boring $n$-categories. An $n$-category is a higher algebraic structure modelling fields on $n$-dimensional spacetime; most of the talk will

be informal about the precise underlying model. An $n$-category is $k$-boring if the lowest $k$ levels of structure are trivial.

In low dimensions, "boringness" seems to be about commutativity: $1$-boring $1$-categories are monoids, and $2$-boring $2$-categories are commutative monoids.

Going up a level, $1$-boring $2$-categories are monoidal categories, $2$-boring $3$-categories are braided monoidal categories, and $3$-boring $4$-categories are symmetrical monoidal categories.

The appearance of braided monoidal categories in that sequence shows that there are "intermediate levels" of commutativity, which are intimately related to topology.

There's a lovely construction, $Z$, simultaneously generalising the centre of monoid, the Drinfeld centre of a monoidal category, and the Mueger centre of a braided category.

It takes in a $k$-boring $n$-category and produces a $(k+1)$-boring $(n+1)$-category, "making it more commutative". We'll spend a while understanding this construction.

There are two amazing features of this construction. First, there is a "stabilisation" phenomenon, that says that for "sufficiently boring" categories, $Z$ is in some sense an equivalence.

There are proofs of this fact in certain models of higher categories, but I don't much like them. Stabilisation kicks in once $k \geq \frac{n}{2} + 1$, and the secret explanation for this

should be based on transversality of subcomplexes of $n$-dimensional space.

Second, there is a "homological" phenomenon, that with some (as yet not known) extra conditions, it seems that $Z^2 = 0$, and that the homology groups of the periodic table are very interesting

algebraic gadgets. I don't understand this at all well!