nLab
connection on a cubical set

Idea

In a cubical set, you are guaranteed for every $n$-cell (which can be drawn as a 1-cell)

$$a\stackrel{f}{\to }b$$a\stackrel{f}{\to}b

that there is the identity $(n+1)$-cell (which can be drawn as a 2-cell) of the form

$$\begin{array}{ccc}a& \stackrel{f}{\to }& b\\ {\downarrow }^{\mathrm{Id}}& {\Downarrow }^{\mathrm{Id}}& {\downarrow }^{\mathrm{Id}}\\ a& \stackrel{f}{\to }& b\end{array}$$\array{ a & \stackrel{f}{\to} & b \\ \darr^{Id} & \Downarrow^{Id} & \darr^{Id} \\ a & \stackrel{f}{\to} & b }

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

$$\begin{array}{ccc}a& \stackrel{f}{\to }& b\\ {\downarrow }^f& \Downarrow & {\downarrow }^{\mathrm{Id}}\\ b& \stackrel{\mathrm{Id}}{\to }& b\end{array}$$\array{ a & \stackrel{f}{\to} & b \\ \darr^{f} & \Downarrow & \darr^{Id} \\ b & \stackrel{Id}{\to} & b }

And so forth. You should think of this as saying that the “thin” cell

$$a\underset{f}{\overset{f}{\rightrightarrows }}b$$a\underoverset{f}{f}{\rightrightarrows}b

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.

Definition

If $K=\{K_n\mid n\ge 0\}$ is a cubical set, then a connection structure on $K$ consists of functions ${\Gamma }_i^{+},{\Gamma }_i^{-}:K_n\to K_{n+1}$, $i=1,\dots ,n;n\ge 1$, satisfying the relations for $\alpha ,\beta =\pm$:

1. ${\Gamma }_i^{\alpha }{\Gamma }_j^{\beta }={\Gamma }_{j+1}^{\beta }{\Gamma }_i^{\alpha }$ if $i<j$;

2. ${\Gamma }_i^{\alpha }{\Gamma }_i^{\alpha }={\Gamma }_{i+1}^{\alpha }{\Gamma }_i^{\alpha }$;

3. ${\partial }_j^{\alpha }{\Gamma }_j^{\alpha }={\partial }_{j+1}^{\alpha }{\Gamma }_j^{\alpha }=\mathrm{id}$;

4. ${\partial }_j^{\alpha }{\Gamma }_j^{-\alpha }={\partial }_{j+1}^{\alpha }{\Gamma }_j^{-\alpha }={\epsilon }_j{\partial }_j^{\alpha }$;

5. ${\partial }_i^{\alpha }{\Gamma }_j^{\beta }=\{\begin{array}{ll}{\Gamma }_{j-1}^{\beta }{\partial }_i^{\alpha }& \text{if}i<j\\ {\Gamma }_j^{\beta }{\partial }_{i-1}^{\alpha }& \text{if}i>j+1;\end{array}$

6. ${\Gamma }_j^{\alpha }{\epsilon }_j={\epsilon }_j^2={\epsilon }_{j+1}{\epsilon }_j$;

7. ${\Gamma }_i^{\alpha }{\epsilon }_j=\{\begin{array}{ll}{\epsilon }_{j+1}{\Gamma }_i^{\alpha }& \text{if}i<j\\ {\epsilon }_j{\Gamma }_{i-1}^{\alpha }& \text{if}i>j;\end{array}$

The connections are to be thought of as “extra degeneracies”. A degenerate cube of type ${\epsilon }_{j}x$ 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 ${\epsilon }_{j}y$ . So this makes the cubical theory nearer to the simplicial. Cubical complexes with this, and other, structures have also been considered by Evrard.

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.

Properties

As a model for homotopy theory

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.

Examples

The prime example of a cubical set with connections is the singular cubical complex $\mathrm{KX}$ 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

$${\gamma }_i^{\alpha }(t_1,t_2,\dots ,t_{n+1})=(t_1,t_2,\dots ,t_{i-1},A(t_i,t_{i+1}),t_{i+2},\dots ,t_{n+1})$$\gamma _i^\alpha (t_1 ,t_2 ,\ldots \, ,t_{n+1} ) = (t_1 ,t_2 ,\ldots\, ,t_{i-1},A(t_i ,t_{i+1}),t_{i+2},\ldots \, ,t_{n+1} )

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 $\mathrm{Equ}(E)$ of $G$-maps between the fibres, and the Moore paths $\Lambda$ on this form a double category $D$ with $\mathrm{Equ}(E)$ 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.

References

• 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.

The statement that cubes with connection form a strict test category is due to

• Georges Maltsiniotis, La catégorie cubique avec connections est une catégorie test stricte, preprint, 2009, 1–16. (web)

based on

• Denis-Charles Cisinski, Les préfaisceaux comme modeèle des types d’homotopie, Astérisque, 308 (2006)