What are the relative advantages of n-fold categories and n-categories?

Question

For example 2-categories seem simpler at first compared to double categories because the latter is a "wider environment" (cf Bertozzini). However when doing calculations, many people prefer to use double categories (cf Brown) and when we describe them in the 2-arrows-only language it seems that double categories have fewer axioms. (Recall the theorem of Brown-Mosa-Spencer that the category of 2-categories is equivalent to the category of edge symmetric double categories with connection, and the theorem by Ehresman that all double categories can be embedded in an edge symmetric double category.)

It seems that more work has been done on n-categories though, why is that? Is it easier to start with n-categories?

Answer 1

I think there are good technical reasons for preferring one particular mode in certain situations, depending on how easily certain concepts are expressed. For me, the main intuition since 1965 was based on the diagram

^(source)

and the idea that the big square should be the composition of all the little squares. This I termed "algebraic inverses to subdivision". Subdivision is an important tool in mathematics for local-to-global problems, which are themselves an important range of problems in mathematics and its applications. I found that Ehresmann's notion of double category, or groupoid, was very suited to express this notion, and was easy to generalise to higher dimensions. This led to proofs of what we now call Higher Homotopy Seifert-van Kampen Theorems, and for which the globular notions were not of any help.

The notion of strict higher cubical category or groupoid is also useful for formulating and proving monoidal closed structures, due to the rule $I^m \times I^n \cong I^{m+n}$, see the final section of this paper in Advances of mathematics, 170 (2002) 71--118..

The paper Ellis, G.~J. and Steiner, R., Higher-dimensional crossed modules and the homotopy groups of $(n+1)$-ads. J. Pure Appl. Algebra 46 (1987) 117--136, relates certain $(n+1)$-fold groupoids, i.e. those in which one structure is a group, to a fascinating structure called crossed $n$-cube of groups, and which is closely related to classical ideas in the homotopy theory of $n$-ads, see particularly Theorems 3.7,3.8, which have not been obtained by other methods.

On the other hand, to discuss the notion of commuting cube in a strict cubical category with connections, the relation with the globular case was crucial, see this paper by Higgins.

The notions of the globular, simplicial, or cubical sites have been well studied. I am not sure that globular sets are very convenient. What seems not to have been well studied, or even studied, is the underlying geometric site for $n$-fold categories, since it is the geometry of cubes in which all the directions are distinct, so the direction $i$ faces of a cube are distinct from the direction $j$ faces if $i \ne j$.

Also weak cubical categories do not seem much studied, though the classical example is the cubical singular complex of a space.

Answer 2

I believe is more a matter of tastes, personally I find easier and simpler n-fold categories than categories.

For me n-fold categories are more natural and so are easier, for different reasons: for start one interesting thing is the various sources and targets of the composition are given just by $k-1$-cells (faces), where for $n$-categories sources are given by a $i$-cells for each $i < k$, this gives an intuitive representation of $k$-cells as $k$-dimensional cubes, with orientation for each pair of opposite faces, and a representation of composition as pasting cubes along the faces coherently with these orientations. To do something similar with $n$-categories you should work with $k$-cells as $k$-globes and see compositions as a sort of pasting of globes which involves also deformations of such globes and so (at least by me) it's a little more difficult to figure.

On the other end this cubical approach has proven to be more easier to write computations: consider the case of fundamental group in which to do computations it usually preferred to use maps from the cubical interval rather then maps from the spheres.

Another point in favor of n-fold categories is that every n-categories can be seen as a n-fold category in which every cells have collapsed faces.

I something else come to my mind I reserve the right to add something later. :)