# In your opinion, what are the relative advantages of n-fold categories and n-categories?

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 less 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?

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I cannot give an answer, but usually what is studied is what appears "in nature": maybe n-cats just appear more often in nature than n-fold cats?... – Qfwfq Jan 28 '13 at 11:53
There can be different (technical) correct answers depending on people's experiences. – Rachel Jan 28 '13 at 13:24
There is quite a lot of mathematics related to your question. In the future, please try look for other titles --- "in your opinion" is a poor way to frame MathOverflow discussions. – Theo Johnson-Freyd Jan 28 '13 at 15:54
OK thanks, i think I'm starting to get the hang of it...if I had deleted the first 3 words and just asked "What are the relative advantages..." then it would have been OK, and following with, "Why is it that since n-fold categories are deemed to be more efficient, more work seems to have been done on n-categories?" – Rachel Jan 29 '13 at 14:39
@Rachel: Yes, I think the title without the first three words would be mildly better. I also think the question has a few places that approach (but by no means surpass) the "argumentative/subjective" threshold. In any case, don't take my comment too strongly --- aesthetics is certainly one of the driving forces in n-category theory, and perhaps in all of mathematics. – Theo Johnson-Freyd Jan 30 '13 at 5:21

## 2 Answers

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

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.

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Ronnie, the link to the Advances paper returns a "not found" error. – Vidit Nanda Jan 28 '13 at 16:26
Thanks Vel. Corrected. (It weas on my computer but had not been copied over.) – Ronnie Brown Jan 28 '13 at 17:12
Adding to Ronnie's comment on crossed n-cubes of groups. These are very very easy to define and give models for homotopy n+1-types. The corresponding n+1-groupoid will have a lot of seemingly extra structure whose details are not that clear once you get to n= 4. The reason, intuitively, is that the crossed n-cube lays things out for you, but at the cost of a lot of repetition of the information, while the n-category folds it all into a small space, so naturally things interact in an apparently more complex way. – Tim Porter Jan 29 '13 at 8:46

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. :)

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One of the matters of taste is to have first definitions which are symmetric and without choices, and this applies to $n$-fold categories. This aesthetic advantage, and others, apply to the use of the homotopy double groupoid of a pair of spaces, rather than the second relative homotopy group as a crossed module. Of course for special reasons, such as I mentioned above, and also for computation, one often needs to move to an asymmetric situation. – Ronnie Brown Jan 28 '13 at 21:01