nLab cubulation

Idea

Where a triangulation of a topological space is its homeomorphic identification with the topological realization of a simplicial set (simplicial complex), the analogous construction with cubical sets may be referred to as “cubulation”.

The category of cubes

The category of cubes, $Cube$, 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 $p \circ i_0 = id_1 = p \circ i_1$. The monoidal unit $1$ in $Cube$ is terminal, hence there is a unique map $!: X \to 1$ for any object $X$. The interval $I$ of $Cube$ monoidally generates $Cube$ in the sense of PROS.

It may be shown that if $m \leq n$, there are $\binom{n}{m}2^{n-m}$ 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 $Cube$, it follows that if Top is considered as a cartesian monoidal category equipped with $I = [0, 1]$ in this sense of interval, we get an induced monoidal functor

$$\Box: Cube \to Top$$

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

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

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

Definition

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

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, $Top$ may be taken to be the category of topological spaces, or otherwise any sufficiently convenient category of topological spaces (completeness and cocompleteness are baseline assumptions).

Simplices as cubical sets

We define a functor

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

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 $\Sigma: \Delta \to Set^{Cube^{op}}$, we mimic the second definition of the affine simplex functor given at triangulation, replacing $Top$ 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

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

where the projection maps $\pi_1$, $\pi_2$ are defined by taking advantage of the fact that the monoidal unit of $\otimes$ is terminal:

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

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

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

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

which plays a role analogous to the affine simplex functor into $Top$.

Cubulating geometric simplices

Observe that geometric realization $R_{cub}: Set^{Cube^{op}} \to Top$ takes cubical simplicial joins to topological simplicial joins, because $R_{cub}$ sends $\otimes$-products to cartesian products, and preserves pushouts because it is cocontinuous. We conclude that both $\sigma: \Delta \to Top$ and $R_{cub} \circ \Sigma: \Delta \to Top$ 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

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

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

$$\sigma(n) \cong \int^m \Sigma_n(m) \cdot \Box(m)$$

Standard cubulation of a triangulated space

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

$$\array{ Y & \cong & \int^n X(n) \cdot \sigma(n) & & \\ & \cong & \int^n X(n) \cdot (\int^m \Sigma_n(m) \cdot \Box(m)) & & cubulation of \sigma(n) \\ & \cong & \int^m (\int^n X(n) \cdot \Sigma_n(m)) \cdot \Box(m) & & interchange of coends }$$

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

$$C(m) = \int^n X(n) \cdot \Sigma_n(m)$$

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

Triangulating a cubulated space

In this section we assume $Top$ is a convenient category of spaces, so that geometric realization of simplicial sets is product-preserving (see there).

Cubes as simplicial sets

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

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

Triangulating geometric cubes

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

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

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

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

Standard triangulation of a cubulated space

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

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

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

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

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

Related concepts

• triangulation