TITLE: What is the intuition of connections for cubical sets?
QUESTION [6 upvotes]: I am beggining to do some work with cubical sets and thought that I should have an understanding of various extra structures that one may put on cubical sets (for purposes of this question, connections). I know that cubical sets behave more nicely when one has an extra set of degeneracies called connections. The question is: Why these particular relations? Why do they show up? Precise references would be greatly appreciated.

REPLY [3 votes]: As far as higher cubical categories are concerned, a connection will allow you to literally rotate a face, i.e. turn a face of one type into a face of another type, in an invertible way.
In short it materializes an equivalence between the different types of faces into special degenerate cubes.
The 2d case for example is fairly simple as one can either turn horizontal arrows into vertical arrows or vice versa.
One advantage of a connection is therefore that it allows one to speak of commutative n-cubes in an n-tuple category with connection. To do so, you can take an n-cube, apply connections until you only have non trivial faces of one type. Then check whether the obtained cube is an identity or not. It turns out that it does not depend on the way you chose to apply the connection, if your cube gives an identity cube with one face rearrangement, it will with another. It is, to my understanding the essence of Brown and AlAlg's equivalence between cubical categories with connections and globular categories.
So for cubical categories it is very restrictive, which is also why they are so friendly. But I am not sure about the impact on cubical sets. You surely will find good material in Tim and Ronnie's references.