In a cubical set, you are guaranteed for every $n$-cell (which can be drawn as a 1-cell)
that there is the identity $(n+1)$-cell (which can be drawn as a 2-cell) of the form
A cubical set is said to have connections if in addition it has for every $n$-cell $a\stackrel{f}{\to}b$ also $(n+1)$-cells of the form
And so forth. You should think of this as saying that the “thin” cell
is regarded as a degenerate cube by the cubical set in all the possible ways.
So it’s a very natural condition, particularly if you think of all these cubical cells as cubical paths in some space.
If $K= \{K_n| n \geq 0\}$ is a cubical set, then a connection structure on $K$ consists of functions $\Gamma^+_i, \Gamma^- _i: K_n \to K_{n+1}$, $i=1, \ldots \, , n; n \geq 1$, satisfying the relations for $\alpha, \beta=\pm$:
$\Gamma^\alpha_i\Gamma^\beta_j= \Gamma^\beta_{j+1} \Gamma^\alpha _i$ if $i \lt j$;
$\Gamma^\alpha_i\Gamma^\alpha_i= \Gamma^\alpha_{i+1} \Gamma^\alpha _i$;
$\partial^\alpha_j \Gamma^\alpha_j= \partial ^\alpha_{j+1} \Gamma ^\alpha_j = id$;
$\partial^\alpha_j \Gamma^{-\alpha}_j= \partial ^\alpha_{j+1} \Gamma ^{-\alpha}_j = \varepsilon _j \partial^\alpha_j$;
$\partial^\alpha_i \Gamma ^\beta_j = \begin{cases}\Gamma^\beta_{j-1} \partial ^\alpha _i & \text{if }\; i \lt j \\ \Gamma^\beta_j \partial ^\alpha _{i-1} & \text{if }\; i \gt j+1; \end{cases}$
$\Gamma^\alpha_j \varepsilon_j = \varepsilon^2_j = \varepsilon_{j+1}\varepsilon _j$;
$\Gamma^\alpha_i \varepsilon _j = \begin{cases} \varepsilon_{j+1} \Gamma^\alpha _i & \text {if }\; i \lt j \\ \varepsilon _j \Gamma^\alpha_{i-1} & \text{if }\; i \gt j ; \end{cases}$
The connections are to be thought of as “extra degeneracies”. A degenerate cube of type $\varepsilon_j x$ has opposite faces equal and all other faces degenerate.
A cube of type $\Gamma_i^\alpha x$ has a pair of adjacent faces equal and all other faces of type $\Gamma_j^\alpha y$ or $\varepsilon_j y$ . So this makes the cubical theory nearer to the simplicial. Cubical complexes with this, and other, structures have also been considered by Evrard.
The first appearance of this notion in dimension $2$ was in the paper by Brown and Spencer listed below, and used to obtain an equivalence between crossed modules and edge symmetric double groupoids with connection.
Such connections on cubical sets were introduced in 1981 by Brown and Higgins in order to obtain the equivalence of their “cubical ω-groupoids” with crossed complexes. They are also essential to allow the notion of “commutative $n$-shell” in such a structure.
It would seem that…
Alternatively, let $\mathcal{P}_{\subset}$ be the category whose objects are finite sets with morphisms from $m$ to $n$ the functions $\mathcal{P}m \to \mathcal{P}n$ preserving $\subseteq$. There is a faithful essentially-surjective functor $J : \box \to \mathcal{P}_{\subset}$ such that restriction of presheaves along $J$ is exactly “forget connection structure”.
There is also a faithful representation of $\mathcal{P}_{\subset}$ in polyhedra-with-boundary and piecewise-linear maps, valued on objects $n\mapsto [0,1]^n$ with the coordinate-ordered simplicial subdivision of $[0,1]^n$. This induces a connection structure on the singular cubical set of a topological space.
The ordinary cube category is a test category. This means that bare cubical sets carry the structure of a category with weak equivalences whose homotopy category is that of ∞-groupoids.
But the category of cubes with connection is even a strict test category (Maltsiniotis, 2008). This means that under geometric realization (see the discussion at homotopy hypothesis) the cartesian product of cubical sets with connection is sent to the correct product homotopy type.
The lack of this property for cubical sets without connection was one of the original reasons reasons for abandoning Kan’s initial cubical approach to combinatorial homotopy theory in favour of the simplicial approach; the implications of this new result have yet to be thought through. Another reason was that cubical groups were in general not Kan complexes; however cubical groups with connection are Kan complexes. See the paper by Tonks listed below.
The prime example of a cubical set with connections is the singular cubical complex $KX$ of a topological space $X$. Here for $n \ge 0$ $K_n$ is the set of singular $n$-cubes in $X$ (i.e. continuous maps $I^n \to X$) and the connection $\Gamma_i^\alpha :K_{n } \to K_{n+1}$ is induced by the map $\gamma_i^\alpha : I^{n+1} \to I^{n}$ defined by
where $A(s,t)=\max(s,t), \min(s,t)$ as $\alpha=-,+$ respectively.
The first hint of such a general structure came in the paper by Brown and Spencer given below. The term “connection” was used there because of a relation of a generalisation of this idea to path-connections in differential geometry. A principal $G$-bundle $E$ over $B$ gives rise to the Ehresmann groupoid $Equ(E)$ of $G$-maps between the fibres, and the Moore paths $\Lambda$ on this form a double category $D$ with $Equ(E)$ and $\Lambda(B)$ as edge categories. A connection $\Gamma$ is then a functor from $\Lambda(B)$ to one of the category structures on $D$ which gives a smooth lifting of paths to transport of the fibres. This is the origin of the term transport law? for the relation of connections to composition.
Discussion
There is a discussion of cubical vs simplicial singular homology and for other aspects at (this mathoverflow).
A complete cubical approach to algebraic topology at the border between homotopy and homology is given in the book discussed in these pages at Nonabelian Algebraic Topology see (here). As an example, the theory shows how the Relative Hurewicz Theorem? follows from a Higher Homotopy Seifert-van Kampen Theorem, withut using singular homology.
Ronnie Brown and C.B. Spencer, “Double groupoids and crossed modules’’, Cah. Top. Géom. Diff. 17 (1976) 343–362.
Evrard, M., “Homotopie des complexes simpliciaux et cubiques”, Preprint(1976).
Brown, R. and Higgins, P.J., “On the algebra of cubes”, J. Pure Appl. Algebra 21 (1981) 233–260.
F. Al-Agl, R. Brown and R. Steiner, “Multiple categories: the equivalence between a globular and cubical approach”, Advances in Mathematics, 170 (2002), 71–118.
M. Grandis and L. Mauri, “Cubical sets and their site”, Theory Applic. Categories, 11 (2003) 185–201.
P.J. Higgins, “Thin elements and commutative shells in cubical $\omega$-categories”, Theory Appl. Categ. 14 (2005) 60–74.
The statement that cubical groups with connections are Kan complexes is due to
Cubical sets with connection are used in the following paper:
and in the following in preference to simplicial methods to take resolutions in an analytic motivic setting:
J. Ayoub - “L’algèbre de Hopf et le groupe de Galois motiviques d’un corps de caractéristique nulle, I” pdf,
A Vezzani - “A motivic version of the theorem of Fontaine and Wintenberger” arXiv:1405.4548,
The statement that cubes with connection form a strict test category is due to
based on