nLab
geometric shapes for higher structures

Handling higher structures such as higher categories usually involves conceiving them as conglomerates of cells of certain shape.

Examples for possible shapes used to model higher categories are

• globes;

• simplices;

• cubes.

• trees.

• opetopes.

There are corresponding categories whose

• objects are the “standard cellular shapes” of the given sort: globes, simplices, cubes, respectively, one for each natural number $n\in \mathbb{N}$ and usually denoted $[n]$

• morphisms are generated from all possible ways of mapping these standard cellular shapes to each other such that their cellular structure is preserved;

• and composition of such morphisms is subject to the relations which are inherited from the geometric meaning of these maps, which says for instance that the left boundary of the top boundary of a cube is the same as the top boundary of its left boundary – these are the globular identities, the simplicial identities and the cubical identities, respectively.

The resulting categories of basic cellular shapes are

• the globe category $\circ$;

• the simplex category $▵$;

• the cube category $\square$.

• the tree category $\Omega$

A higher structure based on these geometric sheapes is a presheaf on one of these categories. These are called

• globular sets;

• simplicial sets;

• cubical sets,

• dendroidal set

respectively.

Other notions of geometric shape which have been found useful in higher order category theory include opetopes (based on operad theory, and invented by Baez-Dolan), multitopes? (due to Hermida-Makkai-Power), and the shapes encapsulated in Joyal’s disk category $\Theta$ (which include globes and simplices as special cases).

In many definitions of higher categories an infinity-category is one of these presheaves

• either equipped with extra properties in the geometric definition of higher categories;

• or equipped with extra structure in the algebraic definition of higher categories.

For instance omega-categories are based on globular sets and n-fold categories on cubical sets, while most geometrically defined higher categories such as quasi-categories are based on the simplicial sets, thought of as a nerve.

However, Ronnie Brown writes: For the work on higher homotopy groupoids and their applications to higher homotopy van Kampen theorems we found cubical methods essential. In the first of the following papers, we use a higher homotopy cubical $\omega$-groupoid with connections of a filtered space, while the second paper uses a fundamental cat${}^n$-group of an $n$-cube of pointed spaces, giving an $n$-fold groupoid in the category of groups. The setting up of these structures is non-trivial, highly geometric and essential for the homotopical applications. The paper with Loday also uses multisimplicial techniques.

References

• R. Brown and P. J. Higgins, Colimit theorems for relative homotopy groups, J. Pure Appl. Algebra, 22 (1981) 11-41.

• R. Brown and J.-L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987) 311-334.

• This Week’s Finds in Mathematical Physics (Week 242) (Discussion at the n-Cafe)