Higher dimensional pasting diagram of cubes

A pasting diagram is the analogue of composition of arrows in higher categories. The nlab page gives an example of a two dimensional diagram and how to compute it as a composite of two morphisms. I am looking for an analogue of this result in higher dimensions. i.e. two n-cubes sharing an n-1 dimensional face.

It's hard to draw the picture, but the guess for the answer, for example in 3 dimensions, is something like (1-morphism $\circ$ 3-morphism) +(2-morphism $\circ$ 2-morphism) +(3-morphism $\circ$ 1-morphism) where the first morphism is given by an edge of the first cube etc.

I didn't find the references on the nlab page really useful.

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The following nlab page should be relevant. ncatlab.org/nlab/show/compositions+in+cubical+sets – Ronnie Brown Nov 29 '13 at 21:39

The use of cubical compositions is developed extensively in the book Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, where a pdf is available, and which gives an exposition of certain cubical methods in algebraic topology developed since the 1970s. A strong advantage of cubical methods is the ease of defining multiple compositions using a matrix notation: one has $2$ partial compositions say $+_1, +_2$ in dimension $2$ and an array $(a_{ij})$ is said to be composable if all compositions $a_{ij} +_1 a_{i+1,j}, a_{ij}+_2 a_{i,j+1}$ are defined for the appropriate $i,j$. Then, assuming the interchange law, one writes $[a_{ij}]$ for the composition. More analysis of this use of the interchange law is available in this paper.

We have not been able to develop in the globular or simplicial contexts proofs analogous to many of the results given in the referenced book.

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I know also this: arxiv.org/abs/1008.1714 – Buschi Sergio Nov 27 '13 at 16:24
Beautiful diagrams! I am completely put to shame with my finger scribbles :-) – Adam Gal Dec 11 '13 at 10:16

There are some graphic calculations in dimension 3 and 4 in my paper Combinatorics of branchings in higher dimensional automata (p348-358 in dimension 3 and p361-362 with a 4-cube).

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If you orient the cubes properly, you can pretty easily see the structure of morphisms on them. See for instance Gray's paper on coherence in bicategories.

E.g. For a pair of three cubes you get almost what you wrote, except the middle term is actually two such (commuting) terms.

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