nLab geometric realization of simplicial topological spaces

Idea

For $X_\bullet$ a simplicial object in Top – a simplicial topological space – its geometric realization is a plain topological space ${|X_\bullet|} \in Top$ obtained by gluing all topological space $X_n$ together, as determined by the face and degeneracy maps.

The construction of ${|X_\bullet|}$ is a direct analog of the ordinary notion of geometric realization of a simplicial set, but taking into account the topology on the spaces of $n$-simplices $X_n$.

The dual concept is that of totalization of cosimplicial topological spaces.

Definition

Let Top in the following denote either

• the category of compactly generated weakly Hausdorff spaces, or

• the category of k-spaces.

Let $\Delta$ denote the simplex category, and write $\Delta_{Top} \colon \Delta \to Top \colon [n] \mapsto \Delta^n_{Top}$ for the standard cosimplicial topological space of topological simplices.

Write equivalently

$$Top^{\Delta^{op}} \coloneqq sTop \coloneqq [\Delta^{op}, Top]$$

for the category of simplicial topological spaces.

Definition

For $X_\bullet \colon \Delta^{op} \to Top$ a simplicial topological space, its geometric realization is the coend

$${|X_\bullet|} \coloneqq \int^{n \in \Delta} X_n \times \Delta^n_{Top}$$

formed in Top.

This operation naturally extends to a functor

$${|-|} \colon sTop \to Top \,.$$

More explicitly, $\vert X_\bullet \vert$ is the topological space given by the quotient

$${\vert X_\bullet \vert} = \coprod_{n} X_n \times \Delta^n_{Top} /\sim$$

where the equivalence relation “$\sim$” identifies, for every morphism $[k] \to [l]$ in $\Delta$, the points $(x,f_* p) \in X_l \times \Delta^l_{Top}$ and $(f^* x,p) \in X_k \times \Delta^k_{Top}$.

This form of geometric realization of simplicial topological spaces goes back to (Segal 1968). An early reference that realizes this construction as a coend is (MacLane).

One also considers geometric realization after restricting to the subcategory $\Delta_+ \overset{j^{op}}{\hookrightarrow} \Delta$ of the simplex category on the strictly increasing maps (that is, the coface maps only—no codegeneracies). (This yields the underlying semi-simplicial set).

Definition

The corresponding coend in Top is called the fat geometric realization

$${\Vert X_\bullet\Vert} \coloneqq \int^{n \in \Delta_+} X_n \times \Delta^n_{Top} \,.$$

This is called fat , because it does not quotient out the relations induced by the degeneracy maps and hence is “bigger” than ordinary geometric realization.

Explicitly, this is the topological space given by the quotient

$${\Vert X_\bullet \Vert} = \coprod_{n} X_n\times \Delta^n_{Top} /{\sim_+}$$

where the equivalence relation “$\sim_+$” identifies $(x,f_* p) \in X_l\times \Delta^l_{Top}$ with $(f^* x,p) \in X_k\times \Delta^k_{Top}$ only when $[k] \to [l]$ is a coface map.

Example

The geometric realization of the point — the simplicial topological space that is in each degree the 1-point topological space — is homeomorphic to the point, but the fat geometric realization of the point is an “infinite dimensional topological ball”: the terminal morphism

$${\vert * \vert} \stackrel{\simeq_{iso}}{\longrightarrow} *$$

is an isomorphism, but the morphism

$${\Vert * \Vert} \stackrel{\simeq_{h.e.}}{\longrightarrow} *$$

is just a homotopy equivalence.

Remark

The ordinary geometric realization can be described as the tensor product of functors ${|X_\bullet|}=\Delta^\bullet \otimes_{\Delta} X_\bullet$, and the fat geometric realization can likewise be described as ${\Vert X_\bullet\Vert} = i^* \Delta^\bullet \otimes_{\Delta_+} i^* X_\bullet$, where $i\colon \Delta_+ \hookrightarrow \Delta$ is the inclusion. By general facts about tensor product of functors (essentially, the associativity of composition of profunctors), it follows that we can also write ${\Vert X_\bullet\Vert} \cong \Delta^\bullet \otimes_{\Delta} \Lan_i i^* X_\bullet \cong {| Lan_i i^* X_\bullet |}$. In other words, the fat geometric realization of $X_\bullet$ is the ordinary geometric realization of a “fattened up” version of $X_\bullet$, which is obtained by forgetting the degeneracy maps of $X_\bullet$ and then “freely throwing in new ones.”

Reminder on nice simplicial topological spaces

Simplicial topological spaces are in homotopy theory presentations for certain topological ∞-groupoids . In this context what matters is not the operation of geometric realization itself, but its derived functor. This is obtained by evaluating ordinary geometric realization on “sufficiently nice” resolutions of simplicial topological spaces. These we discuss now.

Recall the following definitions and facts from nice simplicial topological space.

Definition

Let $X \colon \Delta^{op} \to Top$ be a simplicial topological space.

Such $X$ is called

• good (Segal 1973, App. 2; Segal 1974, Def. A.4)

  if all the degeneracy maps $X_{n-1} \hookrightarrow X_n$ are closed cofibrations;

• proper (May 1972)

  if the inclusion $s X_n \hookrightarrow X_n$ of the degenerate simplices is a closed cofibration, where $s X_n = \bigcup_i s_i(X_{n-1})$.

Noticing that the union of degenerate simplices appearing here is a latching object and that closed cofibrations are cofibrations in the Strøm model structure on Top, the last condition equivalently says:

• $X_\bullet$ is proper if it is cofibrant in the Reedy model structure $[\Delta^{op}, Top_{Strom}]_{Reedy}$ on simplicial objects in Top with respect to the Strøm model structure on Top.

Proposition

A good simplicial topological space is proper:

$$(X_\bullet \; good) \Rightarrow (X_\bullet\; proper) \,.$$

(Roberts & Stevenson 2012, App. A, following hints in Gaunce Lewis 1982)

Bisimplicial sets and good resolutions

We now discuss the resolution of any simplicial topological space by a good one. Write

$$({\vert- \vert} \dashv Sing) \colon Top \stackrel{\overset{{|-|}}{\longleftarrow}}{\underset{Sing}{\longrightarrow}} sSet$$

for the ordinary geometric realization/singular simplicial complex adjunction (see homotopy hypothesis). Recall that the composite ${|Sing(-)|}\colon Top\to Top$ is a cofibrant replacement functor: for any space $X$, the space ${|Sing(X)|}$ is a CW complex and comes with a natural map ${|Sing X|} \to X$ (the counit of the adjunction $({|-|} \dashv Sing)$) which is a weak homotopy equivalence.

Proposition

Let $X_\bullet$ be a simplicial topological space. Then the simplicial topological space

$${|Sing(X_\bullet)|} \in [\Delta^{op}, Top] \,,$$

obtained by applying ${|Sing(-)|} \colon Top \to Top$ degreewise, is good and hence proper. Moreover, we have a natural morphism

$${|Sing X_\bullet|} \to X_\bullet$$

which is degreewise a weak homotopy equivalence.

Proof

Each space $|Sing X_n|$ is a CW-complex, hence in particular a locally equi-connected space. By (Gaunce Lewis 1982, p. 153) inclusions of retracts of locally equi-connected spaces are closed cofibrations, and since degeneracy maps are retracts, this means that the degeneracy maps in $|Sing X_\bullet|$ are closed cofibrations.

The second sentence follows directly by the remarks above.

Note that there is nothing special about ${|Sing(-)|}$ in the proof; any functorial CW replacement would do just as well (such as that obtained by the small object argument, or the constructions denoted $\tau(-)$ and $simp(-)$ in Segal 1974, p. 308-309).

However, ${|Sing(-)|}$ has the advantage that its geometric realization can be computed alternately in terms of diagonals of bisimplicial sets, as we now show.

If $S_{\bullet,\bullet} \colon \Delta^{op} \times \Delta^{op} \to Set$ is a bisimplicial set, we write $d S$ for its diagonal, which is the composite

$$\Delta^{op} \to \Delta^{op} \times \Delta^{op} \stackrel{S}{\longrightarrow} Set.$$

On the other hand, we can also consider a bisimplicial set as a simplicial object in $sSet$ and take its “geometric realization”:

$${|S_{\bullet,\bullet}|} \coloneqq \int^{[n]\in\Delta^{op}} S_{n,\bullet} \times \Delta^n_{sSet}$$

where $\Delta^n_{sSet}$ denotes the $n$-simplex as a simplicial set, i.e. the representable functor $\Delta(-,[n])\colon \Delta^{op}\to Set$.

Lemma

For a bisimplicial set $S_{\bullet,\bullet}$, there is a natural isomorphism of simplicial sets $d S \cong {|S|}$.

Proof

Both functors $d$ and ${|-|}$ are cocontinuous functors between presheaf categories $Set^{\Delta^{op}\times\Delta^{op}} \to Set^{\Delta^{op}}$, so it suffices to verify that they agree on representables. The representable $(\Delta\times\Delta)(-,([n],[m]))$ is called a bisimplex $\Delta^{n,m}$; its diagonal is

$$(d \Delta^{n,m})(-) = \Delta(-,n) \times \Delta(-,m) = \Delta^n_{sSet} \times \Delta^m_{sSet}.$$

Now note that geometric realization of a bisimplicial set is left adjoint to the “singular complex” $Sing\colon sSet \to sSet^{\Delta^{op}}$ defined by

$$Sing(X_\bullet)(n,m) = \underline{Map}(\Delta^n_{sSet}, X_\bullet)_m = sSet(\Delta^n_{sSet} \times \Delta^m_{sSet}, X_\bullet) = sSet(d \Delta^{n,m}, X_\bullet)$$

where $\underline{Map}(-,-)$ denotes the simplicial mapping space between two simplicial sets. But by the Yoneda lemma, $Sing(X_\bullet)(n,m) = sSet^{\Delta^{op}}(\Delta^{n,m},Sing(X_\bullet))$, so this shows that $d \Delta^{n,m}$ has the same universal property as ${|\Delta^{n,m}|}$; hence they are isomorphic.

In particular, the lemma implies that for the two different ways of considering a bisimplicial set as a simplicial simplicial set (“vertically” or “horizontally”), the resulting “geometric realizations” as simplicial sets are isomorphic (since both are isomorphic to the diagonal, which is symmetrically defined).

Note that Lemma can be interpreted as an isomorphism between two profunctors $\Delta ⇸ \Delta\times\Delta$, of which the first is representable by the diagonal functor $\Delta \to \Delta\times\Delta$. It follows that if $S_{\bullet,\bullet}$ is a bisimplicial object in any cocomplete category, we also have

$$d S_{\bullet,\bullet} \cong \int^{[n]\in\Delta} S_{n,\bullet}\,\times \,(\Delta^n_{sSet})_{\bullet}$$

where the right-hand side is a “realization” functor from bisimplicial objects to simplicial objects in any cocomplete category. On the other hand, if $S$ is a bisimplicial space, then we also have the levelwise realization

$$\int^{[n]\in\Delta} S_{n,\bullet} \,\times\, \Delta^n_{Top}$$

which will not, in general, agree with the diagonal and the abstract realization considered above. It does agree, however, after we pass to a further geometric realization as a single topological space.

Proposition

For $S_{\bullet,\bullet}$ any bisimplicial space, there is a homeomorphism between the geometric realizations of the following two simplicial spaces: (1) the diagonal $d S_{\bullet,\bullet}$ of $S$, and (2) the levelwise realization $|S_{\bullet,\bullet}|$ of $S$.

Proof

Applying Lemma as above, we have

$$d S_{\bullet,\bullet} \cong \int^{[n]\in \Delta} S_{n,\bullet} \times \Delta^n_{sSet}$$

If we then take the realization of these simplicial spaces, we find

$${| d S_{\bullet,\bullet} |} \cong \left| \int^{[n]\in\Delta} S_{n,\bullet} \times \Delta^n_{sSet} \right| \cong \int^{[n]\in\Delta} {|S_{\bullet,\bullet}|} \times \Delta^n_{Top} \cong {\vert({\vert S_{\bullet,\bullet} \vert})\vert}$$

using the fact that geometric realization of simplicial sets preserves colimits and products, and ${|\Delta^n_{sSet}|} = \Delta^n_{Top}$.

This fact is attributed to Tornehave by Quillen on page 94 of his ‘Higher Algebraic K-theory I’.

Finally, for any simplicial space $X_\bullet$, we have a bisimplicial set $Sing(X_\bullet)_\bullet$. Applying the previous proposition to this bisimplicial set, regarded as a discrete bisimplicial space, we find a homeomorphism

$${\vert({\vert Sing(X_\bullet) \vert})\vert} \cong {| d Sing(X_\bullet)_\bullet |} \,.$$

This also follows from results of (Gaunce Lewis 1982). Thus, as a resolution of $X_\bullet$, the levelwise realization of the levelwise singular complex ${\vert Sing(X_\bullet) \vert}$ has the pleasant property that its geometric realization, as a simplicial space, can be calculated as the realization of a single simplicial set (the diagonal of $Sing(X_\bullet)_\bullet$).

Properties

General

Remark

We have the following degenerate case of geometric realization of simplicial topological spaces.

• If $X_\bullet \colon \Delta^{op} \to Set \hookrightarrow Top$ is a degreewise discrete space, hence just a simplicial set, the notion of geometric realization above coincides with the notion of geometric realization of simplicial sets.

• If $X_\bullet$ is simplicially constant on a topological space $X_0$, then its geometric realization is homeomorphic to that space:

  $${\vert X_\bullet \vert} \simeq X_0 \,.$$

  (This is not true for the fat geometric realization, only the ordinary one. The fat geometric realization will be homotopy equivalent to $X$.)

Remark

Geometric realization of simplicial topological spaces has a right adjoint functor $\underline{Sing}$:

$$({\vert - \vert} \dashv \underline{Sing}) \colon sTop \stackrel{\overset{{\vert - \vert}}{\longrightarrow}}{\underset{\underline{Sing}} {\longleftarrow}} Top \,,$$

For $X \in$ Top a topological space, we have by definition

$$\underline{Sing}(X) \colon [n] \mapsto [\Delta^n_{Top}, X] \,,$$

where on the right we have the internal hom, or exponential, space from the standard topological $n$-simplex to $X$.

Proposition

For every $X \in Top$ there is a weak homotopy equivalence

$${|\underline{Sing}(X)|} \to X \,.$$

The proof can be found in Neil Strickland‘s answer to this mathoverflow question. (An incorrect argument appears as (Seymour, prop. 3.1) where it is claimed that $\underline{Sing}(X)$ is proper. In fact, the first degeneracy map $X \to [\Delta^1_{Top}, X]$ is not in general a cofibration as explained in the linked question.)

Ordinary geometric realization has the following two disadvantages:

Proposition

• If $X_\bullet$ has in each degree the homotopy type of a CW-complex, its realization ${\vert X_\bullet \vert}$ in general need not.

• If a morphism $f \colon X_\bullet \to Y_\bullet$ is degreewise a homotopy equivalence, its geometric realization ${\vert f \vert}$ need not be a homotopy equivalence.

See (Segal74, appendix A)

This is different for the fat geometric realization.

Proposition

• If $X_\bullet$ is degreewise of the homotopy type of a CW-complex (i.e. is degreewise m-cofibrant), then so is ${\Vert X_\bullet \Vert}$.

• If $f \colon X_\bullet \to Y_\bullet$ is degreewise a homotopy equivalence, then also ${\Vert f \Vert}$ is a homotopy equivalence.

This appears as (Segal74, prop. A.1).

Relation between fat and ordinary geometric realization

Proposition

If the simplicial topological space $X_\bullet$ is good then the natural morphism from its fat geometric realization to its ordinary geometric realization is a homotopy equivalence

$${\Vert X_\bullet \Vert} \stackrel{\simeq}{\longrightarrow} {|X_\bullet|} \,.$$

A direct proof of this (not using that good implies proper) appears as (Segal74, prop. A.1 (iv)) and a more detailed proof, using properness, is due to tom Dieck 1974, Prop. 1.

A recent paper treating the special case where $X_\bullet$ is the nerve of a topological category, and each $X_n$ is of the homotopy type of a CW-complex, is Wang 2017, Wang 18.

Compatibility with limits

We discuss how geometric realization interacts with limits of simplicial topological spaces.

Ordinary geometric realization

Proposition

(geometric realization prerserves pullbacks)

In the category Top of compactly generated topological spaces, geometric realization of simplicial spaces preserves pullbacks: for $X_\bullet \to Y_\bullet \leftarrow Z_\bullet$ a diagram in $Top^{\Delta^{op}}$ there are natural homeomorphisms

$${|X_\bullet|} \,\times_{|Y_\bullet|}\, {|Z_\bullet|} \simeq {|X_\bullet \,\times_{Y_\bullet}\, Z_\bullet|} \,.$$

This appears for instance as (May 1972, Cor. 11.6). See also the proof that geometric realization of simplicial sets preserves pullbacks, at geometric realization (here).

Remark

It is essential in Prop. to work in a category $Top$ such as compactly generated topological spaces (also “k-spaces”): in the category of all topological spaces the proposition would not be true. It works in these cases because product and/or quotient topologies in these categories are slightly different from in the category of all topological spaces.

It follows that:

Proposition

Geometric realization of simplicial compactly-generated topological spaces preserves all finite limits.

Proof

By this Prop. finite limits are preserved as soon as

1. the terminal object is preserved, which here is evidently the case;

2. pullbacks are preserved, which is the case by Prop. .

Fat geometric realization

The operation of fat geometric realization does not preserve fiber products on the nose, in general, but it does preserve all finite limits up to homotopy.

Write ${\Vert * \Vert}$ for the fat geometric realization of the point. Notice that due to the identification of $sTop$ with its overcategory over the point (the simplicial topological space constant on the point), $sTop\simeq sTop/*$, we may regard fat geometric gealization as a functor with values in the overcategory $Top/{\Vert*\Vert}$ over the fat geometric realization of the point.

Proposition

The functor

$${\Vert - \Vert} \colon Top^{\Delta^{op}} \to Top/{\Vert*\Vert}$$

preserves all finite limits.

This is claimed in Gepner-Henriques 07, Remark 2.23.

Preservation of homotopy limits

On homotopy limits:

Proposition

(homotopy pullbacks preserved by geometric realization)
For a morphism of simplicial spaces

(1)$$f_\bullet \,\colon\, X_\bullet \xrightarrow{\;} Y_\bullet \,,$$

the following are sufficient conditions for the geometric realization of any homotopy pullback-square of $f_\bullet$ to be a homotopy pullback-square in topological spaces (assuming throughout that all simplicial spaces are good or that the realization is taken to be fat, so that realization computes the homotopy colimit):

1. Bousfield & Friedlander 1978, Thm. B.4:

   1. on simplicial sets of connected components, $\pi_0(f)_\bullet$ is a Kan fibration,

   2. the simplicial spaces $X_\bullet$ and $Y_\bullet$ satisfy the π-Kan condition;

2. Anderson 1978, p. 2:

   1. on simplicial sets of connected components, $\pi_0(f)_\bullet$ is a Kan fibration,

   2. the component spaces $X_n$, $Y_n$ ($n \in \mathbb{N}$) are each connected or discrete;

3. Rezk 2014, Prop. 5.4:

   • the simplicial set of connected components of the base is constant: $\pi_0(Y)_\bullet \simeq const\big(\pi_0(Y_0)\big)_\bullet$.

   This is also essentially Lurie HA, Lem. 5.5.6.17 (using that simplicial $\infty$-colimits are sifted, by this Prop.), which generalizes the statement to $(\infty,1)$-toposes other than $L_W Top$.

4. Mazel-Gee 2014, Cor. 6.7, Lurie 2011, Prop. 10:

   • the morphism (1) is a homotopy Kan fibration, in that for all positive numbers $n \in \mathbb{N}_+$ and all $0 \leq k \leq n$ the induced map
   (2)$$X(\Delta^n) \xrightarrow{\;\;} X(\Lambda^n_k) \underset { Y(\Lambda^n_k) } {\times^h} Y(\Delta^n)$$

   (into the homotopy fiber product of the space of space of $(n,k)$-horns in $X$ with that of $n$-simplices in $Y$)

   is surjective on connected components,

   hence in that for all solid homotopy-commutative squares as follows, there exists a dashed lift up to homotopy (with the left morphism being the $(n,k)$-horn-inclusion in simplicial sets regarded as degree-wise discrete topological spaces):

Relation to the homotopy colimit

For proper simplicial spaces

Recall that a simplicial topological space is proper if it is Reedy cofibrant relative to the Strøm model structure on Top, in which the weak equivalences are the honest homotopy equivalences. Nevertheless, in certain cases geometric realisation computes the homotopy colimit of the diagram $X_\bullet \colon \Delta^{op} \to Top$ given by the simplicial space, with respect to the standard Quillen model structure on topological spaces in which the weak equivalences are the weak homotopy equivalences.

Lemma

If $X_\bullet \to Y_\bullet$ is an objectwise weak homotopy equivalence between proper simplicial spaces, then the induced map ${|X_\bullet|} \to {|Y_\bullet|}$ is a weak homotopy equivalence.

Proof

The following proof is essentially from (May 1974, A.4); see also (Dugger, prop. 17.4, example 18.2). It relies on two facts relating Hurewicz cofibrations to weak homotopy equivalences:

1. Pushouts along Hurewicz cofibrations preserve weak homotopy equivalences, and

2. Colimits of sequences of Hurewicz cofibrations preserve weak homotopy equivalences.

Let $i_n \colon \Delta_{\le n} \hookrightarrow \Delta$ denote the inclusion of the objects $\le n$, and write ${|X_\bullet|}_n = i_n^* \Delta \otimes_{\Delta_{\le n}} i_n^* X_\bullet$. Writing $L_n X$ for the $n$th latching object (the subspace of degeneracies in $X_n$), we have pushouts

$$\array{ (L_n X \times \Delta^n) \sqcup_{L_n X \times \partial\Delta^n} (X_n \times \partial\Delta^n) & \to & {|X_\bullet|}_{n-1}\\ \downarrow & & \downarrow\\ X_n \times \Delta^n & \to & {|X_\bullet|}_n }$$

Since $X$ is proper, $L_n X \to X_n$ is a cofibration, and of course $\partial\Delta^n \to \Delta^n$ is a cofibration. Thus, by the pushout-product axiom for the Strøm model structure, the left-hand vertical map is a cofibration; hence so is the right-hand vertical map.

Now ${|X_\bullet|}$ is the colimit of the sequence of cofibrations

$${|X_\bullet|}_0 \to {|X_\bullet|}_1 \to {|X_\bullet|}_2 \to \cdots$$

and likewise for ${|Y_\bullet|}$. In other words, the geometric realization is filtered by simplicial degree. Thus, by point (2) above, it suffices to show that each map ${|X_\bullet|}_n \to {|Y_\bullet|}_n$ is a weak homotopy equivalence.

Since ${|X_\bullet|}_0 = X_0$, this is true for $n=0$. Moreover, by the above pushout square, ${|X_\bullet|}_n$ is a pushout of ${|X_\bullet|}_{n-1}$ along a cofibration. Thus, by point (1) above, since $X_n \times \Delta^n \to Y_n \times \Delta^n$ is certainly a weak homotopy equivalence, it will suffice for an induction step to prove that

$$(L_n X \times \Delta^n) \sqcup_{L_n X \times \partial\Delta^n} (X_n \times \partial\Delta^n) \to (L_n Y \times \Delta^n) \sqcup_{L_n Y \times \partial\Delta^n} (Y_n \times \partial\Delta^n)$$

is a weak homotopy equivalence. However, by definition we have a pushout

$$\array{ L_n X \times \partial\Delta^n & \to & X_n \times \partial\Delta^n \\ \downarrow & & \downarrow\\ L_n X \times \Delta^n & \to & (L_n X \times \Delta^n) \sqcup_{L_n X \times \partial\Delta^n} (X_n \times \partial\Delta^n) }$$

This is also a pushout along a Hurewicz cofibration, and cartesian product preserves weak homotopy equivalences, so it will suffice to show that $L_n X \to L_n Y$ is a weak homotopy equivalence.

Recall that $L_n X$ can be written as $\colim^{[n]\to [k]} X_{[k]}$, where the colimit is over all codegeneracy maps $[n]\to [k]$ in $\Delta$ except the identity $[n] \to [n]$. For $0\le m \le n$, write $L^m_n X$ for the corresponding colimit over all codegeneracies which factor through $\sigma_{p}\colon [n]\to [n-1]$ for some $0\le p\le m$. Then $L^0_n X = X_{n-1}$ and $L^n_n X = L_n X$, and for $0\lt m \le n$ we have a pushout square

$$\array{ L^{m-1}_{n-1} X & \to & L^{m-1}_n X \\ \downarrow & & \downarrow \\ X_{n-1} & \to & L^m_n X. }$$

We claim that $L^{m-1}_n X \to L^m_n X$ is a cofibration for all $0\le m \le n$, and we prove it by induction on $n$. For $n=0$ it is obvious. If it holds for $(n-1)$ (and all $m$ with $0\le m \le n-1$), then by composition and properness of $X$, each map $L^{m-1}_{n-1} X \to X_{n-1}$ is a cofibration. Hence, by the above pushout square, so is $L^{m-1}_n X \to L^m_n X$. This proves the claim.

Now, using the above pushout square again and point (1) above, we can prove by induction on $n$, and for fixed $n$, by induction on $m$, that each map $L^m_n X \to L^m_n Y$ is a weak homotopy equivalence. In particular, taking $m=n$, we find that $L_n X \to L_n Y$ is a weak homotopy equivalence, as desired.

Recall that one way to compute the homotopy colimit of a diagram $X\colon D^{op}\to Top$, with respect to the standard (Quillen) model structure, is as the tensor product

$$hocolim X \coloneqq N(D/ -) \otimes_D Q X,$$

where $(D/ -)\colon D \to Cat$ sends each object of $D$ to its overcategory, $N$ denotes the nerve of a small category, and $Q$ denotes a functorial cofibrant replacement in the Quillen model structure on topological spaces (e.g. CW replacement via singular nerve and realization). When $D=\Delta$, there is a canonically defined map $N(\Delta / -) \to \Delta$ (where the second $\Delta$ denotes the canonical cosimplicial simplicial set) called the Bousfield-Kan map. This map induces, for each simplicial space $X$, a map

$$N(D/ -) \otimes_D X_\bullet \to \Delta \otimes_{\Delta} X_\bullet = {|X_\bullet|}$$

which is also called the Bousfield-Kan map.

Since the Strøm model structure is a simplicial model category, standard arguments involving Reedy model structures imply that the Bousfield-Kan map is a Strøm weak equivalence (i.e. a weak homotopy equivalence) whenever $X$ is Strøm Reedy cofibrant (i.e. proper). Thus we have (see also Arkhipov & Ørsted 2018, Exp. 6.4):

Proposition

Let $X_\bullet$ be a proper simplicial space. Then the composite

$$\hocolim X_\bullet = N(D/ -) \otimes_D Q X_\bullet \xrightarrow{Bousfield-Kan} {|Q X_\bullet|} \longrightarrow {|X_\bullet|}$$

is a weak homotopy equivalence.

Proof

Since the definition of $\hocolim$ doesn’t depend on the choice of an objectwise-cofibrant replacement of $X$, we may as well take $Q$ to be, instead of the composite with a functorial cofibrant replacement in $Top_{Quillen}$, rather a cofibrant replacement in the Reedy model structure on $Top^{\Delta^{op}}$ with respect to the Quillen model structure on $Top$. (Any Reedy cofibrant diagram is in particular objectwise cofibrant.) Then $Q X_\bullet$ is Quillen-Reedy cofibrant, hence also proper, and so the Bousfield-Kan map is a homotopy equivalence. On the other hand, $Q X_\bullet \to X_\bullet$ is a levelwise weak homotopy equivalence, while $Q X_\bullet$ and $X_\bullet$ are proper, so by Lemma its realization is a weak homotopy equivalence.

By naturality, the above composite is also equal to the composite

$$\hocolim X_\bullet = N(D/ -) \otimes_D Q X_\bullet \to N(D/ -) \otimes_D X_\bullet \xrightarrow{Bousfield-Kan} {|X_\bullet|};$$

hence this composite is also a weak homotopy equivalence.

For degreewise cofibrant simplicial spaces

Proposition

Let $X_\bullet$ be a simplicial topological space which is degreewise a cofibrant object in the classical model structure on topological spaces, hence which is degreewise a retract of a cell complex (for instance: degreewise a CW-complex).

Then its fat geometric realization models the homotopy colimit over $X_\bullet \;\colon\; \Delta^{op} \longrightarrow Top_{Quillen}$ in the classical model structure on topological spaces:

$$\left\Vert X_\bullet \right\Vert \;\simeq_{whe}\; hocolim X_\bullet \,.$$

This is claimed in Wang 18, Theorem 4.3, Remark 4.4.

Examples and Applications

Topological principal $\infty$-bundles

We discuss aspects of principal ∞-bundles equipped with topological cohesion and their geometric realization to principal bundles in Top.

For $X,Y \in sTop$, write

$$sTop(X,Y) \coloneqq \int_{[k] \in \Delta} [X_k, Y_k] \;\; \in Top$$

for the Top-hom-object, where in the integrand of the end $[-,-] \colon Top^{op} \times Top^{op} \to Top$ is the internal hom of topological spaces.

Definition

We say a morphism $f \colon X \to Y$ of simplicial topological spaces is a global Kan fibration if for all $n \in \mathbb{N}$ and $0 \leq i \leq n$ the canonical morphism

$$X_n \to Y_n \;\times_{sTop(\Lambda[n]_i, Y)}\; sTop(\Lambda[n]_i, X)$$

in Top has a section, where $\Lambda[n]_i \in$ sSet $\hookrightarrow Top^{\Delta^{op}}$ is the $i$th $n$-horn regarded as a discrete simplicial topological space.

We say a simplicial topological space $X_\bullet \in Top^{\Delta^{op}}$ is (global) Kan simplicial space if the unique morphism $X_\bullet \to *$ is a global Kan fibration, hence if for all $n \in \mathbb{N}$ and all $0 \leq i \leq n$ the canonical continuous function

$$X_n \to sTop(\Lambda[n]_i, X)$$

into the topological space of $i$th $n$-horns admits a section (in Top, hence a global, continuous section).

Remark

This global notion of topological Kan fibration is considered in (Brown & Szczarba, def. 2.1, def. 6.1). In fact there a stronger condition is imposed: a Kan complex in Set automatically has the lifting property not only against all full horn inclusions but also against sub-horns; and in (Brown & Szczarba) all these fillers are required to be given by global sections. This ensures that with $X$ globally Kan also the internal hom $[Y,X] \in sTop$ is globally Kan, for any simplicial topological space $Y$. This is more than we need and want to impose here. For our purposes it is sufficient to observe that if $f$ is globally Kan in the sense of (Brown & Szczarba, def. 6.1), then it is so also in the above sense.

Recall from the discussion at universal principal ∞-bundle that for $G$ a simplicial topological group the universal simplicial principal bundle $\mathbf{E}G \to \mathbf{B}G$ is presented by the morphism of simplicial topological spaces traditionally denoted $W G \to \overline{W} G$ (see also at universal principal simplicial complex).

Proposition

Let $G$ be a simplicial topological group. Then:

1. $G$ is a globally Kan simplicial topological space;

2. $\overline{W} G$ is a globally Kan simplicial topological space;

3. $W G \to \overline{W} G$ is a global Kan fibration.

Proof

The first statement appears as (Brown & Szczarba, theorem 3.8), the second is noted in (Roberts & Stevenson), the third appears as (Brown & Szczarba, lemma 6.7).

Proposition

If $G$ is a well-pointed simplicial topological group then both $W G$ and $\overline{W} G$ are good simplicial topological spaces.

For $\bar W G$ this is (RobertsStevenson, prop. 19). For $W G$ this follows with (RobertsStevenson, lemma 10, lemma 11) which says that $W G = Dec_0 \bar W G$ and the observations in the proof of (RobertsStevenson, prop. 16) that $Dec_0 X$ is good if $X$ is.

Proposition

For $G$ a well-pointed simplicial topological group, the geometric realization of the universal simplicial principal bundle $W G \to \bar W G$

$${\vert W G \vert} \to {\vert \bar W G \vert}$$

is a fibration resolution in $Top_{Quillen}$ of the point inclusion $* \to B{|G|}$ into the classifying space of the geometric realization of $G$.

This is (RobertsStevenson, prop. 14).

Proposition

Let $X_\bullet$ be a good simplicial topological space and $G$ a well-pointed simplicial topological group. Then for every morphism

$$\tau \colon X \to \bar W G$$

the corresponding topological simplicial principal bundle $P$ over $X$ is itself a good simplicial topological space.

Proof

The bundle is the pullback $P = X \times_{\overline{W} G} W G$ in $sTop$

$$\array{ P &\longrightarrow& W G \\ \big\downarrow && \big\downarrow \\ X &\stackrel{\tau}{\longrightarrow}& \overline{W} G \mathrlap{\,.} }$$

By assumption on $X$ and $G$ and using prop. we have that $X$, $\bar W G$ and $W G$ are all good simplicial spaces.

This means that the degeneracy maps of $P_\bullet$ are induced degreewise by morphisms between pullbacks in Top that are degreewise closed cofibrations, where one of the morphisms in each pullback is a fibration. By the properties discussed at closed cofibration (here), this implies that also these degeneracy maps of $P_\bullet$ are closed cofibrations.

Let for the following $Top_s \hookrightarrow Top$ be any small full subcategory.

Definition

Under the degreewise Yoneda embedding $sTop_s \hookrightarrow [Top_s^{op}, sSet]$ simplicial topological spaces embed into the category of simplicial presheaves on $Top_s$.

We equip this with the projective model structure on simplicial presheaves $[Top_s^{op}, sSet]_{proj}$, and we speak of homotopy limits in $sTop$ under this embedding.

Proposition

Under this embedding a global Kan fibration $f \colon X \to Y$ in $sTop_s$ maps to a fibration in $[Top_s^{op}, sSet]_{proj}$.

Proof

By definition, a morphism $f \colon X \to Y$ in $[Top_s^{op}, sSet]_{proj}$ is a fibration if for all $U \in Top$ and all $n \in \mathbb{N}$ and $0 \leq i \leq n$ diagrams of the form

$$\array{ \Lambda[n]_i \cdot U &\to& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ \Delta[n] \cdot U &\to& Y }$$

have a lift. This is equivalent to saying that the function

$$Hom(\Delta[n]\cdot U, X) \to Hom(\Delta[n]\cdot U,Y) \times_{Hom(\Lambda[n]_i \cdot U, Y)} Hom(\Lambda[n]_i \cdot U, X)$$

is surjective. Notice that we have

$$\begin{aligned} Hom_{[Top_s^{op}, sSet]}(\Delta[n]\cdot U, X) & = Hom_{sTop}(\Delta[n]\cdot U, X) \\ & = \int_{[k] \in \Delta} Hom_{Top}( \Delta[n]_k \times U, X_k) \\ & = \int_{[k] \in \Delta} Hom_{Top}(U, [\Delta[n]_k, X_k]) \\ & = Hom_{Top}(U, \int_{[k] \in \Delta} [\Delta[n]_k, X_k]) \\ & = Hom_{Top}(U, sTop(\Delta[n], X)) \\ & = Hom_{Top}(U, X_n) \end{aligned}$$

and analogously for the other factors in the above morphism. Therefore the lifting problem equivalently says that the function

$$Hom_{Top}(U, \; X_n \to Y_n \times_{sTop(\Lambda[n]_i, Y)} sTop(\Lambda[n]_i,X) \;)$$

is surjective. But by the assumption that $f \colon X \to Y$ is a global Kan fibration of simplicial topological spaces, def. , we have a section $\sigma \colon Y_n \times_{sTop(\Lambda[n]_i), Y} sTop(\Lambda[n]_i,X) \to X_n$. Therefore $Hom_{Top}(U, \sigma)$ is a section of our function.

Proposition

The homotopy colimit operation

$$sTop \hookrightarrow [Top_s^{op}, sSet]_{proj} \stackrel{hocolim}{\to} Top_{Quillen}$$

preserves homotopy fibers of morphisms $\tau \colon X \to \bar W G$ with $X$ good and globally Kan and $G$ well-pointed.

Proof

By prop. and prop. we have that $W G \to \bar W G$ is a fibration resolution of the point inclusion $* \to \bar W G$ in $[Top^{op}, sSet]_{proj}$. By the general discussion at homotopy limit this means that the homotopy fiber of a morphism $\tau \colon X \to \bar W G$ is computed as the ordinary pullback $P$ in

$$\array{ P &\to& W G \\ \downarrow && \downarrow \\ X &\stackrel{\tau}{\to}& \bar W G }$$

(since all objects $X$, $\bar W G$ and $W G$ are fibrant and at least one of the two morphisms in the pullback diagram is a fibration) and hence

$$hofib(\tau) \simeq P \,.$$

By prop. and prop. it follows that all objects here are good simplicial topological spaces. Therefore by prop. we have

$$hocolim P_\bullet \simeq {|P_\bullet|}$$

in Ho(Top). By prop. we have that

$$\cdots = {|X_\bullet|} \times_{|\bar W G|} {|W G|} \,.$$

But prop. says that this is again the presentation of a homotopy pullback/homotopy fiber by an ordinary pullback

$$\array{ {|P|} &\to& {|W G|} \\ \downarrow && \downarrow \\ {|X|} &\stackrel{\tau}{\to}& {|\bar W G|} } \,,$$

because $|W G| \to |\bar W G|$ is again a fibration resolution of the point inclusion. Therefore

$$hocolim P_\bullet \simeq hofib( {|\tau|} ) \,.$$

Finally by prop. and using the assumption that $X$ and $\bar W G$ are both good, this is

$$\cdots \simeq hofib (hocolim \tau) \,.$$

In total we have shown

$$hocolim (hofib \tau) \simeq hofib (hocolim \tau) \,.$$

Classifying spaces

For $G$ a topological group, write $\mathbf{B}G$ for its delooping topological groupoid: the topological groupoid with a single object and $Mor_{\mathbf{B}G}(*,*) \coloneqq G$, with composition given by the product on $G$.

The nerve $N \mathbf{B}G$ of this topological groupoid is naturally a simplicial topological space, with

$$N \mathbf{B}G \colon [n] \mapsto G^{\times_n} \,.$$

Proposition

The geometric realization of $N \mathbf{B}G$ is a model for the classifying space $B G$ of $G$-principal bundles

$${|N \mathbf{B}G|} \simeq B G \,.$$

An early reference for this classical fact is (Segal68).

Corollary

Let $G$ be a well-pointed topological group, $B G$ its classifying space and $\Omega B G$ the loop space of the classifying space.

There is a weak homotopy equivalence

$$\Omega B G \stackrel{\simeq}{\to} G \,.$$

Proof

By prop. we have that $B G \simeq |\mathbf{B}G|$ (we suppress the nerve operation notationally, which injects groupoids into ∞-groupoids).

By standard facts about the $\bar W$-functor (see simplicial principal bundle) we have a pullback square of simplicial topological spaces

$$\array{ G &\to& W G \\ \downarrow && \downarrow \\ * &\to& \bar W G }$$

exhibiting the homotopy pullback

$$\array{ G &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ * &\to& \mathbf{B}G } \,.$$

Under geoemtric realization this maps to

$$\array{ G &\to& * \\ \downarrow &\swArrow_{\simeq}& \downarrow \\ * &\to& B G }$$

in Top. By prop this is still a homotopy pullback, and hence exhibits $G$ as the loop space of $B G$.

Cech nerves

Proposition

Let $X,Y$ be topological spaces and $\pi \colon Y \to X$ a continuous function that admits local sections. Write $C(\pi) \in sTop$ for the Cech nerve.

Then the canonical map

$${|C(\pi)|} \to X$$

from the geometric realization of $X$ back to $X$ is a weak homotopy equivalence. If $X$ is a paracompact topological space then it is even a homotopy equivalence.

For paracompact $X$ this goes back to (Segal68). The general case is discussed in (DuggerIsaksen). A generalization to parameterized spaces is in (RobertsStevenson, lemma 22).

Related concepts

• simplicial topological space, nice simplicial topological space

• simplicial topological group

• geometric realization

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

References

General

The realization construction of simplicial topological spaces is implicit in classical discussion of classifying spaces $B G$, in particular

• R. James Milgram, The bar construction and abelian H-spaces, Illinois J. Math. 11(2): 242-250 (1967) (doi:10.1215/ijm/1256054662)

which may be understood as the realization of the nerves of their topological delooping groupoids $G \rightrightarrows \ast$,

More explicit allusion to realization at least of semi-simplicial nerves of topological categories appeared, without details, in:

• Graeme Segal, Section 2 of: Classifying spaces and spectral sequences, Publications Mathématiques de l’IHÉS, 34 (1968), p. 105-112 (numdam:PMIHES_1968__34__105_0)

The detailed coend formula for and the relevance of compactly generated topological spaces for preservation at least of finite products is due (with reference to Milgram 1967) to:

• Saunders MacLane, Section 6 of: The Milgram bar construction as a tensor product of functors, In: F.P. Peterson (eds.) The Steenrod Algebra and Its Applications: A Conference to Celebrate N.E. Steenrod’s Sixtieth Birthday Lecture Notes in Mathematics 168, Springer, 1970 (doi:10.1007/BFb0058523, pdf)

Further details, including proof of the preservation not just of binary products but also of pullbacks appears in:

• Peter May, Section 11 of: The geometry of iterated loop spaces, Springer 1972 (pdf, doi:10.1007/BFb0067491)

The notion of good simplicial spaces is due to

• Graeme Segal, Appendix 2 of: Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. (doi:10.1007/BF01390197, pdf, MR 0331377)

and the notion of fat geometric realization together with its homotopy equivalence to ordinary realization for good simplicial spaces is due to:

• Graeme Segal, Appendix A of: Categories and cohomology theories, Topology, 13:293–312, 1974 (doi:10.1016/0040-9383(74)90022-6)

• Tammo tom Dieck, On the homotopy type of classifying spaces, Manuscripta Math 11, 41–49 (1974) (doi:10.1007/BF01189090)

Discussion of fat geometric realization as representing the homotopy colimit (as a homotopy coend):

• Sergey Arkhipov, Sebastian Ørsted, Exp. 6.4 in: Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories (arXiv:1807.03266)

More discussion of the comparison of different realization functors is in:

• Yi-Sheng Wang, Fat realization and Segal’s classifying space (arXiv:1710.03796)

• Yi-Sheng Wang, Geometric realization and its variants (arXiv:1804.00345)

A proof that good simplicial spaces are proper is implicit in the proof of Segal 1974, Lemma A.5. It appears explicitly in

• L. Gaunce Lewis, Jr., When is the natural map $X\to \Omega \Sigma X$ a cofibration? , Trans. Amer. Math. Soc. 273 (1982) no. 1, 147–155 (jstor:1999197)

A generalization of the statement that good implies proper to other topological concrete categories and a discussion of the geometric realization of $W G \to \bar W G$ for $G$ a simplicial topological group is in

• David Roberts, Danny Stevenson, Simplicial principal bundles in parametrized spaces, New York Journal of Mathematics Volume 22 (2016) 405-440 (arXiv:1203.2460, nyjm:22-19)

• Danny Stevenson, Classifying theory for simplicial parametrized groups (arXiv:1203.2461)

Comments on the relation between properness and cofibrancy in the Reedy model structure on $[\Delta^{op}, Set]$ are made in

• Paul Goerss, Kristen Schemmerhorn, Model Categories and Simplicial Methods (pdf).

The relation between (fat) geometric realization and homotopy colimits is considered as prop. 17.5 and example 18.2 of

• Daniel Dugger, A primer on homotopy colimits (pdf)

The proof that geometric realization of proper simplicial spaces preserves weak equivalences is from

• Peter May, $E_\infty$-spaces, group completions, and permutative categories, London Math. Soc. Lecture Notes No. 11, 1974, 61-94 (doi:10.1017/CBO9780511662607.008, pdf)

A definition of the Bousfield-Kan map, and the Reedy model category theory necessary to show that it is a weak equivalence, can be found in

• Hirschhorn, Model categories and their localizations, AMS Mathematical Surveys and Monographs No. 99, 2003

On the (fat) geometric realization of (nerves of) topological groupoids:

• David Gepner, André Henriques, Section 2.3 in: Homotopy theory of orbispaces (arXiv:math/0701916)

See also

• David Carchedi, section 3.2 of On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces (arXiv:1504.02394)

On globally Kan simplicial spaces:

• Edgar H. Brown, R. H. Szczarba, Continuous cohomology and real homotopy type , Trans. Amer. Math. Soc. 311 (1989), no. 1, 57 (pdf)

On the right adjoint to geometric realization of simplicial topological spaces

• R. M. Seymour, Kan fibrations in the category of simplicial spaces Fund. Math., 106(2):141-152, 1980.

On geometric relatization of semi-simplicial topological spaces:

• Johannes Ebert, Oscar Randal-Williams, Semi-simplicial spaces, Algebr. Geom. Topol. 19 (2019) 2099-2150 (arXiv:1705.03774)

On geometric realization of general Cech nerves:

• Dan Dugger, Dan Isaksen, Topological hypercovers and $\mathbb{A}^1$- realizations, Math. Z. 246 (2004) no. 4

On geometric realization of simplicial spaces as a left Quillen functor on the Reedy model category of Top:

• Lars Hesselholt, Reedy model structure on the category of simplicial spaces; geometric realization is a left Quillen functor, Lecture 11 in Algebraic Topology I, 2008 (pdf, pdf)

(Non-)Compatibility with homotopy pullbacks

Discussion of sufficient conditions (cf. π-Kan condition) for geometric realization to be compatible with homotopy pullbacks:

• Aldridge Bousfield, Eric Friedlander, Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. (pdf, pdf)

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

• Jacob Lurie, Simplicial spaces, Lecture 7 of: Algebraic L-theory and Surgery, 2011 (pdf)

• 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)

• Aaron Mazel-Gee, Model $\infty$-categories I: some pleasant properties of the $\infty$-category of simplicial spaces (arXiv:1412.8411)

• Jacob Lurie, around Lem. 5.5.6.17 in: Higher Algebra (~2017)

• Jacob Lurie, Thms. A.5.4.1, Spectral Algebraic Geometry (2018) [pdf]

• Daniel Berwick-Evans, Pedro Boavida, Dmitri Pavlov, §3 of: Classifying spaces of infinity-sheaves, Algebraic & Geometric Topology [arXiv:1912.10544]

Realization of topological stacks

On geometric realizations of topological stacks:

• Johannes Ebert, The homotopy type of a topological stack (arXiv:0901.3295)