nLab
cubulation

The category of cubes

The category $\mathrm{Cube}$ may be defined universally to be a walking interval: it is initial among monoidal categories that are equipped with an object $I$, two maps $i_0,i_1:1\to I$ (where $1$ is the monoidal unit) and a map $p:I\to 1$ such that $p\circ i_0={\mathrm{id}}_1=p\circ i_1$. The monoidal unit $1$ in $\mathrm{Cube}$ is terminal, hence there is a unique map $!:X\to 1$ for any object $X$. The interval $I$ of $\mathrm{Cube}$ monoidally generates $\mathrm{Cube}$ in the sense of PROS.

It may be shown that if $m\le n$, there are $\left(\genfrac{}{}{0ex}{}{n}{m}\right)2^{n-m}$ injections $I^{\otimes m}\to I^{\otimes n}$, the same as the number of $m$-dimensional faces of the geometric $n$-cube. There are no diagonal maps in the category of cubes as defined here.

• A different possibility is to consider the Lawvere theory of two constants, which gives a different category of cubes with diagonal maps.

Standard geometric cube functor

From the universal property of $\mathrm{Cube}$, it follows that if $\mathrm{Top}$ is considered as a cartesian monoidal category equipped with $I=[0,1]$ in this sense of interval, we get an induced monoidal functor

$$\square :\mathrm{Cube}\to \mathrm{Top}$$\Box: Cube \to Top

The monoidal product on $\mathrm{Cube}$ induces a monoidal product $\otimes$ on ${\mathrm{Set}}^{{\mathrm{Cube}}^{\mathrm{op}}}$ by Day convolution. The cubical realization functor $R_{\mathrm{cub}}:{\mathrm{Set}}^{{\mathrm{Cube}}^{\mathrm{op}}}\to \mathrm{Top}$ is, up to isomorphism, the unique cocontinuous monoidal functor which extends the monoidal functor $\square$ along the Yoneda embedding; therefore $R_{\mathrm{cub}}$ takes $\otimes$-products of cubical sets to the corresponding cartesian products of spaces.

In terms of an explicit formula, the cubical realization of a cubical set $C:{Cube}^{\mathrm{op}}\to \mathrm{Set}$ is given by the coend formula

$$R_{\mathrm{cub}}C={\int }^{m\in \mathrm{Ob}(\mathrm{Cube})}C(m)\times \square (m)$$R_{cub} C = \int^{m \in Ob(Cube)} C(m) \times \Box(m)

Definition

A cubulation of a topological space $Y$ is a cubical set $C:{\mathrm{Cube}}^{\mathrm{op}}\to \mathrm{Set}$ together with a homeomorphism $h:R_{\mathrm{cub}}C\to Y$.

Relation between triangulation and cubulation

In rough terms, a space can be triangulated if and only if it can be cubulated. This can be shown by simple conceptual arguments, as follows.

Cubulating a triangulated space

In this section, $\mathrm{Top}$ may be taken to be the category of topological spaces, or otherwise any sufficiently convenient category of spaces (completeness and cocompleteness are baseline assumptions).

Simplices as cubical sets

We define a functor

$$\Sigma :\Delta \to {\mathrm{Set}}^{{\mathrm{Cube}}^{\mathrm{op}}}$$\Sigma: \Delta \to Set^{Cube^{op}}

The functor $\Sigma$ effectively regards an $n$-simplex as an iterated join of simplicial sets and then produces the analogous join in the category of cubical sets. This for instance regards the 2-simplex as a square with one degenerate edge.

To define $\Sigma :\Delta \to {\mathrm{Set}}^{{\mathrm{Cube}}^{\mathrm{op}}}$, we mimic the second definition of the affine simplex functor given at triangulation, replacing $\mathrm{Top}$ by cubical sets and the topological simplicial join by a suitable “cubical simplicial join”. Formally, we define a monoidal structure on cubical sets by taking $X\star Y$ to be the pushout of the diagram

$$X\stackrel{{\pi }_1}{\leftarrow }X\otimes Y\stackrel{1_X\otimes i_0\otimes 1_Y}{\to }X\otimes I\otimes Y\stackrel{1_X\otimes i_1\otimes 1_Y}{\leftarrow }X\otimes Y\stackrel{{\pi }_2}{\to }Y$$X \stackrel{\pi_1}{\leftarrow} X \otimes Y \stackrel{1_X \otimes i_0 \otimes 1_Y}{\to} X \otimes I \otimes Y \stackrel{1_X \otimes i_1 \otimes 1_Y}{\leftarrow} X \otimes Y \stackrel{\pi_2}{\to} Y

where the projection maps ${\pi }_1$, ${\pi }_2$ are defined by taking advantage of the fact that the monoidal unit of $\otimes$ is terminal:

$${\pi }_1=(X\otimes Y\stackrel{1_X\otimes !}{\to }X\otimes 1\cong X)$$\pi_1 = (X \otimes Y \stackrel{1_X \otimes !}{\to} X \otimes 1 \cong X)

$${\pi }_2=(X\otimes Y\stackrel{!\otimes 1_Y}{\to }1\otimes Y\cong Y)$$\pi_2 = (X \otimes Y \stackrel{! \otimes 1_Y}{\to} 1 \otimes Y \cong Y)

The terminal cubical set is of course a monoid with respect to this monoidal product, so by the walking monoid property we obtain a monoidal functor

$$\Sigma :\Delta \to {\mathrm{Set}}^{{\mathrm{Cube}}^{\mathrm{op}}}$$\Sigma: \Delta \to Set^{Cube^{op}}

which plays a role analogous to the affine simplex functor into $\mathrm{Top}$.

Cubulating geometric simplices

Observe that geometric realization $R_{\mathrm{cub}}:{\mathrm{Set}}^{{\mathrm{Cube}}^{\mathrm{op}}}\to \mathrm{Top}$ takes cubical simplicial joins to topological simplicial joins, because $R_{\mathrm{cub}}$ sends $\otimes$-products to cartesian products, and preserves pushouts because it is cocontinuous. We conclude that both $\sigma :\Delta \to \mathrm{Top}$ and $R_{\mathrm{cub}}\circ \Sigma :\Delta \to \mathrm{Top}$ take monoidal products in $\Delta$ to topological simplicial joins, and both take the walking monoid of $\Delta$ to the one-point space. By the universal property of $\Delta$, it follows that there is a natural isomorphism

$$\sigma \cong R_{\mathrm{cub}}\circ \Sigma$$\sigma \cong R_{cub} \circ \Sigma

(as monoidal functors), giving the canonical cubulation of affine simplices. In terms of an explicit formula, we have

$$\sigma (n)\cong {\int }^m{\Sigma }_n(m)\cdot \square (m)$$\sigma(n) \cong \int^m \Sigma_n(m) \cdot \Box(m)

Standard cubulation of a triangulated space

Given a triangulation $(X,h:RX\to Y)$ of a space $Y$, we have isomorphisms

$$\begin{array}{ccccc}Y& \cong & {\int }^{n}X(n)\cdot \sigma (n)& & \\ & \cong & {\int }^{n}X(n)\cdot ({\int }^m{\Sigma }_n(m)\cdot \square (m))& & \mathrm{cubulation}\mathrm{of}\sigma (n)\\ & \cong & {\int }^m({\int }^{n}X(n)\cdot {\Sigma }_n(m))\cdot \square (m)& & \mathrm{interchange}\mathrm{of}\mathrm{coends}\end{array}$$\array{ Y & \cong & \int^n X(n) \cdot \sigma(n) & & \\ & \cong & \int^n X(n) \cdot (\int^m \Sigma_n(m) \cdot \Box(m)) & & cubulation of \sigma(n) \\ & \cong & \int^m (\int^n X(n) \cdot \Sigma_n(m)) \cdot \Box(m) & & interchange of coends }

where in the last line we used the coend Fubini theorem?. Thus, defining the cubical set $C$ by

$$C(m)={\int }^{n}X(n)\cdot {\Sigma }_n(m)$$C(m) = \int^n X(n) \cdot \Sigma_n(m)

we have a homeomorphism $Y\cong {\int }^{m}C(m)\cdot \square (m)=R_{\mathrm{cub}}C$, i.e., we obtain a cubulation of $Y$.

Triangulating a cubulated space

In this section we assume $\mathrm{Top}$ is a convenient category of spaces, so that geometric realization of simplicial sets is product-preserving.

Cubes as simplicial sets

Define a monoidal functor ${\square }_{\delta }:\mathrm{Cube}\to {\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}$ as follows: regard the category of simplicial sets as a cartesian monoidal category equipped with the representable $\mathrm{hom}(-,[1])$ as an interval (with two face maps from and a projection to the terminal object $\mathrm{hom}(-,[0])$). By the walking interval property of $\mathrm{Cube}$, there is an induced functor

$${\square }_{\delta }:\mathrm{Cube}\to {\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}$$\Box_{\delta}: Cube \to Set^{\Delta^{op}}

Triangulating geometric cubes

Next, because $R:{\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}\to \mathrm{Top}$ is preserves cartesian products and preserves the interval objects, we have an isomorphism

$$(\square :\mathrm{Cube}\to \mathrm{Top})\cong (\mathrm{Cube}\stackrel{{\square }_{\delta }}{\to }{\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}\stackrel{R}{\to }\mathrm{Top})(2)$$(\Box: Cube \to Top) \cong (Cube \stackrel{\Box_\delta}{\to} Set^{\Delta^{op}} \stackrel{R}{\to} Top) \qquad (2)

by the universal property of $\mathrm{Cube}$. In terms of an explicit formula, we have

$$\square (m)\cong {\int }^n{\square }_{\delta m}(n)\cdot \sigma (n)$$\Box(m) \cong \int^n \Box_{\delta m}(n) \cdot \sigma(n)

Standard triangulation of a cubulated space

Given a cubulation $(C,h:R_{\mathrm{cub}}X\to Y)$ of a space $Y$, we have isomorphisms

$$\begin{array}{ccccc}Y& \cong & {\int }^{m}C(m)\cdot \square (m)& & \\ & \cong & {\int }^{m}C(m)\cdot ({\int }^n{\square }_{\delta m}(n)\cdot \sigma (n))& & \mathrm{triangulation}\mathrm{of}\square (m)\\ & \cong & {\int }^n({\int }^{m}C(m)\cdot {\square }_{\delta m}(n))\cdot \sigma (n)& & \mathrm{interchange}\mathrm{of}\mathrm{coends}\end{array}$$\array{ Y & \cong & \int^m C(m) \cdot \Box(m) & & \\ & \cong & \int^m C(m) \cdot (\int^n \Box_{\delta m}(n) \cdot \sigma(n)) & & triangulation of \Box(m) \\ & \cong & \int^n (\int^m C(m) \cdot \Box_{\delta m}(n)) \cdot \sigma(n) & & interchange of coends }

where in the last line we used the coend Fubini theorem?. Thus, defining the simplicial set $X$ by

$$X(n)={\int }^{m}C(m)\cdot {\square }_{\delta m}(n)$$X(n) = \int^m C(m) \cdot \Box_{\delta m}(n)

we have a homeomorphism $Y\cong {\int }^{n}X(n)\cdot \sigma (n)=RX$, i.e., we obtain a triangulation of $Y$.