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.

9$\begingroup$ cubical sets, presumably, unless you're forming connections between graduate students. $\endgroup$ – Noam D. Elkies Nov 27 '11 at 3:14

$\begingroup$ Thank you. I think that I have fixed all of the offenders. $\endgroup$ – Spice the Bird Nov 27 '11 at 3:22

1$\begingroup$ Just to add a comment to the points below: in a simplicial set, a degenerate simplex has some adjacent faces the same. In a cubical set, a degenerate cube has opposite faces the same. The extra structure of connections brings cubical sets nearer to simplicial sets, but keeping other advantages, such as easily understood definitions of compositions. For another application, see Higgins, P.J. Thin elements and commutative shells in cubical {$\omega$}categories. Theory Appl. Categ. 14 (2005) No. 4, 6074 $\endgroup$ – Ronnie Brown Jan 31 '12 at 22:43
A list of precise references for connections on cubical sets has to start with :
R. Brown, P. J. Higgins and R. Sivera, 2011, Nonabelian Algebraic Topology: Filtered spaces, crossed complexes, cubical homotopy groupoids , volume 15 of EMS Monographs in Mathematics , European Mathematical Society.
as in there Brown, Higgins and Sivera have written out and explored the theory in detail. There are several introductory sections on connections both in double categories and in cubical sets. The intuitions come back to the structure of the singular cubical complex of a space in which there are cubes that are degenerate in an intuitive sense but are not of the 'constant in direction $i$' type. The typical example is a square with two adjacent sides constant and the other two copies of the same path. (I cannot draw it here!)
Ronnie Brown has numerous introductory articles on his website and I will give you a link to the handout for a talk on higher dimensional group theory in which there is some discussion of the connections from a group theoretic viewpoint.(http://groupoids.org.uk/pdffiles/liverpoolbeamerhandout.pdf) The discussion is fairly far near the end, so have a look for diagrams with cubes and hieroglyphic pictures!
The point made there is that if you want to say that the top face of a cube is the composite of its other faces, then on expanding the cube as a cross shape collection of five squares, there will be holes to fill in the corners, but connection squares are just the right form to fill them. (It is worth roaming around on Ronnie Browns site including http://groupoids.org.uk/brownpr.html, as there are several other chatty papers and Beamer presentations that may help.)
You can go back to the original BrownHiggins papers, but as they have been used as the base for the new book, they may not give you anything extra.

$\begingroup$ That URL doesn't seem to work, Tim  do you mean this one: pages.bangor.ac.uk/~mas010/publicfull.htm ? $\endgroup$ – Finn Lawler Nov 27 '11 at 19:04

$\begingroup$ @Finn, Strange. The link does work for me,. There is another :pages.bangor.ac.uk/~mas010/index.html and the original one was some way down that page. $\endgroup$ – Tim Porter Nov 28 '11 at 6:22

$\begingroup$ I got a 404 error when I clicked on it yesterday, but now it works. Odd. Never mind; lots of nice stuff there to get distracted by! $\endgroup$ – Finn Lawler Nov 28 '11 at 12:21

$\begingroup$ Those bangor.ac.uk addresses are all dead now. If someone wants to see equivalent content look at the second link in the body of myanswer or at groupoids.org.uk/publicfull.html $\endgroup$ – Tim Porter Nov 4 '18 at 17:15
The point I want to make was that the notion of connection on cubical set was forced on us in the following way.
Go back to
[21]. (with C.B. SPENCER), ``Double groupoids and crossed modules'', Cah. Top. G\'{e}om. Diff. 17 (1976) 343362.
The problem we started with in 1971 was: since double groupoids were putative codomains for a 2d van Kampen type theorem, were there interesting examples of double groupoids?
We easily found functors
(1) (double groupoids) $\to $ (crossed modules)
We eventually found a functor
(2) (crossed modules) $\to$ (double groupoids)
which nicely tied in double groupoids with classical ideas, but which double groupoids arose in this way? A concurrent question was: what is a commutative cube in a double groupoid? (An answer was needed for the conjectured proof of the 2d vKT.)
It was great that both questions were resolved with the notion of connection! (our first perhaps rambling exposition was turned down by JPAA as a result of negative referee reports, and because the 2d van Kampen theorem, an explicit aim, was not yet achieved). As explained in [21] the transport law was borrowed from a paper of Virsik on path connections, hence the name `connection', see also [21] for a general definition.
It was not too hard to formulate the higher dimensional laws on connections, since they involved the monoid structure max in the unit interval, but the verification of the equivalence corresponding to (2) was carried out by Philip Higgins, (phew!), stated in
[22]. (with P.J. HIGGINS), ``Sur les complexes crois\'es, $\omega$groupo\"{\i}des et Tcomplexes'', C.R. Acad. Sci. Paris S\'er. A. 285 (1977) 997999.
and published in
[31]. (with P.J. HIGGINS), ``On the algebra of cubes'', J. Pure Appl. Algebra 21 (1981) 233260.
I hope the early pages of the book partially titled Nonabelian Algebraic Topology (EMS Tract vol 15, 2011) (pdf with hyperref downloadable from my web site, with permission of EMS) will help to explain the background. Look at particularly the notion of algebraic inverse to subdivision, which necessitated the cubical approach.
Another relevant paper is
Higgins, P.J. Thin elements and commutative shells in cubical $\omega$categories. Theory Appl. Categ. 14 (2005) No. 4, 6074.
.

$\begingroup$ Also relevant are: Grandis, Marco; Cubical monads and their symmetries. Proceedings of the Eleventh International Conference of Topology (Trieste, 1993). Rend. Istit. Mat. Univ. Trieste 25 (1993), no. 12, 223–262 (1994). Grandis, Marco; Mauri, Luca Cubical sets and their site. Theory Appl. Categ. 11 (2003), No. 8, 185–211 $\endgroup$ – Ronnie Brown Nov 28 '11 at 15:00
A very concrete instance where you can see the meaning and usefulness of connections is this article by Brown and Mosa: They show that double categories (which do have an underlying (truncated) cubical set) with connections are the same as (globular) 2categories.
The reason is that the connection allows to fold the four different edges of a 2cell in the cubical double category structure into just two edges, leaving degenerate edges at the other sides, and this can as well be captured in the data of a globular 2category where 2cells have just one source and one target 1cell  see the definition of he folding map right before Proposition 5.1 in the above article.
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 ncubes in an ntuple category with connection. To do so, you can take an ncube, 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.

$\begingroup$ With regard to the impact on cubical sets, see Tonks, A. Cubical groups which are Kan. J. Pure Appl. Algebra 81 (1992) 8387. Maltsiniotis, G. La cat\'egorie cubique avec connexions est une cat\'egorie test stricte. Homology, Homotopy Appl. 11 (2009) 309326. The first paper shows that cubical groups with connections are Kan complexes. The second shows that the geometric realisation of the cartesian product of cubical sets with connection is homotopy equivalent to the cartesian proiduct of the realisations. This is good, though not quite as convenient as the simplicial case. $\endgroup$ – Ronnie Brown Nov 30 '11 at 11:33

$\begingroup$ I'll add a couple of later references. Homology Homotopy Appl. Volume 14, Number 1 (2012), 133158. Cubical approach to derived functors Irakli Patchkoria $\endgroup$ – Ronnie Brown Jun 8 '15 at 9:26

$\begingroup$ Vezzani (arXiv:1405.4508) and cited papers there for relevance to motivic theory. In this area the corresponding simplicial methods have disadvantages! See also 116. (with F.A. ALAGL and R. STEINER), `Multiple categories: the equivalence between a globular and cubical approach', Advances in Mathematics, 170 (2002) 71118. $\endgroup$ – Ronnie Brown Jun 8 '15 at 9:33