TITLE: Is it possible to approach higher categories from the point of view of the arrow functor?
QUESTION [7 upvotes]: I have not read anything thoroughly about higher categories. I am only aware that in higher categories, we have higher dimensional cells, after adjusting for the intuition that 0-, 1-, 2-dimensional cells are respectively, objects, morphisms, and commuting squares. This terminology is similar to what we from arrow construction, as it follows.

Given a category $\mathcal{C}$, the objects of an arrow category $\mathcal{C^{\rightarrow}}$ are morphisms of $\mathcal{C}$ and morphisms of $\mathcal{C}$ are commuting squares. Another way to think of an arrow category is by considering  $\mathcal{C^{\rightarrow}}$ as  $\mathcal{C}^{2}$, the category of functors from $2$ to $\mathcal{C}$, where $2$ is the category $0 \longrightarrow1$ and a morphisms of $F: 2 \longrightarrow \mathcal{C}$ and $G: 2 \longrightarrow \mathcal{C}$ is a natural transformations of $F$ and $G$; this is actually a functor $\phi:2 \times 2 \longrightarrow \mathcal{C}$. 
  Moreover, this construction is functorial, i.e., it yields an endofunctor $A$r: Cat $\rightarrow$  Cat, which I call the arrow functor.

Now, for a given category $\mathcal{C}$ (just for the purpose of this question) let $\mathcal{C} ^ {\stackrel{\rightarrow} {n}}$ denote the category which is obtained by an $n$-times iteraion of Ar, with the convention that $\mathcal{C}^{\stackrel{\rightarrow} {1}}$ is the usual arrow category and $\mathcal{C}^{\stackrel{\rightarrow} {0}}$ is $\mathcal{C}$ itself. So, for example the objects and morphisms of $\mathcal{C}^{\stackrel{\rightarrow} {2}}$ are respectively pairs of morphisms and commuting squares in $\mathcal{C}^{\stackrel{\rightarrow} {1}}$. But, as morphisms of $\mathcal{C}^{\stackrel{\rightarrow} {1}}$ are again commuting squares in $\mathcal{C}^{\stackrel{\rightarrow} {0}}$, we can think of objects and morphisms of $\mathcal{C}^{\stackrel{\rightarrow} {2}}$ respectively as commuting squares and commuting cubes in $\mathcal{C}$. Generally, objects and morphisms of $\mathcal{C}^{\stackrel{\rightarrow} {n}}$ are respectively n-dimensional and n+1-dimensional cubes in $\mathcal{C}$, where an $i$-dimensional cube is a functor $$F:2 \times 2 \times ...\times 2 \longrightarrow \mathcal{C}$$ 

Question: Can $\mathcal{C}^{\stackrel{\rightarrow}{n}}$ model higher categories? What about the category $\mathcal{C}^{\stackrel{\rightarrow}{\infty}}$ which has all $\mathcal{C}^{\stackrel{\rightarrow}{i}}$ (for $i \in \mathbb{N}$), while  as objects and functors between them as morphisms. (Not too sure about this. What would be an appropriate class of morphisms between $\mathcal{C}^{\stackrel{\rightarrow}{n}}$ and $\mathcal{C}^{\stackrel{\rightarrow}{m}}$ anyway?) Please, let me know if this construction is of any relevance to higher category theory. This might help me learning some higher category theory.

REPLY [3 votes]: (source)
@David: An advantage of strict $n$-fold categories is that they can express in a simple way the above diagram in the form of "the big square is the composition of the little squares"; namely one defines a composable array $(a_{ij})$ to be an array of elements such that each is composable with its immediate neighbours. Then by associativity and the interchange law, the composite $[a_{ij}]$ is well defined. This "algebraic inverse to subdivision", which is easily extended to higher dimensions,  is used extensively by Higgins and me in proving a Higher Homotopy Seifert-van Kampen theorem, in the paper available here. I do not know how to develop analogous methods in the globular or simplicial contexts; simplicial methods, but not of the higher categorical type,  are used by Brown and Loday to prove their version of a HHSvKT.  All these bring nonabelian colimit methods into homotopy theory. 
I'll also mention that Brown and Higgins proved a groupoid result on the equivalence between strict cubical and globular $\omega$-groupoids, and the analogous but more difficult category result was proved by Al-Agl, Brown, and Steiner. The references are on the (updated)  ncatlab page given by David. 
Later: for complete clarity, though I expect it will be clear, I just add that if $a_{ij}$ is not at the edge of the square, then its immediate neighbours are: $a_{i-1,j}, a_{i+1,j}, a_{i,j-1}, a_{i,j+1}$, and the rule for composability of two squares is that their appropriate edges agree. My papers with Higgins contain lots of 2-dimensional rewriting, using the interchange law; and the paper with Al-Agl and Steiner contains a 3-dimensional rewriting argument to obtain a braid relation.