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.



What is the intuition of connections for cubical sets?
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/liverpool-beamer-handout.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 Brown-Higgins papers, but as they have been used as the base for the new book, they may not give you anything extra.


