nLab geometric realization

Idea

What is commonly called geometric realization (or, less commonly but more accurately: topological realization,see Rem. ) is the operation that builds from a simplicial set $X$ a topological space $|X|$ obtained by interpreting each element in $X_n$ – each abstract $n$-simplex in $X$ – as one copy of the standard topological $n$-simplex $\Delta^n_{Top}$ 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 $X$ on how these simplices are supposed to be stuck together. This procedure generalises the geometric realization of simplicial complexes as described at that entry.

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

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 classical model structure on topological spaces and the classical 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 .

Remark

(alternative terminology: “topological realization”)
While the term geometric realization is classical (Quillen 1968) and in common use by algebraic topologists (e.g. Goerss & Jardine 2009, §VII.3), it is somewhat inaccurate or at least misleading, in that the bare topological spaces produced by the realization construction are not yet what geometers would commonly regard as reflecting geometric structure: Geometry on a topological space is instead typically taken to be the extra structure of a differentiable manifold (e.g. smooth structure) together with some kind of G-structure (such as Riemannian structure, conformal structure, etc.). This terminological mismatch is quite stark in the main application to homotopy theory, where one cares only about the (weak) homotopy type of the realization, which is the most non-geometric aspect of any notion of “space”. This becomes a real issue in contexts of differential topology and more generally of geometric homotopy theory (such as for differential cohomology theories) where realizations of simplicial sets to genuine geometric spaces richer than topological spaces do play a role (e.g. already for the notion of smooth/equivariant triangulations of smooth manifolds).

In conclusion, a more accurate and more descriptive term for the realization operation discussed here would be topological realization. References which use this terminology include Simpson 1996, Jardine 2004, Lackenby 2008, §I.2. In Fresse 2017 the term “topological realization” is used in the pdf draft but is replaced by “geometric realization” in the published version.

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

Up to homotopy, this is a special case of a general notion of

• Geometric realization of cohesive ∞-groupoids.

At the point-set level, it is also a special case of a general notion of

• Geometric realization of simplicial objects.

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

$st : S \to$ 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 $st([n]) := \{ (x_1, \cdots, x_n) | \sum_{i=1}^n x_i = 1, x_i \geq 0 \} \subset \mathbb{R}^{n}$ ) with the obvious induced face and boundary maps.

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

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

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

For the case of cubical sets, see cubical geometric realisation.

Of simplicial topological spaces

See

• geometric realization of simplicial topological spaces

Of cohesive $\infty$-groupoids

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

For the choice $\mathbf{H} =$ 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

Of simplicial objects in a category

Let $M$ be a cocomplete simplicially enriched category with copowers. A simplicial object in $M$ is a functor $X:\Delta^{op}\to M$, where $\Delta$ is the simplex category. Its geometric realization is defined similarly to the classical case as a coend:

$$|X| = \int^{[n]\in\Delta} \Delta[n] \odot X_n$$

where $\odot$ denotes the copower in $M$. This operation is a left adjoint which is even a simplicially enriched functor; see simplicial object for more details.

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

Proposition

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

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

Theorem: Geometric realization is left exact

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

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

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

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

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

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 $k \colon Haus \to CGHaus$ to Hausdorff and compactly generated topological spaces. This gives the correct isomorphism in the case $Space = CGHaus$, where we have that ${|X \times Y|} \cong {|X|} \times_k {|Y|} \coloneqq k({|X|} \times {|Y|})$; the product on the right has been “kelleyfied” to the product appropriate for $CGHaus$.)

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

Theorem

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

Proof

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

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

can be described as the functor

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

where $Int$ 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 $U \sigma \colon \Delta \to Set$ 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

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

is left exact.

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

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 CW-complex. In particular, taking $X = \emptyset$, $R(Y)$ is a $CW$-complex.

Proof

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

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

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

Corollary

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

Proof

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

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

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

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

Geometric realization preserves finite products

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

First, an easy result on products of simplices.

Lemma

The realization of a product of two representables $\Delta(-, m) \times \Delta(-, n)$ is compact.

Proof

It suffices to observe that $\Delta[m] \times \Delta[n]$ has finitely many non-degenerate simplices. That is clear since non-degenerate $k$-simplices in the nerve of a poset $P$ are exactly injective order preserving maps $[k] \to P$.

Lemma

The canonical map

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

is a homeomorphism.

Proof

The canonical map is continuous, and a bijection at the underlying set level by theorem . The codomain is the compact Hausdorff space $\sigma(m) \times \sigma(n)$, and the domain is compact by Lemma . 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 $\leq$ on the interval $I$ defines a closed subset of $I \times I$. These properties ensure that the affine $n$-simplex $\{(x_1, \ldots, x_n) \in I^n: x_1 \leq \ldots \leq x_n\}$ is itself compact Hausdorff, so that the proof of lemma 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 $R: Set^{\Delta^{op}} \to Space$ preserves products.

Proof

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

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

and so we calculate

$$\array{ R(X \times Y) & \cong & R((\int^m X(m) \cdot \Delta(-, m)) \times (\int^n Y(n) \cdot \Delta(-, n))) \\ & \cong & R(\int^m \int^n X(m) \cdot Y(n) \cdot (\Delta(-, m) \times \Delta(-, n))) \\ & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot R(\Delta(-, m) \times \Delta(-, n)) \\ & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot (R(\Delta(-, m)) \times R(\Delta(-, n)) \\ & \cong & \int^m \int^n X(m) \cdot Y(n) \cdot (\sigma(m) \times \sigma(n)) \\ & \cong & (\int^m X(m) \cdot \sigma(m)) \times (\int^n Y(n) \cdot \sigma(n)) \\ & \cong & R(X) \times R(Y) }$$

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 $Space$ is cartesian closed), and the remaining lines used the fact that $R$ preserves colimits, and also products of representables by lemma .

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

A construction of Drinfeld of geometric realization as Hom([0,1],-)

Drinfeld, On the notion of geometric realization provides a conceptual explanation of preserving finite limits, and “reformulates the definitions so that the following facts become obvious:

• geometric realization commutes with finite projective limits (e.g., with Cartesian products);

• the geometric realization of a simplicial set … (resp. cyclic set) is equipped with an action of the group of orientation preserving homeomorphisms of the segment $I := [0, 1]$ … (resp. the circle $S^ 1$).“ (quote from the paper)

A draft M.Gavrilovich, K.Pimenov. Geometric realisation as a Skorokhod semi-continuous path space endofunctor attempts to further reformulate this by showing that, in a certain precise sense, geometric realisation is an endofunctor of a certain category $sF$ of simplicial sets equipped with extra structure of topological nature (a notion of smallness). The underlying endofunctor of $sSets$ is

$$HHom ( Hom_{preorders}(-, [0,1]_\leq), Y_.)$$

The category $sF$ contains simplicial sets, topological and uniform spaces as full subcategories, and has forgetful functors $sF\to sSets$, $sF\to Top$, and $sF\to UniformSpaces$ such that the following compositions are identity: $sSets\to sF\to sSets$, $Top\to sF\to Top$, and $UniformSpaces\to sF\to UniformSpaces$. Moreover, this endofunctor seems to have the right adjoint, defined by the usual construction.

Here are some details. The category $sF$ may be thought as the category of simplicial sets with extra structure of topological nature, a notion of smallness. Formally it is just the category of simplicial objects in the category of filters. The endofunctor $HHom ( Hom_{preorders}(-, [0,1]_\leq), Y_.):sF\to sF$ above is the inner hom of sSets equipped with an extra structure motivated by Skorokhod/Levi-Prokhorov convergence. The precise claim is that the geometric realisation of sSets factors as

$$sSets \to sF \to sF \to Top$$

To gain some intuition, consider $Y_.=\Delta_n=Hom(-,[n])$ the standard simplex. Then by Remark 2.4.1-2 of Grayson the standard geometric simplex is the set of monotone functions $[0.1]\to [n]$

$$Hom_{preorders}([0,1]_\leq), [n] ) = Hom ( Hom_{preorders}(-, [0,1]_\leq), \Delta_n )$$

equipped with a metric

$$\dist(f,g)=\inf \{ \epsilon: \forall x \exists y ( f(x)=f(y) ) \wedge \forall y \exists x ( f(x)=g(y) ) \}$$

reminiscent of Skorokhod metric in probability theory. Now instead of $\Delta_n$ take an arbitrary simplicial set, and rewrite the definition of Skorokhod convergence in terms of the notion of smallness in $sF$.

Geometric realization preserves fibrations

Theorem

The geometric realization of a Kan fibration is a Serre fibration.

Proof

This is shown in Quillen 68.

This result implies that the geometric realization functor preserves all five classes of maps in a model category: weak equivalences, cofibrations, acyclic cofibrations, fibrations, and acyclic fibrations.

In fact the geometric realization of a Kan fibration is even a Hurewicz fibration (at least relative to a convenient category of spaces in which it lives). This follows from the fact that a Serre fibration between CW-complexes is a Hurewicz fibration; a direct proof along the lines of Quillen’s can be found in Fritch and Piccinini, Theorem 4.5.25.

Induced properties of the fibrant replacement

The previous two sections show that the geometric realization preserves finite limits and fibrations. Since its right adjoint, the singular complex functor $Top \to sSet$, also preserves both (much more trivially), and since all objects of $Top$ are fibrant and the adjunction is simplicially enriched, it follows that the composite $sSet \to Top \to sSet$ is a simplicially enriched fibrant replacement functor on $sSet$ that additionally preserves both finite limits and fibrations.

Geometric realization of barycentric subdivisions

Theorem

(Fritsch–Puppe, 1967; Fritsch 1974.) There is a homeomorphism

$$|Sd X| \to |X|$$

from the geometric realization of the barycentric subdivision of a simplicial set $X$ to the geometric realization of $X$. This homeomorphism is homotopic to the geometric realization of the last vertex map. The homeomorphism turns the CW-complex $|Sd X|$ into a subdivision of the CW-complex $|X|$. The statement also holds relative a simplicial subset $A\subset X$.

For an expository account, see Fritsch–Piccinini.

Examples

• For $G$ a group, $\mathbf{B}G$, its one-object groupoid obtained by delooping, $N(\mathbf{B}G)$ the corresponding simplicial nerve Kan complex, we have that the geometric realization
  $$\mathcal{B}G = |N\mathbf{B}G|$$

  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, simplicial topological spaces, cohesive ∞-groupoids, cubical sets.

• totalization

• singular complex functor?

References

General

Original articles:

• Daniel Quillen, The geometric realization of a Kan fibration is a Serre fibration. Proc. Amer. Math. Soc. 19 1968 1499–1500. pdf

Monographs:

• Paul Goerss, J. F. Jardine, Section VII.3 of: Simplicial homotopy theory, Progress in Mathematics, Birkhäuser (1999)

  Modern Birkh[er Classics (2009) [doi:10.1007/978-3-0346-0189-4]

• Rudolf Fritsch, Renzo A. Piccinini §4.2 and §4.3 in: Cellular structures in topology, Cambridge studies in advanced mathematics 19 Cambridge University Press (1990) [doi:10.1017/CBO9780511983948, pdf]

Lecture notes:

• Marc Lackenby, Section I.2 of: Topology and Groups (2008, 2018) [pdf, pdf]

Generalization to realization of simplicial presheaves:

• Carlos Simpson, The topological realization of a simplicial presheaf [arXiv:q-alg/9609004]

Compatibility with homotopy limits

Discussion of sufficient conditions for homotopy geometric realization to be compatible with homotopy pullback (see also at geometric realization of simplicial topological spaces):

• D. Anderson, Fibrations and geometric realization , Bull. Amer. Math. Soc. Volume 84, Number 5 (1978), 765-788. (euclid:1183541139)

• Charles Rezk, When are homotopy colimits compatible with homotopy base change?, 2014 (pdf, pdf)

• Edoardo Lanari, Compatibility of homotopy colimits and homotopy pullbacks of simplicial presheaves (pdf, pdf)

  (expanded version of Rezk 14)

Geometric realization of barycentric subdivisions

• Rudolf Fritsch, Dieter Puppe, Die Homöomorphie der geometrischen Realisierungen einer semisimplizialen Menge und ihrer Normalunterteilung, Arch. Math. (Basel) 18 (1967), 508–512.

• Relative semisimpliziale Approximation, Arch. Math. (Basel) 25 (1974), 75–78.

• Rudolf Fritsch, Renzo A. Piccinini, Cellular structures in topology, CUP, 1990.