The category of cubes

The category CubeCube may be defined universally to be a walking interval: it is initial among monoidal categories that are equipped with an object II, two maps i 0,i 1:1Ii_0, i_1: 1 \to I (where 11 is the monoidal unit) and a map p:I1p: I \to 1 such that pi 0=id 1=pi 1p \circ i_0 = id_1 = p \circ i_1. The monoidal unit 11 in CubeCube is terminal, hence there is a unique map !:X1!: X \to 1 for any object XX. The interval II of CubeCube monoidally generates CubeCube in the sense of PROS.

It may be shown that if mnm \leq n, there are (nm)2 nm\binom{n}{m}2^{n-m} injections I mI nI^{\otimes m} \to I^{\otimes n}, the same as the number of mm-dimensional faces of the geometric nn-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 CubeCube, it follows that if TopTop is considered as a cartesian monoidal category equipped with I=[0,1]I = [0, 1] in this sense of interval, we get an induced monoidal functor

:CubeTop\Box: Cube \to Top

The monoidal product on CubeCube induces a monoidal product \otimes on Set Cube opSet^{Cube^{op}} by Day convolution. The cubical realization functor R cub:Set Cube opTopR_{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 cubR_{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 opSetC: \Cube^{op} \to Set is given by the coend formula

R cubC= mOb(Cube)C(m)×(m)R_{cub} C = \int^{m \in Ob(Cube)} C(m) \times \Box(m)


A cubulation of a topological space YY is a cubical set C:Cube opSetC: Cube^{op} \to Set together with a homeomorphism h:R cubCYh: 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, TopTop 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

Σ:ΔSet Cube op\Sigma: \Delta \to Set^{Cube^{op}}

The functor Σ\Sigma effectively regards an nn-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 Σ:ΔSet Cube op\Sigma: \Delta \to Set^{Cube^{op}}, we mimic the second definition of the affine simplex functor given at triangulation, replacing TopTop 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 XYX \star Y to be the pushout of the diagram

Xπ 1XY1 Xi 01 YXIY1 Xi 11 YXYπ 2YX \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 π 1\pi_1, π 2\pi_2 are defined by taking advantage of the fact that the monoidal unit of \otimes is terminal:

π 1=(XY1 X!X1X)\pi_1 = (X \otimes Y \stackrel{1_X \otimes !}{\to} X \otimes 1 \cong X)
π 2=(XY!1 Y1YY)\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

Σ:ΔSet Cube op\Sigma: \Delta \to Set^{Cube^{op}}

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

Cubulating geometric simplices

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

σR cubΣ\sigma \cong R_{cub} \circ \Sigma

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

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

Standard cubulation of a triangulated space

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

Y nX(n)σ(n) nX(n)( mΣ n(m)(m)) cubulationofσ(n) m( nX(n)Σ n(m))(m) interchangeofcoends\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 CC by

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

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

Triangulating a cubulated space

In this section we assume TopTop is a convenient category of spaces, so that geometric realization of simplicial sets is product-preserving.

Cubes as simplicial sets

Define a monoidal functor δ:CubeSet Δ op\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])hom(-, [1]) as an interval (with two face maps from and a projection to the terminal object hom(,[0])hom(-, [0])). By the walking interval property of CubeCube, there is an induced functor

δ:CubeSet Δ op\Box_{\delta}: Cube \to Set^{\Delta^{op}}

Triangulating geometric cubes

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

(:CubeTop)(Cube δSet Δ opRTop)(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 CubeCube. In terms of an explicit formula, we have

(m) n δm(n)σ(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 cubXY)(C, h: R_{cub} X \to Y) of a space YY, we have isomorphisms

Y mC(m)(m) mC(m)( n δm(n)σ(n)) triangulationof(m) n( mC(m) δm(n))σ(n) interchangeofcoends\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 XX by

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

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

Revised on October 28, 2010 12:22:21 by Todd Trimble (