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
opetopes, aka multitopes.
There are corresponding categories whose
objects are the “standard cellular shapes” of the given sort: globes, simplices, cubes, respectively. For globes, simplices, and cubes, there is one object for each natural number $n \in \mathbb{N}$, so they are usually denoted $[n]$. For other sorts of shapes, there can be multiple shapes of each dimension.
morphisms are generated from (some of) the 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 $\mathbb{G}$
the simplex category $\Delta$
the cube category $\square$
the tree category $\Omega$
the opetope category? $\mathbb{O}$
In general, there are two types of morphism in such categories, called face inclusions (or cofaces) and degeneracies. Face inclusions exhibit a lower-dimensional cell as a “face” of a higher-dimensional one, while degeneracies collapse a higher-dimensional cell down to a lower-dimensional one. All such categories have face inclusions (these are the really basic “structural” maps that determine the shapes of the cells), but some (such as $\Delta$, $\square$, and $\Omega$) include degeneracies while some (such as $\mathbb{G}$ and $\mathbb{O}$) do not usually.
A general notion which includes all of these categories is that of a generalized Reedy category. If degeneracies are excluded, then we obtain the simpler notion of a (generalized) direct category.
A higher structure based on these geometric shapes is a presheaf on one of these categories. These are called
respectively.
There are also other notions of geometric shape which have been found useful in higher category theory, such as the shapes encapsulated in Joyal’s disk category $\Theta$ (which include both globes and simplices as special cases).
Many definitions of higher categories are given by 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, the usual definition of strict ∞-categories is based on giving globular sets extra structure, as is the weak version of Batanin ∞-category?. Similarly, n-fold categories give extra structure to cubical sets. On the other hand, the notion of quasi-categories is based on simplicial sets, while other more “homotopical” notions of higher category are based on cubical or multisimplicial techniques. In general, the “geometric” definitions can usually be thought of as a nerve of the “algebraic” ones.
Sometimes, two definitions that use different kinds of shapes nevertheless capture equivalent notions. This is hoped to be true for all proposed definitions of n-category, although they use many different kinds of shapes. On the other hand, sometimes the use of different shapes indicates a fundamentally different structure being considered. For instance, edge-symmetric $n$-fold categories with connections give the same notion as globular $n$-categories, while arbitrary $n$-fold categories are something quite different. Moreover, even in the case where different kinds of shapes produce “equivalent” notions, it can sometimes be markedly easier to work with one kind of shape than another for some particular application.
See for instance
directed complex?
This Week’s Finds in Mathematical Physics (Week 242) (Discussion at the n-Cafe)
On a type-theoretic approach to formalizing higher structures: Nicolai Kraus: seminar slides, 2017