I am beggining to do some work with cubicle cubical sets and thought that I should have an understanding of various extra structures that one may put on cubicle cubical sets (for purposes of this question, connections). I know that cubicle 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.

I am beggining to do some work with cubicle sets and thought that I should have an understanding of various extra structures that one may put on cubicle sets (for purposes of this question, connections). I know that cubicle 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.