Higher-Category Periodic Table

The Baez–Dolan periodic table (Baez–Dolan 1995; see also the nLab entry) organises the higher-categorical world along two axes and reveals that many familiar algebraic structures are the same structure seen at different heights. Its slogan is degeneracy. A structure with only one object of the lowest dimension is really a structure one level up. A monoidal category is a -category with a single object, since its objects become endomorphisms and its tensor becomes composition, and iterating this idea generates a table whose entries stabilise.

Write for the categorical dimension and for the number of times the structure is tuply monoidal, so that a -tuply monoidal -category is a -category with a single cell in each dimension below . Reading up a column adds monoidal layers, and the layers eventually stop making a difference.

	set	category	-category
	monoid	monoidal cat.	monoidal -cat.
	comm. monoid	braided monoidal	braided monoidal
	comm. monoid	symmetric monoidal	sylleptic monoidal
	comm. monoid	symmetric monoidal	symmetric monoidal

Thus a braided monoidal category is a -category with one object and one -morphism, and a symmetric monoidal category is the stabilised entry of its column.

Proposition (stabilisation hypothesis)

Along each row of fixed , the entries stabilise once . Adding a further monoidal layer beyond this bound changes nothing up to equivalence. The stabilised structure is the symmetric monoidal -category, and the transition is where extra swaps stop buying new information.

Remark (enrichment reading)

Each downward step increasing can be read as one-object degeneracy, and each rightward step increasing as raising the level of the hom-data. From the enriched-category viewpoint the columns are towers of hom-objects. A monoidal category is a one-object category enriched so that its single hom carries the tensor as composition.