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connection on a cubical set

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Contents

Idea

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

afba\stackrel{f}{\to}b

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

a f b Id Id Id a f b\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 nn-cell afba\stackrel{f}{\to}b also (n+1)(n+1)-cells of the form

a f b f Id b Id b\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

affba\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 nn0}K= \{K_n| n \geq 0\} is a cubical set, then a connection structure on KK consists of functions Γ i +,Γ i :K nK n+1\Gamma^+_i, \Gamma^- _i: K_n \to K_{n+1}, i=1,,n;n1i=1, \ldots \, , n; n \geq 1, satisfying the relations for α,β=±\alpha, \beta=\pm:

  1. Γ i αΓ j β=Γ j+1 βΓ i α\Gamma^\alpha_i\Gamma^\beta_j= \Gamma^\beta_{j+1} \Gamma^\alpha _i if i<ji \lt j;

  2. Γ i αΓ i α=Γ i+1 αΓ i α\Gamma^\alpha_i\Gamma^\alpha_i= \Gamma^\alpha_{i+1} \Gamma^\alpha _i;

  3. j αΓ j α= j+1 αΓ j α=id\partial^\alpha_j \Gamma^\alpha_j= \partial ^\alpha_{j+1} \Gamma ^\alpha_j = id;

  4. j αΓ j α= j+1 αΓ j α=ε j j α\partial^\alpha_j \Gamma^{-\alpha}_j= \partial ^\alpha_{j+1} \Gamma ^{-\alpha}_j = \varepsilon _j \partial^\alpha_j;

  5. i αΓ j β={Γ j1 β i α if i<j Γ j β i1 α if i>j+1; \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}

  6. Γ j αε j=ε j 2=ε j+1ε j \Gamma^\alpha_j \varepsilon_j = \varepsilon^2_j = \varepsilon_{j+1}\varepsilon _j;

  7. Γ i αε j={ε j+1Γ i α if i<j ε jΓ i1 α if i>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 ε jx\varepsilon_j x has opposite faces equal and all other faces degenerate.

A cube of type Γ i αx \Gamma_i^\alpha x has a pair of adjacent faces equal and all other faces of type Γ j αy\Gamma_j^\alpha y or ε jy\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 22 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 nn-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. 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.

Examples

The prime example of a cubical set with connections is the singular cubical complex KXKX of a topological space XX. Here for n0n \ge 0 K nK_n is the set of singular nn-cubes in XX (i.e. continuous maps I nXI^n \to X) and the connection Γ i α:K nK n+1 \Gamma_i^\alpha :K_{n } \to K_{n+1} is induced by the map γ i α:I n+1I n\gamma_i^\alpha : I^{n+1} \to I^{n} defined by

γ i α(t 1,t 2,,t n+1)=(t 1,t 2,,t i1,A(t i,t i+1),t i+2,,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)A(s,t)=\max(s,t), \min(s,t) as α=,+\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 GG-bundle EE over BB gives rise to the Ehresmann groupoid Equ(E)Equ(E) of GG-maps between the fibres, and the Moore paths Λ\Lambda on this form a double category DD with Equ(E)Equ(E) and Λ(B)\Lambda(B) as edge categories. A connection Γ\Gamma is then a functor from Λ(B)\Lambda(B) to one of the category structures on DD 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.

  • P.J. Higgins, “Thin elements and commutative shells in cubical ω\omega-categories”, Theory Appl. Categ. 14 (2005) 60–74.

The statement that cubical groups with connections are Kan complexes is due to

  • A. Tonks, “Cubical groups which are Kan”, J. Pure Appl. Algebra, 81 (1992) 83–87.

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

Revised on November 15, 2013 02:19:20 by Ronnie Brown (81.152.207.246)