nLab
compositions in cubical sets

The cubical singular complex $KX$, or $S^{\square }X$, of a topological space $X$ has an additional structure of compositions.

Let $(m)=(m_1,\dots ,m_n)$ be an $n$-tuple of positive integers and

$${\varphi }_{(m)}:I^n\to [0,m_1]\times \cdots \times [0,m_n]$$\phi_{(m)} : I^n \rightarrow [0, m_1] \times \cdots \times [0,m_n]

be the map $(x_1,\dots ,x_n)\mapsto (m_1{x}_1,\dots ,m_n{x}_n).$ Then a subdivision of type $(m)$ of a map $\alpha :I^n\to X$ is a factorisation $\alpha =\alpha \prime \circ {\varphi }_{(m)}$; its parts are the cubes ${\alpha }_{(r)}$ where $(r)=(r_1,\dots ,r_n)$ is an $n$-tuple of integers with $1\le r_i\le m_i$, $i=1,\dots ,n,$ and where ${\alpha }_{(r)}:I^n\to X$ is given by

$$(x_1,\dots ,x_n)\mapsto \alpha \prime (x_1+r_1-1,\dots ,x_n+r_n-1).$$(x_1, \ldots \, , x_n) \mapsto\alpha'(x_1 + r_1 - 1, \ldots \, , x_n + r_n - 1).

We then say that $\alpha$ is the composite of the cubes ${\alpha }_{(r)}$ and write $\alpha =[{\alpha }_{(r)}]$. The domain of ${\alpha }_{(r)}$ is then the set $\{(x_1,\dots ,x_n)\in I^n:r_i-1\le x_i\le r_i,1\le i\le n\}$.

The composite is in direction $j$ if $m_j$ is the only $m_i>1,$ and we then write $\alpha =[{\alpha }_1,\dots ,{\alpha }_m{}_j{]}_j;$ the composite is in the directions $j$, $k$ $(j\ne k)$ if $m_j$, $m_k$ are the only $m_i>1,$ and we then write

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

for $r=1,\dots ,m_j$ and $s=1,\dots ,m_k.$ 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 $K$ with connections in which each $K_n$ has $n$ partial compositions ${+}_i$ and $n$ unary operations ${-}_i$, $i=\mathrm{1,2},\dots ,n$ satisfying the following axioms.

If $a,b\in K_n$, then $a{+}_{i}b$ is defined if and only if ${\partial }_i^{+}a={\partial }_i^{-}b$, and then for $\alpha =\pm$:

$$\{\begin{array}{ll}{\partial }_i^{-}(a{+}_{i}b)={\partial }_i^{-}a& \\ {\partial }_i^{+}(a{+}_{i}b)={\partial }_i^{+}b& \end{array}$$\begin{cases} \partial^-_i (a+_i b) = \partial^-_i a & \\ \partial^+_i (a+_i b) = \partial^+_i b & \end{cases}

$${\partial }_i^{\alpha }(a{+}_{j}b)=\{\begin{array}{ll}{\partial }_i^{\alpha }a{+}_{j-1}{\partial }_i^{\alpha }b& (i<j)\\ {\partial }_i^{\alpha }a{+}_j{\partial }_i^{\alpha }b& (i>j),\end{array}$$\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 $a\in K_n$, then ${-}_{i}a$ is defined and

$$\{\begin{array}{ll}{\partial }_i^{-}({-}_{i}a)={\partial }_i^{+}a& \\ {\partial }_i^{+}({-}_{i}a)={\partial }_i^{-}a& \end{array}$$\begin{cases}\partial^-_i (-_i a)=\partial^+_i a & \\ \partial^+_i(-_i a)=\partial^-_i a & \end{cases}

$${\partial }_i^{\alpha }({-}_{j}a)=\{\begin{array}{ll}{-}_{j-1}{\partial }_i^{\alpha }a& (i<j)\\ {-}_j{\partial }_i^{\alpha }a& (i>j)\end{array}$$\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}

$${\epsilon }_i(a{+}_{j}b)=\{\begin{array}{ll}{\epsilon }_{i}a{+}_{j+1}{\epsilon }_{i}b& (i\le j)\\ {\epsilon }_{i}a{+}_j{\epsilon }_{i}b& (i>j)\end{array}$$\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}

$${\epsilon }_i({-}_{j}b)=\{\begin{array}{ll}{-}_{j+1}{\epsilon }_{i}a& (i\le j)\\ {-}_j{\epsilon }_{i}a& (i>j)\end{array}$$\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 $i\ne j$ and whenever both sides are defined:

$$(a{+}_{i}b){+}_j(c{+}_{i}d)=(a{+}_{j}c){+}_i(b{+}_{j}d)$$(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:

$$\left[\begin{array}{cc}a& c\\ b& d\end{array}\right]{}_i{\downarrow }^{{\to }^j}$$\begin{bmatrix} a& c\\ b & d \end{bmatrix} \quad _{i}\!\downarrow ^{\textstyle\to ^j}

Further:

${-}_i(a{+}_{j}b)=({-}_{i}a){+}_j({-}_{i}b)$ and ${-}_i({-}_{j}a)={-}_j({-}_{i}a)$ if $i\ne j$

${-}_j(a{+}_{j}b)=({-}_{j}b){+}_j({-}_{j}a)$ and ${-}_j({-}_{j}a)=a$.

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

$${\Gamma }_i^{\alpha }(a{+}_{j}b)=\{\begin{array}{ll}{\Gamma }_i^{\alpha }a{+}_{j+1}{\Gamma }_i^{\alpha }b& (i<j)\\ {\Gamma }_i^{\alpha }a{+}_j{\Gamma }_i\alpha b& (i<j)\end{array}$$\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}

$${\Gamma }_j^{+}(a{+}_{j}b)=({\Gamma }_j^{+}a{+}_j{\epsilon }_{j}a){+}_{j+1}({\epsilon }_{j+1}a{+}_j{\Gamma }_j^{+}b)$$\Gamma_j^+(a+_j b)= (\Gamma_j^+ a +_{j} \varepsilon_j a) +_{j+1} (\varepsilon_{j+1} a +_{j} \Gamma_j^+ b)

$${\Gamma }_j^{-}(a{+}_{j}b)=({\Gamma }_j^{-}a{+}_j{\epsilon }_{j+1}b){+}_{j+1}({\epsilon }_{j}b{+}_j{\Gamma }_j^{-}b)$$\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

$$\left[\begin{array}{cc}{\Gamma }_j^{+}a& {\epsilon }_{j}a\\ {\epsilon }_{j+1}a& {\Gamma }_j^{+}b\end{array}\right]{}_{j+1}{\downarrow }^{{\to }^j}$$\begin{bmatrix} \Gamma^+_j a & \varepsilon_j a \\\varepsilon_{j+1} a & \Gamma^+_j b \end{bmatrix} \quad _{j+1}\!\downarrow ^{\textstyle\to ^j}

$$\left[\begin{array}{cc}{\Gamma }_j^{-}a& {\epsilon }_{j+1}b\\ {\epsilon }_{j}b& {\Gamma }_j^{-}b\end{array}\right]{}_{j+1}{\downarrow }^{{\to }^j}$$\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 $\mathrm{KX}$ of a space $X$ satisfies these axioms if ${+}_j,{-}_j$ are defined by

$$(a{+}_{j}b)(t_1,t_2,\dots ,t_n)=\{\begin{array}{ll}a(t_1,\dots ,t_{j-1}\mathrm{,2}{t}_j,t_{j+1},\dots ,,t_n)& (t_j\le \frac{1}{2})\\ b(t_1,\dots ,t_{j-1}\mathrm{,2}{t}_j-1,t_{j+1},\dots ,t_n)& (t_j\ge \frac{1}{2})\\ \end{array}$$(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 ${\partial }_j^{+}a={\partial }_j^{-}b$; and

$$({-}_{j}a)(t_1,t_2,\dots ,t_n)=a(t_1,\dots ,t_{j-1}\mathrm{,1}-t_j,t_{j+1},\dots ,t_n).$$(-_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 ${+}_i$ give a category structure with source and target maps ${\partial }_i^{-},{\partial }_i^{+}:K_n\to K_{n-1}$ and identity maps ${\epsilon }_i:K_{n-1}\to K_n$, and also for all $a$

$${\Gamma }_i^{+}a{\circ }_i{\Gamma }_i^{-}a={\epsilon }_{i+1}a,{\Gamma }_i^{+}a{\circ }_{i+1}{\Gamma }_i^{-}a={\epsilon }_{i}a.$$\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 ${\Gamma }_i^{+}$ can be recovered from the ${\Gamma }_i^{-}$, and vice versa, by using the inverses, assumed to arise from the groupoid structure.

The main result of the first 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.

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

References

• 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.

• 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.