nLab
geometric realization

Idea

Geometric realization is the operation that builds from a simplicial set $X$ a topological space $\mid X\mid$ obtained by interpreting each element in $X_n$ – each abstract $n$-simplex in $X$ – as one copy of the standard topological $n$-simplex ${\Delta }_{\mathrm{Top}}^n$ 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 $X$ 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 $[n]\mapsto {\Delta }_{\mathrm{Top}}^n$. (N.B.: in this article, $[n]$ denotes the ordinal with $n+1$ elements. The corresponding contravariant representable is denoted $\Delta (-,n)$.)

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

• For cell complexes such as simplicial sets.

A generalization of this of central importance is the

• Geometric realization of simplicial topological spaces

This is a special case of a general notion of

• Geometric realization of cohesive ∞-groupoids.

Of cell complexes such as simplicial sets

Let $S$ 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

$\mathrm{st}:S\to$ Top

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

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

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

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

• nerve, homotopy hypothesis.

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

• geometric realization of categories

Of simplicial topological spaces

See

• geometric realization of simplicial topological spaces

Of cohesive $\mathrm{\infty }$-groupoids

Every cohesive (∞,1)-topos $H$ (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 $H=$ ∞Grpd this reproduces the geometric realization of simplicial sets, see at discrete ∞-groupoid the section

For the choice $H=$ 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 $\mid X\mid$ is a CW complex (see lemma 1 below), and so geometric realization $\mid (-)\mid :{\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}\to \mathrm{Top}$ takes values in the full subcategory of CW complexes, and therefore in any convenient category of topological spaces, for example in the category $\mathrm{CGHaus}$ of compactly generated Hausdorff spaces. Let $\mathrm{Space}$ be any convenient category of topological spaces, and let $i:\mathrm{Space}\to \mathrm{Top}$ denote the inclusion.

Proposition

For any simplicial set $X$, there is a natural isomorphism $i({\int }^{n:\Delta }X(n)\cdot \sigma (n))\cong \mid X\mid$, where the coend on the left is computed in $\mathrm{Space}$.

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

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

Theorem: Geometric realization is left exact

We continue to assume $\mathrm{Space}$ is any convenient category of topological spaces. In this section we prove that geometric realization

$$R:{\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}\to \mathrm{Space}$$R: 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

$$\mid (-)\mid :{\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}\to \mathrm{Top},$${|(-)|}: 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 $k:\mathrm{Haus}\to \mathrm{CGHaus}$. This gives the correct isomorphism in the case $\mathrm{Space}=\mathrm{CGHaus}$, where we have that $\mid X\times Y\mid \cong \mid X\mid {\times }_k\mid Y\mid ≔k(\mid X\mid \times \mid Y\mid )$; the product on the right has been “kelleyfied” to the product appropriate for $\mathrm{CGHaus}$.)

We reiterate that $R$ denotes the geometric realization functor considered as valued in a convenient category of spaces, whereas $\mid (-)\mid$ is geometric realization viewed as taking values in $\mathrm{Top}$.

Theorem

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

Proof

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

$$\Delta \stackrel{\sigma }{\to }\mathrm{Space}\stackrel{U}{\to }\mathrm{Set}$$\Delta \stackrel{\sigma}{\to} Space \stackrel{U}{\to} Set

can be described as the functor

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

where $\mathrm{Int}$ is the category of intervals (linearly ordered sets with distinct top and bottom). Because every interval, in particular $I$, is a filtered colimit of finite intervals, and because finite intervals are finitely presentable intervals, it follows that $U\sigma :\Delta \to \mathrm{Set}$ 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

$$UR=-{\otimes }_{\Delta }U\sigma :{\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}\to \mathrm{Set}$$U R = - \otimes_\Delta U \sigma: Set^{\Delta^{op}} \to Set

is left exact.

Obviously the preceding proof is not sensitive to whether we use $\mathrm{Space}$ or $\mathrm{Top}$.

Geometric realization preserves equalizers

Lemma

If $i:X\to Y$ is a monomorphism of simplicial sets, then $R(i):R(X)\to R(Y)$ is a closed subspace inclusion, in fact a relative $\mathrm{CW}$-complex. In particular, taking $X=\varnothing$, $R(Y)$ is a $\mathrm{CW}$-complex.

Proof

Any monomorphism $i:X\to Y$ in ${\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}$ can be seen as the result of iteratively adjoining nondegenerate $n$-simplices. In other words, there is a chain of inclusions $X=F(0)\hookrightarrow F(1)\hookrightarrow \dots Y={\mathrm{colim}}_{i}F(i)$, where $F:\kappa \to {\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}$ is a functor from some ordinal $\kappa =\{0\le 1\le \dots \}$ (as preorder) that preserves directed colimits, and each inclusion $F(\alpha \le \alpha +1):F(\alpha )\to F(\alpha +1)$ fits into a pushout diagram

$$\begin{array}{ccc}\partial \Delta (-,n)& \to & F(\alpha )\\ i\downarrow & & \downarrow \\ \Delta (-,n)& \to & F(\alpha +1)\end{array}$$\array{ \partial \Delta(-, n) & \to & F(\alpha) \\ \mathllap{i} \downarrow & & \downarrow \\ \Delta(-, n) & \to & F(\alpha+1) }

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

Corollary

$R:{\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}\to \mathrm{Space}$ preserves equalizers.

Proof

The equalizer of a pair of maps in $\mathrm{Top}$ is computed as the equalizer on the level of underlying sets, equipped with the subspace topology. So if

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

is an equalizer diagram in ${\mathrm{Set}}^{{\Delta }^{\mathrm{op}}}$, then $\mid i\mid$ is the equalizer of the pair $\mid f\mid$, $\mid g\mid$, because the underlying function $U(\mid i\mid )$ is the equalizer of $U(\mid f\mid )$, $U(\mid g\mid )$ on the underlying set level by the preceding theorem, and because $\mid i\mid$ is a (closed) subspace inclusion by lemma 1. But this $\mathrm{Top}$-equalizer $\mid i\mid :\mid E\mid \to \mid X\mid$ lives in the full subcategory $\mathrm{Space}$, and therefore $R(i)=\mid i\mid$ is the equalizer of the pair $R(f)=\mid f\mid$, $R(g)=\mid g\mid$.

As the proof indicates, that realization preserves equalizers is not at all sensitive to whether we use $\mathrm{Top}$ or a convenient category of spaces $\mathrm{Space}$.

Geometric realization preserves finite products

That geometric realization preserves products is sensitive to whether we think of it as valued in $\mathrm{Top}$ or in a convenient category $\mathrm{Space}$. In particular, the proof uses cartesian closure of $\mathrm{Space}$ 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 $\Delta (-,m)\times \Delta (-,n)$ 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 $p_i:\Delta (-,m+n)\to \Delta (-,m)\times \Delta (-,n)$ (there are $\left(\genfrac{}{}{0ex}{}{m+n}{m}\right)$ many to be exact) which collectively define an epimorphism

$$(p_i):\sum _{1\le i\le \left(\genfrac{}{}{0ex}{}{m+n}{m}\right)}\Delta (-,m+n)\to \Delta (-,m)\times \Delta (-,n).$$(p_i): \sum_{1 \leq i \leq \binom{m+n}{m}} \Delta(-, m+n) \to \Delta(-, m) \times \Delta(-, n).

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

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

$$\Delta (-,k)\to \Delta (-,m+n)\stackrel{i_j}{\hookrightarrow }\sum _{1\le i\le \left(\genfrac{}{}{0ex}{}{m+n}{m}\right)}\Delta (-,m+n)$$\Delta(-, k) \to \Delta(-, m+n) \stackrel{i_j}{\hookrightarrow} \sum_{1 \leq i \leq \binom{m+n}{m}} \Delta(-, m+n)

(where $i_j$ is the $j^{\mathrm{th}}$ inclusion) provides the desired lift of $f$, thus completing the proof.

Lemma

The canonical map

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

Theorem

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

Proof

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

$$X\cong {\int }^{m}X(m)\cdot \Delta (-,m)Y\cong {\int }^{n}Y(n)\cdot \Delta (-,n)$$X \cong \int^m X(m) \cdot \Delta(-, m) \qquad Y \cong \int^n Y(n) \cdot \Delta(-, n)

and so we calculate

$$\begin{array}{ccc}R(X\times Y)& \cong & R(({\int }^{m}X(m)\cdot \Delta (-,n))\times ({\int }^{n}Y(n)\cdot \Delta (-,n)))\\ & \cong & R({\int }^m{\int }^{n}X(m)\cdot Y(n)\cdot (\Delta (-,m)\times \Delta (-,n)))\\ & \cong & {\int }^m{\int }^{n}X(m)\cdot Y(n)\cdot R(\Delta (-,m)\times \Delta (-,n))\\ & \cong & {\int }^m{\int }^{n}X(m)\cdot Y(n)\cdot (R(\Delta (-,m))\times R(\Delta (-,n))\\ & \cong & {\int }^m{\int }^{n}X(m)\cdot Y(n)\cdot (\sigma (m)\times \sigma (n))\\ & \cong & ({\int }^{m}X(m)\cdot \sigma (m))\times ({\int }^{n}Y(n)\cdot \sigma (n))\\ & \cong & R(X)\times R(Y)\end{array}$$\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 $\mathrm{Space}$ is cartesian closed), and the remaining lines used the fact that $R$ preserves colimits, and also products of representables by lemma 3.

• A slightly higher-level rendition of the proof might look like this:

  $$\begin{array}{ccc}R(X\times Y)& \cong & R((X{\otimes }_{\Delta }\mathrm{hom})\times (Y{\otimes }_{\Delta }\mathrm{hom}))\\ & \cong & R((X\times Y){\otimes }_{\Delta \times \Delta }(\mathrm{hom}\times \mathrm{hom}))\\ & \cong & (X\times Y){\otimes }_{\Delta \times \Delta }R(\mathrm{hom}\times \mathrm{hom})\\ & \cong & (X\times Y){\otimes }_{\Delta \times \Delta }(R(\mathrm{hom})\times R(\mathrm{hom}))\\ & \cong & (X{\otimes }_{\Delta }R(\mathrm{hom}))\times (Y{\otimes }_{\Delta }R(\mathrm{hom}))\\ & \cong & R(X{\otimes }_{\Delta }\mathrm{hom})\times R(Y{\otimes }_{\Delta }\mathrm{hom})\\ & \cong & R(X)\times R(Y)\end{array}$$\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

• For $G$ a group, $BG$, its one-object groupoid obtained by delooping, $N(BG)$ the corresponding simplicial nerve Kan complex, we have that the geometric realization

  $$ℬG=\mid NBG\mid$$\mathcal{B}G = |N\mathbf{B}G|

  is the topological space that is the classifying space for $G$-principal bundles (covering spaces), as long as we give $G$ the discrete topology.

Related concepts

• geometric realization

  • of categories, of simplicial topological spaces, of cohesive ∞-groupoids

• totalization