compositions in cubical sets

The cubical singular complex KXK X, or S XS^\square X, of a topological space XX has an additional structure of compositions.

Let (m)=(m 1,,m n)(m) = (m_1, \ldots\, , m_n) be an nn-tuple of positive integers and

ϕ (m):I n[0,m 1]××[0,m n]\phi_{(m)} : I^n \rightarrow [0, m_1] \times \cdots \times [0, m_n]

be the map (x 1,,x n)(m 1x 1,,m nx n).(x_1 , \ldots \, , x_n) \mapsto (m_1 x_1, \ldots \, , m_n x_n). Then a subdivision of type (m)(m) of a map α:I nX\alpha : I^n \rightarrow X is a factorisation α=αϕ (m)\alpha = \alpha' \circ \phi_{(m)}; its parts are the cubes α (r)\alpha_{(r)} where (r)=(r 1,,r n)(r) = (r_1, \ldots \,, r_n) is an nn-tuple of integers with 1r im i1 \leq r_i \leq m_i, i=1,,n,i = 1, \ldots\, , n, and where α (r):I nX\alpha_{(r)} : I^n \rightarrow X is given by

(x 1,,x n)α(x 1+r 11,,x n+r n1).(x_1, \ldots \, , x_n) \mapsto \alpha'(x_1 + r_1 - 1, \ldots \, , x_n + r_n - 1).

We then say that α\alpha is the composite of the cubes α (r)\alpha_{(r)} and write α=[α (r)]\alpha = [\alpha_{(r)}]. The domain of α (r)\alpha_{(r)} is then the set {(x 1,,x n)I n:r i1x ir i,1in}\{(x_1,\ldots,x_n) \in I^n : r_i-1 \leq x_i \leq r_i, 1 \leq i \leq n\}.

The composite is in direction jj if m jm_j is the only m i>1,m_i \gt 1, and we then write α=[α 1,,α mj] j;\alpha = [\alpha_1 , \ldots\, , \alpha_m{_j} ]_j; the composite is in the directions jj, kk (jk)(j \neq k) if m jm_j, m km_k are the only m i>1,m_i \gt 1, and we then write

α=[α rs] j,kor[α rs] j k\alpha = [ \alpha_{rs}]_{j,k} \quad or \quad [ \alpha_{rs}] \quad _{j}\!\downarrow ^{\textstyle\to ^k}

for r=1,,m jr = 1 , \ldots\, , m_j and s=1,,m k.s = 1, \ldots \,, m_k. The aim is to follow matrix conventions in writing double compositions.

These definitions and notations are useful for showing how the singular cubical complex allows expression for algebraic inverses to subdivision, something seemingly very difficult either simplicially or globularly. The implications of this advantage for weak category theory seem not to have been investigated.

A cubical set with connections and compositions and inverses is a cubical set KK with connections in which each K nK_n has nn partial compositions + i+_i and nn unary operations i-_i, i=1,2,,ni = 1,2,\ldots \, ,n satisfying the following axioms.

If a,bK na,b \in K_n, then a+ iba+_i b is defined if and only if i +a= i b \partial^+_i a=\partial^-_i b , and then for α=±\alpha=\pm:

{ i (a+ ib)= i a i +(a+ ib)= i +b \begin{cases} \partial^-_i (a+_i b) = \partial^-_i a & \\ \partial^+_i (a+_i b) = \partial^+_i b & \end{cases}
i α(a+ jb)={ i αa+ j1 i αb (i<j) i αa+ j i αb (i>j), \partial^\alpha_i (a +_j b) = \begin{cases} \partial^\alpha_i a+_{j-1}\partial^\alpha_i b &(i \lt j) \\ \partial^\alpha_i a+_j \partial^\alpha_i b& (i \gt j), \end{cases}

If aK na \in K_n, then ia-_i a is defined and

{ i ( ia)= i +a i +( ia)= i a \begin{cases}\partial^-_i (-_i a)=\partial^+_i a & \\ \partial^+_i(-_i a)=\partial^-_i a & \end{cases}
i α( ja)={ j1 i αa (i<j) j i αa (i>j) \partial^\alpha_i(-_j a) = \begin{cases} -_{j-1}\partial^\alpha_i a & (i\lt j) \\ -_{j}\partial^\alpha_i a & (i \gt j) \end{cases}
ε i(a+ jb)={ε ia+ j+1ε ib (ij) ε ia+ jε ib (i>j) \varepsilon_i(a+_j b) = \begin{cases} \varepsilon_i a +_{j+1} \varepsilon_i b & (i \leq j) \\ \varepsilon_i a +_j\varepsilon_i b & (i \gt j) \end{cases}
ε i( jb)={ j+1ε ia (ij) jε ia (i>j) \varepsilon_i (-_j b) = \begin{cases} -_{j+1} \varepsilon_i a & (i \leq j) \\ -_j \varepsilon_i a & (i \gt j) \end{cases}

We have for iji \ne j and whenever both sides are defined:

(a+ ib)+ j(c+ id)=(a+ jc)+ i(b+ jd) (a+_i b) +_j (c+_i d) = (a+_j c) +_i (b+_j d)

These relations are called the interchange laws, and both sides of this equation may be written:

[a c b d] i j\begin{bmatrix} a& c\\ b & d \end{bmatrix} \quad _{i}\!\downarrow ^{\textstyle\to ^j}


i(a+ jb)=( ia)+ j( ib) -_i (a+_j b) = (-_i a) +_j (-_i b) and i( ja)= j( ia) -_i (-_j a) = -_j (-_i a) if iji \ne j

j(a+ jb)=( jb)+ j( ja)-_j(a+_j b) = (-_j b)+_j(-_j a) and j( ja)=a -_j (-_j a) =a.

If further KK is a cubical set with connections then we also require

Γ i α(a+ jb)={Γ i αa+ j+1Γ i αb (i<j) Γ i αa+ jΓ iαb (i<j) \Gamma^\alpha _i (a+_j b) = \begin{cases} \Gamma_i^\alpha a +_{j+1} \Gamma_i^\alpha b & (i \lt j) \\ \Gamma_i^\alpha a +_j\Gamma_i\alpha b & (i \lt j) \end{cases}
Γ j +(a+ jb)=(Γ j +a+ jε ja)+ j+1(ε j+1a+ jΓ j +b) \Gamma_j^+(a+_j b)= (\Gamma_j^+ a +_{j} \varepsilon_j a) +_{j+1} (\varepsilon_{j+1} a +_{j} \Gamma_j^+ b)
Γ j (a+ jb)=(Γ j a+ jε j+1b)+ j+1(ε jb+ jΓ j b) \Gamma_j^-(a+_j b)= (\Gamma_j^- a +_{j} \varepsilon_{j+1} b) +_{j+1} (\varepsilon_{j} b +_{j} \Gamma_j^- b)

These last two equations are called the transport laws and the right hand sides are also written respectively

[Γ j +a ε ja ε j+1a Γ j +b] j+1 j \begin{bmatrix} \Gamma^+_j a & \varepsilon_j a \\ \varepsilon_{j+1} a & \Gamma^+_j b \end{bmatrix} \quad _{j+1}\!\downarrow ^{\textstyle\to ^j}
[Γ j a ε j+1b ε jb Γ j b] j+1 j \begin{bmatrix} \Gamma^-_j a & \varepsilon_{j+1} b \\ \varepsilon_{j} b & \Gamma^-_j b \end{bmatrix} \quad _{j+1}\!\downarrow ^{\textstyle\to ^j}

They can be interpreted as saying that turning left, or right, with your arm outstretched, is the same as turning left, or right.

It is easily verified that the singular cubical set KXKX of a space XX satisfies these axioms if + j, j+_j , -_j are defined by

(a+ jb)(t 1,t 2,,t n)={a(t 1,,t j1,2t j,t j+1,,,t n) (t j12) b(t 1,,t j1,2t j1,t j+1,,t n) (t j12) (a+_j b)(t_1 ,t_2 ,\ldots \, ,t_n ) = \begin{cases} a(t_1 ,\ldots \, , t_{j-1},2t_j,t_{j+1} ,\ldots, ,t_n) &(t_j \leq \frac{1}{2})\\ b(t_1 ,\ldots \, , t_{j-1},2t_j-1,t_{j+1} ,\ldots \, ,t_n) &(t_j \geq \frac{1}{2})\\ \end{cases}

whenever j +a= j b \partial^+_j a=\partial^-_j b; and

( ja)(t 1,t 2,,t n)=a(t 1,,t j1,1t j,t j+1,,t n). (-_j a)(t_1 ,t_2 ,\ldots,t_n ) = a(t_1 ,\ldots, t_{j-1},1-t_j,t_{j+1} ,\ldots,t_n).

The above list of relations may seem formidable, but they all express simple geometric ideas most of which have been well used in some form or another in algebraic topology.

Notice also that in the singular cubical complex of a space, the interchange and transport laws hold exactly.

We also get a (strict) notion of cubical omega-category with connections by assuming that all compositions + i+_i give a category structure with source and target maps i , i +:K nK n1\partial^-_i, \partial^+_i:K_n \to K_{n-1} and identity maps ε i:K n1K n\varepsilon_i: K_{n-1} \to K_n, and also for all aa

Γ i +a iΓ i a=ε i+1a,Γ i +a i+1Γ i a=ε ia.\Gamma^+_i a \circ_i\Gamma^-_i a = \varepsilon _{i+1} a, \quad \Gamma^+_i a \circ_{i+1}\Gamma^-_i a = \varepsilon_{i}a.

These are important cancellation laws for the connections. They can be interpreted as saying that turning left and then right, or vice versa, leaves you facing the same way. They were introduced by C.B. Spencer for double categories (see below).

In the omega-groupoid case, the Γ i +\Gamma^+_i can be recovered from the Γ i \Gamma^-_i, and vice versa, by using the inverses, assumed to arise from the groupoid structure.

The main result of the second paper below is that (strict) cubical omega-groupoids with connections are equivalent to crossed complexes. It is easy to construct a functor from the former to the latter; the hard work is to show that such an omega-groupoid may be functorially reconstructed from the crossed complex it contains.

This work is used in the third paper to construct and apply a strict cubical homotopy groupoid of a filtered space.

The main result of the fourth paper below is that (strict) cubical omega-categories with connections are equivalent to strict globular omega-categories.

For more information, see Nonabelian Algebraic Topology..


  • C.B. Spencer, “An abstract setting for homotopy pushouts and pullbacks”, Cahiers Topologie G'eom. Diff'erentielle, 18 (1977), 409-429.

  • R. Brown and P.J. Higgins, The algebra of cubes, J. Pure Appl. Alg.+, 21 (1981), 233–260.

  • Brown, R. and Higgins, P.~J. Colimit theorems for relative homotopy groups. J. Pure Appl. Algebra 22~(1) (1981) 11–41.

  • 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.

Last revised on December 1, 2013 at 01:32:54. See the history of this page for a list of all contributions to it.