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Triangulation

A triangulation of a topological space YY is a simplicial set XX together with a homeomorphism h:RXYh: R X \to Y, where RR denotes the geometric realization functor.

(Usually, in classical algebraic and geometric topology, the XX here is taken to be a simplicial complex, but the difference does not really matter if one is considering uses in homotopy theoretic contexts. For the reason, see the discussion at subdivision. When considering polyhedral structure as such, for instance for PL manifolds, the simplical complex version would be needed. In such a case we may refer to a classical triangulation.)

Explicitly, RXR X is given by a coend formula

nΔX(n)σ(n)\int^{n \in \Delta} X(n) \cdot \sigma(n)

where σ:ΔTop\sigma: \Delta \to Top is the standard affine simplex functor. Provided that TopTop is interpreted as a nice category of spaces (such as CGHausCGHaus – see the discussion at geometric realization), the functor RR is left exact, and in particular preserves products.

Standard affine simplex functor

There are various ways of understanding the affine simplex functor σ:ΔTop\sigma: \Delta \to Top from a categorical perspective. (Note: in this article we will be working with the algebraist’s version of the simplex category Δ\Delta, namely the category of finite ordinals and order-preserving maps, including the initial or empty object which represents a (-1)-dimensional simplex. The nn-element ordinal is conventionally, but perhaps unfortunately, denoted [n1][n-1], to indicate the dimension.)

First definition

One way begins by regarding Δ op\Delta^{op} as isomorphic to the category of nonempty ordinals, for which maps are functions that preserve order and the top and bottom elements. In other words, as the category of finite intervals, where an interval is a totally ordered set with a top and bottom element. Indeed, for each ordinal [n1][n-1], the hom-set hom([n1],[1])\hom([n-1], [1]) inherits from [1][1] an interval structure under the pointwise definitions, and

hom(,[1]):Δ opFinInt\hom(-, [1]) \colon \Delta^{op} \to FinInt

is an equivalence (this can also be seen as a restriction of the Stone duality between finite posets and finite distributive lattices).

Under this contravariant equivalence, the nn-element object [n1][n-1] of Δ\Delta corresponds to the (n+1)(n+1)-element finite interval (again denoted [n1][n-1]). Consider the functor that arises by homming into the standard unit interval II:

Δ(FinInt) opInt ophom(,I)Set\Delta \simeq (FinInt)^{op} \hookrightarrow Int^{op} \stackrel{\hom(-, I)}{\to} Set

This functor lifts to a functor hom Int(,I):ΔTophom_{Int}(-, I): \Delta \to Top that takes [n1][n-1] to the space of interval maps [n1]I[n-1] \to I,

{(x 0,x 1,,x n):0=x 0x 1x n=1},\{(x_0, x_1, \ldots, x_n): 0 = x_0 \leq x_1 \leq \ldots \leq x_n = 1\},

topologized as a subspace {0x 1x n11}\{0 \leq x_1 \leq \ldots \leq x_{n-1} \leq 1\} of I n1I^{n-1}. This gives the affine simplex functor σ:ΔTop\sigma: \Delta \to Top.

The category of intervals is an ω\omega-accessible category where the finitely presentable objects are the finite intervals. It follows that each representable on IntInt, in particular hom(,I):FinInt opSet\hom(-, I): FinInt^{op} \to Set, is a filtered colimit of representable presheaves on FinIntFinInt.

Second definition

A second way of understanding σ\sigma is by taking advantage of the fact that the algebraist’s Δ\Delta is the walking monoid. This means that given a monoidal structure on TopTop and a monoid MM therein, there is a unique monoidal functor σ:ΔTop\sigma: \Delta \to Top which sends the generic monoid [0][0] to the monoid MM. To this end, take the monoidal product on TopTop to be “topological simplicial join”: the join XYX \star Y of two spaces XX, YY may be defined to be the pushout of the diagram

Xπ 1X×Y1 X×{0}×1 YX×I×Y1 X×{1}×1 YX×Yπ 2YX \stackrel{\pi_1}{\leftarrow} X \times Y \stackrel{1_X \times \{0\} \times 1_Y}{\to} X \times I \times Y \stackrel{1_X \times \{1\} \times 1_Y}{\leftarrow} X \times Y \stackrel{\pi_2}{\to} Y

and now take the monoid in TopTop to be the 1-point space 11 with its unique monoid structure.

The induced monoidal functor is the affine simplex functor σ:ΔTop\sigma: \Delta \to Top. In effect, it identifies the nn-dimensional simplex with an iterated simplicial join of n+1n+1 copies of 11:

σ(n)=11\sigma(n) = 1 \star \ldots \star 1

because [n][n] is itself the (n+1) st(n+1)^{st} monoidal power of the 1-element ordinal [0][0]. Equivalently, it can be regarded as the result of applying the cone functor CX=1XC X = 1 \star X nn times to 11.

Cubulation

A cubulation of a topological space YY is a cubical set CC together with a homeomorphism h:R cubCYh: R_{cub}C \to Y where R cubR_{cub} denotes the realization functor for cubical sets Set opSet^{\Box^{op}}. Explicitly, R cubCR_{cub}C is given by a coend formula

R cubC= mCubeC(m)(m)R_{cub}C = \int^{m \in Cube} C(m) \cdot \Box(m)

where :CubeTop\Box: Cube \to Top is the standard geometric cube functor.

Standard geometric cube functor

The category CubeCube may be regarded as a “walking interval” in a sense slightly different to the sense of interval above: 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 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.

Relation between triangulation and cubulation

As explained below, there is a “cubulation” functor for standard simplices, Σ:ΔSet Cube op\Sigma: \Delta \to Set^{Cube^{op}}, such that the affine simplex functor σ:ΔTop\sigma: \Delta \to Top is naturally isomorphic to the composite

ΔΣSet opR cubTop\Delta \stackrel{\Sigma}{\to} Set^{\Box^{op}} \stackrel{R_{cub}}{\to} Top

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)) m( nX(n)Σ n(m))(m)\array{ Y & \cong & \int^n X(n) \cdot \sigma(n) \\ & \cong & \int^n X(n) \cdot (\int^m \Sigma_n(m) \cdot \Box(m)) \\ & \cong & \int^m (\int^n X(n) \cdot \Sigma_n(m)) \cdot \Box(m) }

where in the last line we used the coend Fubini theorem? for interchange of coends. 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.

There is also a triangulation functor for standard cubes, :CubeSet Δ op\Box: Cube \to Set^{\Delta^{op}}, which can be used to triangulate the realizations of cubical sets.

Cubulating simplices and triangulating cubes

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.

In other words, to define Σ:ΔSet Cube op\Sigma: \Delta \to Set^{Cube^{op}}, we mimic the second construction of the affine simplex functor given above, 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.

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), which is what we want.

Similarly, we can easily define a monoidal functor δ:CubeSet Δ op\Box_{\delta}: Cube \to Set^{\Delta^{op}} such that

(: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)

In detail, 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}}

Finally, because R:Set Δ opTopR: Set^{\Delta^{op}} \to Top is product-preserving and preserves the interval objects, the isomorphism (2) obtains by the universal property of CubeCube.

Revised on November 29, 2012 19:48:24 by Todd Trimble (67.81.93.16)