nLab
geometric realization

Contents

Idea

Geometric realization is the operation that builds from a simplicial set XX a topological space X|X| obtained by interpreting each element in X nX_n – each abstract nn-simplex in XX – as one copy of the standard topological nn-simplex Δ Top n\Delta^n_{Top} and then gluing together all these along their boundaries to a big topological space, using the information encoded in the face and degeneracy maps of XX on how these simplices are supposed to be stuck together. It generalises the geometric realization of simplicial complexes as described at that entry.

This is the special case of the general notion of nerve and realization that is induced from the standard cosimplicial topological space [n]Δ Top n[n] \mapsto \Delta^n_{Top}. (N.B.: in this article, [n][n] denotes the ordinal with n+1n+1 elements. The corresponding contravariant representable is denoted Δ(,n)\Delta(-, n).)

In the context of homotopy theory geometric realization plays a notable role in the homotopy hypothesis, where it is part of the Quillen equivalence between the model structure on topological spaces and the standard model structure on simplicial sets.

The construction generalizes naturally to a map from simplicial topological spaces to plain topological spaces. For more on that see geometric realization of simplicial spaces.

The dual concept is totalization .

Definition

There are various levels of generality in which the notion of (topological) geometric realization makes sense. The basic definition is

A generalization of this of central importance is the

This is a special case of a general notion of

Of cell complexes such as simplicial sets

Let SS be one of the categories of geometric shapes for higher structures, such as the globe category or the simplex category or the cube category.

There is an obvious functor

st:S st : S \to Top

which sends the standard cellular shape [n][n] (the standard cellular globe, simplex or cube, respectively) to the corresponding standard topological shape (for instance the standard nn-simplex st([n]):={(x 1,,x n)x ix i+1} n st([n]) := \{ (x_1, \cdots, x_n) | x_i \leq x_{i+1} \} \subset \mathbb{R}^{n} ) with the obvious induced face and boundary maps.

Using this, in cases where TopTop can be regarded as enriched over and tensored over a base category VV, the geometric realization of a presheaf K :S opVK^\bullet : S^{op} \to V on SS – e.g., of a globular set, a simplicial set or a cubical set, respectively (when V=SetV= Set) – is the topological space given by the coend or weighted colimit

K = [n]Sst([n])K n. |K^\bullet| = \int^{[n] \in S} st([n]) \cdot K^n \,.

In the case of simplicial sets, see for more discussion also

Via simplicial nerve functors geometric realization of simplicial sets induces geometric realizations of many other structures, for instance

Of simplicial topological spaces

See

Of cohesive \infty-groupoids

Every cohesive (∞,1)-topos H\mathbf{H} (in fact every locally ∞-connected (∞,1)-topos) comes with its intrinsic notion of geometric realization.

The general abstract definition is at cohesive (∞,1)-topos in the section Geometric homotopy.

For the choice H=\mathbf{H} = ∞Grpd this reproduces the geometric realization of simplicial sets, see at discrete ∞-groupoid the section

For the choice H=\mathbf{H} = ETop∞Grpd and Smooth∞Grpd this reproduces geometric realization of simplicial topological spaces. See the sections ETop∞Grpd – Geometric homotopy and Smooth ∞-groupoid – Geometric homotopy

Properties

In this section we consider topological geometric realization of simplicial sets, which is the best studied and perhaps most significant case.

Realizations as CW complexes

Each X{|X|} is a CW complex (see lemma 1 below), and so geometric realization ():Set Δ opTop{|(-)|}: Set^{\Delta^{op}} \to Top takes values in the full subcategory of CW complexes, and therefore in any convenient category of topological spaces, for example in the category CGHausCGHaus of compactly generated Hausdorff spaces. Let SpaceSpace be any convenient category of topological spaces, and let i:SpaceTopi \colon Space \to Top denote the inclusion.

Proposition

For any simplicial set XX, there is a natural isomorphism i( n:ΔX(n)σ(n))Xi(\int^{n: \Delta} X(n) \cdot \sigma(n)) \cong {|X|}, where the coend on the left is computed in SpaceSpace.

This is obvious: more generally, if F:JAF: J \to A is a diagram and i:ABi: A \hookrightarrow B is a full replete subcategory, and if the colimit in BB of iFi \circ F lands in AA, then this is also the colimit of FF in AA. (The dual statement also holds, with limits instead of colimits.)

Below, we let R:Set Δ opSpaceR: Set^{\Delta^{op}} \to Space denote the geometric realization when considered as landing in SpaceSpace.

Theorem: Geometric realization is left exact

We continue to assume SpaceSpace is any convenient category of topological spaces. In this section we prove that geometric realization

R:Set Δ opSpaceR: Set^{\Delta^{op}} \to Space

is a left exact functor in that it preserves finite limits.

It is important that we use some such “convenience” assumption, because for example

():Set Δ opTop,{|(-)|}: Set^{\Delta^{op}} \to Top,

valued in general topological spaces, does not preserve products. (To get a correct statement, one usual procedure is to “kelley-fy” products by applying the coreflection k:HausCGHausk: Haus \to CGHaus. This gives the correct isomorphism in the case Space=CGHausSpace = CGHaus, where we have that X×YX× kYk(X×Y){|X \times Y|} \cong {|X|} \times_k {|Y|} \coloneqq k({|X|} \times {|Y|}); the product on the right has been “kelleyfied” to the product appropriate for CGHausCGHaus.)

We reiterate that RR denotes the geometric realization functor considered as valued in a convenient category of spaces, whereas (){|(-)|} is geometric realization viewed as taking values in TopTop.

Theorem

Let U=hom(1,):SpaceSetU = \hom(1, -): Space \to Set be the underlying-set functor. Then the composite UR:Set Δ opSetU R: Set^{\Delta^{op}} \to Set is left exact.

Proof

As described at the nLab article on triangulation here, the composite

ΔσSpaceUSet\Delta \stackrel{\sigma}{\to} Space \stackrel{U}{\to} Set

can be described as the functor

ΔFinInt opInt opInt(,I)Set\Delta \cong FinInt^{op} \hookrightarrow Int^{op} \stackrel{Int(-, I)}{\to} Set

where IntInt is the category of intervals (linearly ordered sets with distinct top and bottom). Because every interval, in particular II, is a filtered colimit of finite intervals, and because finite intervals are finitely presentable intervals, it follows that Uσ:ΔSetU \sigma: \Delta \to Set is a flat functor (a filtered colimit of representables). But on general grounds, tensoring with a flat functor is left exact, which in this case means

UR= ΔUσ:Set Δ opSetU R = - \otimes_\Delta U \sigma: Set^{\Delta^{op}} \to Set

is left exact.

Obviously the preceding proof is not sensitive to whether we use SpaceSpace or TopTop.

Geometric realization preserves equalizers

Lemma

If i:XYi: X \to Y is a monomorphism of simplicial sets, then R(i):R(X)R(Y)R(i): R(X) \to R(Y) is a closed subspace inclusion, in fact a relative CWCW-complex. In particular, taking X=X = \emptyset, R(Y)R(Y) is a CWCW-complex.

Proof

Any monomorphism i:XYi \colon X \to Y in Set Δ opSet^{\Delta^{op}} can be seen as the result of iteratively adjoining nondegenerate nn-simplices. In other words, there is a chain of inclusions X=F(0)F(1)Y=colim iF(i)X = F(0) \hookrightarrow F(1) \hookrightarrow \ldots Y = colim_i F(i), where F:κSet Δ opF: \kappa \to Set^{\Delta^{op}} is a functor from some ordinal κ={01}\kappa = \{0 \leq 1\leq \ldots\} (as preorder) that preserves directed colimits, and each inclusion F(αα+1):F(α)F(α+1)F(\alpha \leq \alpha + 1): F(\alpha) \to F(\alpha + 1) fits into a pushout diagram

Δ(,n) F(α) i Δ(,n) F(α+1)\array{ \partial \Delta(-, n) & \to & F(\alpha) \\ \mathllap{i} \downarrow & & \downarrow \\ \Delta(-, n) & \to & F(\alpha+1) }

where ii is the inclusion. Now R(i)R(i) is identifiable as the inclusion S n1D nS^{n-1} \to D^n, and since RR preserves pushouts (which are calculated as they are in TopTop), we see by this lemma that RF(α)RF(α+1)R F(\alpha) \to R F(\alpha+1) is a closed subspace inclusion and evidently a relative CW-complex. By another lemma, it follows that XYX \to Y is also a closed inclusion and indeed a relative CW-complex.

Corollary

R:Set Δ opSpaceR: Set^{\Delta^{op}} \to Space preserves equalizers.

Proof

The equalizer of a pair of maps in TopTop is computed as the equalizer on the level of underlying sets, equipped with the subspace topology. So if

EiXgfYE \stackrel{i}{\to} X \stackrel{\overset{f}{\to}}{\underset{g}{\to}} Y

is an equalizer diagram in Set Δ opSet^{\Delta^{op}}, then i{|i|} is the equalizer of the pair f{|f|}, g{|g|}, because the underlying function U(i)U({|i|}) is the equalizer of U(f)U({|f|}), U(g)U({|g|}) on the underlying set level by the preceding theorem, and because i{|i|} is a (closed) subspace inclusion by lemma 1. But this TopTop-equalizer i:EX{{|i|}}: {{|E|}} \to {{|X|}} lives in the full subcategory SpaceSpace, and therefore R(i)=iR(i) = {|i|} is the equalizer of the pair R(f)=fR(f) = {|f|}, R(g)=gR(g) = {|g|}.

As the proof indicates, that realization preserves equalizers is not at all sensitive to whether we use TopTop or a convenient category of spaces SpaceSpace.

Geometric realization preserves finite products

That geometric realization preserves products is sensitive to whether we think of it as valued in TopTop or in a convenient category SpaceSpace. In particular, the proof uses cartesian closure of SpaceSpace in an essential way (in the form that finite products distribute over arbitrary colimits).

First, a small technical result about simplicial sets.

Lemma

The product of two representables Δ(,m)×Δ(,n)\Delta(-, m) \times \Delta(-, n) is the colimit of a finite diagram of representables, i.e., is the quotient of a finite coproduct of representables.

Proof

We describe a finite collection of monomorphisms p i:Δ(,m+n)Δ(,m)×Δ(,n)p_i: \Delta(-, m+n) \to \Delta(-, m) \times \Delta(-, n) (there are (m+nm)\binom{m+n}{m} many to be exact) which collectively define an epimorphism

(p i): 1i(m+nm)Δ(,m+n)Δ(,m)×Δ(,n).(p_i): \sum_{1 \leq i \leq \binom{m+n}{m}} \Delta(-, m+n) \to \Delta(-, m) \times \Delta(-, n).

The product of ordinals [m]×[n][m] \times [n] can be pictured as a rectangular grid consisting of m×nm \times n rectangles, and an order-preserving monomorphism [k][m]×[n][k] \to [m] \times [n] can be pictured as a path of length kk along the grid, traveling north and east. A maximal such path is of length m+nm+n, traveling from (0,0)(0, 0) to (n,m)(n, m) say, and is specified by a choice of which mm among m+nm+n many steps are steps north. These maximal paths give the monos p ip_i.

To show the collective map (p i)(p_i) is epic, we must show that any map f:Δ(,k)Δ(,m)×Δ(,n)f: \Delta(-, k) \to \Delta(-, m) \times \Delta(-, n) lifts through (p i)(p_i). By factoring ff as an epi followed by a mono, it is sufficient to prove this in the case where ff is monic. In that case, ff is described by a path of length kk, which is embedded in a maximal path p jp_j. Such an embedding is given by a monic [k][m+n][k] \to [m+n], whereupon the composite

Δ(,k)Δ(,m+n)i j 1i(m+nm)Δ(,m+n)\Delta(-, k) \to \Delta(-, m+n) \stackrel{i_j}{\hookrightarrow} \sum_{1 \leq i \leq \binom{m+n}{m}} \Delta(-, m+n)

(where i ji_j is the j thj^{th} inclusion) provides the desired lift of ff, thus completing the proof.

Lemma

The canonical map

Δ(,m)×Δ(,n)Δ(,m)×Δ(,n){|\Delta(-, m) \times \Delta(-, n)|} \to {|\Delta(-, m)|} \times {|\Delta(-, n)|}

is a homeomorphism.

Proof

The canonical map is continuous, and a bijection at the underlying set level by theorem 1. The codomain is the compact Hausdorff space σ(m)×σ(n)\sigma(m) \times \sigma(n), and the domain is also compact: by lemma 2, and using the fact that realization preserves finite colimits, the left side is the topological quotient of a coproduct of finitely many simplices, hence compact. But a continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

Remark

The key properties of II needed for this subsection are (1) the fact it is compact Hausdorff, and (2) the order relation \leq on the interval II defines a closed subset of I×II \times I. These properties ensure that the affine nn-simplex {(x 1,,x n)I n:x 1x n}\{(x_1, \ldots, x_n) \in I^n: x_1 \leq \ldots \leq x_n\} is itself compact Hausdorff, so that the proof of lemma 3 goes through. The point is that in place of II, we can really use any interval LL that satisfies these properties, thus defining an LL-based geometric realization instead of the standard (II-based) geometric realization being developed here.

Theorem

The functor R:Set Δ opSpaceR: Set^{\Delta^{op}} \to Space preserves products.

Proof

The proof is purely formal. Let XX and YY be simplicial sets. By the co-Yoneda lemma, we have isomorphisms

X mX(m)Δ(,m)Y nY(n)Δ(,n)X \cong \int^m X(m) \cdot \Delta(-, m) \qquad Y \cong \int^n Y(n) \cdot \Delta(-, n)

and so we calculate

R(X×Y) R(( mX(m)Δ(,n))×( nY(n)Δ(,n))) R( m nX(m)Y(n)(Δ(,m)×Δ(,n))) m nX(m)Y(n)R(Δ(,m)×Δ(,n)) m nX(m)Y(n)(R(Δ(,m))×R(Δ(,n)) m nX(m)Y(n)(σ(m)×σ(n)) ( mX(m)σ(m))×( nY(n)σ(n)) R(X)×R(Y)\array{ R(X \times Y) & \cong & R((\int^m X(m) \cdot \Delta(-, n)) \times (\int^n Y(n) \cdot \Delta(-, n))) \\ & \cong & R(\int^m \int^n X(m) \cdot Y(n) \cdot (\Delta(-, m) \times \Delta(-, n))) \\ & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot R(\Delta(-, m) \times \Delta(-, n)) \\ & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot (R(\Delta(-, m)) \times R(\Delta(-, n)) \\ & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot (\sigma(m) \times \sigma(n)) \\ & \cong & (\int^m X(m) \cdot \sigma(m)) \times (\int^n Y(n) \cdot \sigma(n)) \\ & \cong & R(X) \times R(Y) }

where in each of the second and penultimate lines, we twice used the fact that ×- \times - preserves colimits in its separate arguments (i.e., the fact that the nice category SpaceSpace is cartesian closed), and the remaining lines used the fact that RR preserves colimits, and also products of representables by lemma 3.

  • A slightly higher-level rendition of the proof might look like this:
    R(X×Y) R((X Δhom)×(Y Δhom)) R((X×Y) Δ×Δ(hom×hom)) (X×Y) Δ×ΔR(hom×hom) (X×Y) Δ×Δ(R(hom)×R(hom)) (X ΔR(hom))×(Y ΔR(hom)) R(X Δhom)×R(Y Δhom) R(X)×R(Y)\array{ R(X \times Y) & \cong & R((X \otimes_{\Delta} \hom) \times (Y \otimes_{\Delta} \hom)) \\ & \cong & R((X \times Y) \otimes_{\Delta \times \Delta} (\hom \times \hom)) \\ & \cong & (X \times Y) \otimes_{\Delta \times \Delta} R(\hom \times \hom) \\ & \cong & (X \times Y) \otimes_{\Delta \times \Delta} (R(\hom) \times R(\hom)) \\ & \cong & (X \otimes_{\Delta} R(\hom)) \times (Y \otimes_{\Delta} R(\hom)) \\ & \cong & R(X \otimes_{\Delta} \hom) \times R(Y \otimes_{\Delta} \hom) \\ & \cong & R(X) \times R(Y) }

Examples

Revised on December 2, 2013 12:38:00 by Urs Schreiber (77.251.114.72)