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geometric realization of simplicial topological spaces

Contents

Idea

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

The construction of X {|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 nn-simplices X nX_n.

Definition

Let Top in the following denote either

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

Write equivalently

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

for the category of simplicial topological spaces.

Definition

For X :Δ opTopX_\bullet \colon \Delta^{op} \to Top a simplicial topological space, its geometric realization is the coend

X nΔX n×Δ Top n {|X_\bullet|} \coloneqq \int^{n \in \Delta} X_n \times \Delta^n_{Top}

formed in Top.

This operation naturally extends to a functor

:sTopTop. {|-|} \colon sTop \to Top \,.

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

X = nX n×Δ Top n/ {\vert X_\bullet \vert} = \coprod_{n} X_n \times \Delta^n_{Top} /\sim

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

This form of geometric realization of simplicial topological spaces goes back to (Segal68). An early reference that realizes this construction as a coend is (MacLane).

One also considers geometric realization after restricting to the subcategory Δ +Δ\Delta_+ \hookrightarrow \Delta of the simplex category on the strictly increasing maps (that is, the coface maps only—no codegeneracies).

Definition

The corresponding coend in Top is called the fat geometric realization

X nΔ +X n×Δ Top n. {\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

X = nX n×Δ Top n/ + {\Vert X_\bullet \Vert} = \coprod_{n} X_n\times \Delta^n_{Top} /{\sim_+}

where the equivalence relation +\sim_+” identifies (x,f *p)X l×Δ Top l(x,f_* p) \in X_l\times \Delta^l_{Top} with (p,f *x)X k×Δ Top k(p,f^* x) \in X_k\times \Delta^k_{Top} only when [k][l][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

* iso* {\vert * \vert} \stackrel{\simeq_{iso}}{\to} *

is an isomorphism, but the morphism

* h.e.* {\Vert * \Vert} \stackrel{\simeq_{h.e.}}{\to} *

is just a homotopy equivalence.

Remark

The ordinary geometric realization can be described as the tensor product of functors X =Δ ΔX {|X_\bullet|}=\Delta^\bullet \otimes_{\Delta} X_\bullet, and the fat geometric realization can likewise be described as X =i *Δ Δ +i *X {\Vert X_\bullet\Vert} = i^* \Delta^\bullet \otimes_{\Delta_+} i^* X_\bullet, where i:Δ +Δi\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 X Δ ΔLan ii *X Lan ii *X {\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 X_\bullet is the ordinary geometric realization of a “fattened up” version of X X_\bullet, which is obtained by forgetting the degeneracy maps of X 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:Δ opTopX \colon \Delta^{op} \to Top be a simplicial topological space.

Such XX is called

  • good if all the degeneracy maps X n1X nX_{n-1} \hookrightarrow X_n are all closed cofibrations;

  • proper if the inclusion sX nX ns X_n \hookrightarrow X_n of the degenerate simplices is a closed cofibration, where sX n= is i(X n1)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:

The notion of good simplicial topological space goes back to (Segal73), that of proper simplicial topological space to (May).

Proposition

A good simplicial topological space is proper:

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

A proof appears as (Lewis, corollary 2.4 (b)). A generalization of this result to more general topological categories is (RobertsStevenson, prop. 16).

Bisimplicial sets and good resolutions

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

(Sing):TopSingsSet ({\vert- \vert} \dashv Sing) \colon Top \stackrel{\overset{{|-|}}{\leftarrow}}{\underset{Sing}{\to}} sSet

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

Proposition

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

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

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

SingX X {|Sing X_\bullet|} \to X_\bullet

which is degreewise a weak homotopy equivalence.

Proof

Each space SingX n|Sing X_n| is a CW-complex, hence in particular a locally equi-connected space. By (Lewis, 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 SingX |Sing X_\bullet| are closed cofibrations.

The second sentence follows directly by the remarks above.

Note that there is nothing special about Sing(){|Sing(-)|} in the proof; any functorial CW replacement would do just as well (such as that obtained by the small object argument). However, Sing(){|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 ,:Δ op×Δ opSetS_{\bullet,\bullet} \colon \Delta^{op} \times \Delta^{op} \to Set is a bisimplicial set, we write dSd S for its diagonal, which is the composite

Δ opΔ op×Δ opSSet.\Delta^{op} \to \Delta^{op} \times \Delta^{op} \stackrel{S}{\to} Set.

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

S , nΔ opS n,×Δ sSet n {|S_{\bullet,\bullet}|} \coloneqq \int^{n\in\Delta^{op}} S_{n,\bullet} \times \Delta^n_{sSet}

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

Lemma

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

Proof

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

(dΔ n,m)()=Δ(,n)×Δ(,m)=Δ sSet n×Δ sSet m.(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:sSetsSet Δ opSing\colon sSet \to sSet^{\Delta^{op}} defined by

Sing(X )(n,m)=Map̲(Δ sSet n,X ) m=sSet(Δ sSet n×Δ sSet m,X )=sSet(dΔ n,m,X )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 Map̲(,)\underline{Map}(-,-) denotes the simplicial mapping space between two simplicial sets. But by the Yoneda lemma, Sing(X )(n,m)=sSet Δ op(Δ n,m,Sing(X ))Sing(X_\bullet)(n,m) = sSet^{\Delta^{op}}(\Delta^{n,m},Sing(X_\bullet)), so this shows that dΔ n,md \Delta^{n,m} has the same universal property as Δ n,m{|\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 1 can be interpreted as an isomorphism between two profunctors ΔΔ×Δ\Delta ⇸ \Delta\times\Delta, of which the first is representable by the diagonal functor ΔΔ×Δ\Delta \to \Delta\times\Delta. It follows that if S ,S_{\bullet,\bullet} is a bisimplicial object in any cocomplete category, we also have

dS , nΔS n,×(Δ sSet n) 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 SS is a bisimplicial space, then we also have the levelwise realization

nΔS n,×Δ Top n \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 ,S_{\bullet,\bullet} any bisimplicial space, there is a homeomorphism between the geometric realizations of the following two simplicial spaces: (1) the diagonal dS ,d S_{\bullet,\bullet} of SS, and (2) the levelwise realization S ,|S_{\bullet,\bullet}| of SS.

Proof

Applying Lemma 1 as above, we have

dS , nΔS n,×Δ sSet n 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

dS , nΔS n,×Δ sSet n nΔS ,×Δ Top n(S ,) {| 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 Δ sSet n=Δ Top n{|\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 X_\bullet, we have a bisimplicial set Sing(X ) Sing(X_\bullet)_\bullet. Applying the previous proposition to this bisimplicial set, regarded as a discrete bisimplicial space, we find a homeomorphism

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

This also follows from results of (Lewis). Thus, as a resolution of X X_\bullet, the levelwise realization of the levelwise singular complex Sing(X ){\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 ) Sing(X_\bullet)_\bullet).

Properties

General

Remark

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

  • If X :Δ opSetTopX_\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 X_\bullet is simplicially constant on a topological space X 0X_0, then its geometric realization is homeomorphic to that space:

    X X 0. {\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 XX.)

Remark

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

(Sing̲):sTopSing̲Top, ({\vert - \vert} \dashv \underline{Sing}) \colon sTop \stackrel{\overset{{\vert - \vert}}{\to}}{\underset{\underline{Sing}} {\leftarrow}} Top \,,

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

Sing̲(X):[n][Δ Top n,X], \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 nn-simplex to XX.

Proposition

For every XTopX \in Top there is a weak homotopy equivalence

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

This appears as (Seymour, prop. 3.1).

Ordinary geometric realization has the following two disadvantages:

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

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

See (Segal74, appendix A)

This is different for the fat geometric realization.

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

  • If f:X Y f \colon X_\bullet \to Y_\bullet is degreewise a homotopy equivalence, then also f{\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 X_\bullet is good then the natural morphism from its fat geometric realization to its ordinary geometric realization is a homotopy equivalence

X X . {\Vert X_\bullet \Vert} \stackrel{\simeq}{\to} {|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 has been given by Tammo tom Dieck.

Compatibility with limits

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

Ordinary geometric realization

Proposition

Geometric realization preserves pullbacks: for X Y Z X_\bullet \to Y_\bullet \leftarrow Z_\bullet a diagram in Top Δ opTop^{\Delta^{op}} there are natural homeomorphisms

X × Y Z X × Y Z . {|X_\bullet|} \,\times_{|Y_\bullet|}\, {|Z_\bullet|} \simeq {|X_\bullet \,\times_{Y_\bullet}\, Z_\bullet|} \,.

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

It is essential here that we are working in a category TopTop such as compactly generated spaces or k-spaces: in the category of all topological spaces this 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.

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

Proposition

The functor

:Top Δ opTop/* {\Vert - \Vert} \colon Top^{\Delta^{op}} \to Top/{\Vert*\Vert}

preserves all finite limits.

See (GepnerHenriques, remark 2.23).

Relation to the homotopy colimit

Recall that a simplicial space is proper if it is Reedy cofibrant relative to the Strøm model structure on TopTop, in which the weak equivalences are the honest homotopy equivalences. Nevertheless, in certain cases geometric realisation computes the homotopy colimit of the diagram X :Δ opTopX_\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 Y X_\bullet \to Y_\bullet is an objectwise weak homotopy equivalence between proper simplicial spaces, then the induced map X Y {|X_\bullet|} \to {|Y_\bullet|} is a weak homotopy equivalence.

Proof

The following proof is essentially from (May74, 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:Δ nΔi_n \colon \Delta_{\le n} \hookrightarrow \Delta denote the inclusion of the objects n\le n, and write X n=i n *Δ Δ ni n *X {|X_\bullet|}_n = i_n^* \Delta \otimes_{\Delta_{\le n}} i_n^* X_\bullet. Writing L nXL_n X for the nnth latching object (the subspace of degeneracies in X nX_n), we have pushouts

(L nX×Δ n) L nX×Δ n(X n×Δ n) X n1 X n×Δ n X n\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 XX is proper, L nXX nL_n X \to X_n is a cofibration, and of course Δ nΔ n\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 {|X_\bullet|} is the colimit of the sequence of cofibrations

X 0X 1X 2 {|X_\bullet|}_0 \to {|X_\bullet|}_1 \to {|X_\bullet|}_2 \to \cdots

and likewise for Y {|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 nY n{|X_\bullet|}_n \to {|Y_\bullet|}_n is a weak homotopy equivalence.

Since X 0=X 0{|X_\bullet|}_0 = X_0, this is true for n=0n=0. Moreover, by the above pushout square, X n{|X_\bullet|}_n is a pushout of X n1{|X_\bullet|}_{n-1} along a cofibration. Thus, by point (1) above, since X n×Δ nY n×Δ nX_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 nX×Δ n) L nX×Δ n(X n×Δ n)(L nY×Δ n) L nY×Δ n(Y n×Δ n)(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

L nX×Δ n X n×Δ n L nX×Δ n (L nX×Δ n) L nX×Δ n(X n×Δ n) \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 nXL nYL_n X \to L_n Y is a weak homotopy equivalence.

Recall that L nXL_n X can be written as colim [n][k]X [k]\colim^{[n]\to [k]} X_{[k]}, where the colimit is over all codegeneracy maps [n][k][n]\to [k] in Δ\Delta except the identity [n][n][n] \to [n]. For 0mn0\le m \le n, write L n mXL^m_n X for the corresponding colimit over all codegeneracies which factor through σ p:[n][n1]\sigma_{p}\colon [n]\to [n-1] for some 0pm0\le p\le m. Then L n 0X=X n1L^0_n X = X_{n-1} and L n nX=L nXL^n_n X = L_n X, and for 0<mn0\lt m \le n we have a pushout square

L n1 m1X L n m1X X n1 L n mX. \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 n m1XL n mXL^{m-1}_n X \to L^m_n X is a cofibration for all 0mn0\le m \le n, and we prove it by induction on nn. For n=0n=0 it is obvious. If it holds for (n1)(n-1) (and all mm with 0mn10\le m \le n-1), then by composition and properness of XX, each map L n1 m1XX n1L^{m-1}_{n-1} X \to X_{n-1} is a cofibration. Hence, by the above pushout square, so is L n m1XL n mXL^{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 nn, and for fixed nn, by induction on mm, that each map L n mXL n mYL^m_n X \to L^m_n Y is a weak homotopy equivalence. In particular, taking m=nm=n, we find that L nXL nYL_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:D opTopX\colon D^{op}\to Top, with respect to the standard (Quillen) model structure, is as the tensor product

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

where (D/):DCat(D/ -)\colon D \to Cat sends each object of DD to its overcategory, NN denotes the nerve of a small category, and QQ denotes a functorial cofibrant replacement in the Quillen model structure (e.g. CW replacement). When D=ΔD=\Delta, there is a canonically defined map N(Δ/)ΔN(\Delta / -) \to \Delta (where the second Δ\Delta denotes the canonical cosimplicial simplicial set) called the Bousfield-Kan map. This map induces, for each simplicial space XX, a map

N(D/) DX Δ ΔX =X 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 homotopy equivalence) whenever XX is Strøm Reedy cofibrant (i.e. proper). Thus we have:

Proposition

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

hocolimX =N(D/) DQX BousfieldKanQX X \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\hocolim doesn’t depend on the choice of an objectwise-cofibrant replacement of XX, we may as well take QQ to be, instead of the composite with a functorial cofibrant replacement in Top QuillenTop_{Quillen}, rather a cofibrant replacement in the Reedy model structure on Top Δ opTop^{\Delta^{op}} with respect to the Quillen model structure on TopTop. (Any Reedy cofibrant diagram is in particular objectwise cofibrant.) Then QX Q X_\bullet is Quillen-Reedy cofibrant, hence also proper, and so the Bousfield-Kan map is a homotopy equivalence. On the other hand, QX X Q X_\bullet \to X_\bullet is a levelwise weak homotopy equivalence, while QX Q X_\bullet and X X_\bullet are proper, so by Lemma 2 its realization is a weak homotopy equivalence.

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

hocolimX =N(D/) DQX N(D/) DX BousfieldKanX ; \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.

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,YsTopX,Y \in sTop, write

sTop(X,Y) [k]Δ[X k,Y k]Top 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 [,]:Top op×Top opTop[-,-] \colon Top^{op} \times Top^{op} \to Top is the internal hom of topological spaces.

Definition

We say a morphism f:XYf \colon X \to Y of simplicial topological spaces is a global Kan fibration if for all nn \in \mathbb{N} and 0in0 \leq i \leq n the canonical morphism

X nY n× sTop(Λ[n] i,Y)sTop(Λ[n] i,X) X_n \to Y_n \;\times_{sTop(\Lambda[n]_i, Y)}\; sTop(\Lambda[n]_i, X)

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

We say a simplicial topological space X Top Δ opX_\bullet \in Top^{\Delta^{op}} is (global) Kan simplicial space if the unique morphism X *X_\bullet \to * is a global Kan fibration, hence if for all nn \in \mathbb{N} and all 0in0 \leq i \leq n the canonical continuous function

X nsTop(Λ[n] i,X) X_n \to sTop(\Lambda[n]_i, X)

into the topological space of iith nn-horns admits a section (in Top, hence a global, continuous section).

Remark

This global notion of topological Kan fibration is considered in (BrownSzczarba, 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 (BrownSzczarba) all these fillers are required to be given by global sections. This ensures that with XX globally Kan also the internal hom [Y,X]sTop[Y,X] \in sTop is globally Kan, for any simplicial topological space YY. This is more than we need and want to impose here. For our purposes it is sufficient to observe that if ff is globally Kan in the sense of (BrownSzczarba, def. 6.1), then it is so also in the above sense.

Recall from the discussion at universal principal ∞-bundle that for GG a simplicial topological group the universal simplicial principal bundle EGBG\mathbf{E}G \to \mathbf{B}G is presented by the morphism of simplicial topological spaces traditionally denoted WGW¯GW G \to \bar W G.

Proposition

Let GG be a simplicial topological group. Then

  1. GG is a globally Kan simplicial topological space;

  2. W¯G\bar W G is a globally Kan simplicial topological space;

  3. WGW¯GW G \to \bar W G is a global Kan fibration.

Proof

The first statement appears as (BrownSzczarba, theorem 3.8), the second is noted in (RobertsStevenson), the third appears as (BrownSzczarba, lemma 6.7).

Proposition

If GG is a well-pointed simplicial topological group then both WGW G and W¯G\bar W G are good simplicial topological space.

For W¯G\bar W G this is (RobertsStevenson, prop. 19). For WGW G this follows with (RobertsStevenson, lemma 10, lemma 11) which says that WG=Dec 0W¯GW G = Dec_0 \bar W G and the observations in the proof of (RobertsStevenson, prop. 16) that Dec 0XDec_0 X is good if XX is.

Proposition

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

WGW¯G {\vert W G \vert} \to {\vert \bar W G \vert}

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

This is (RobertsStevenson, prop. 14).

Proposition

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

τ:XW¯G \tau \colon X \to \bar W G

the corresponding topological simplicial principal bundle PP over XX is itself a good simplicial topological space.

Proof

The bundle is the pullback P=X× W¯GWGP = X \times_{\bar W G} W G in sTopsTop

P W¯G X τ W¯G. \array{ P &\to& \bar W G \\ \downarrow && \downarrow \\ X &\stackrel{\tau}{\to}& \bar W G } \,.

By assumption on XX and GG and using prop. 12 we have that XX, W¯G\bar W G and WGW G are all good simplicial spaces.

This means that the degeneracy maps of P 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, this implies that also these degeneracy maps of P P_\bullet are closed cofibrations.

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

Definition

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

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

Proposition

Under this embedding a global Kan fibration f:XYf \colon X \to Y in sTop ssTop_s maps to a fibration in [Top s op,sSet] proj[Top_s^{op}, sSet]_{proj}.

Proof

By definition, a morphism f:XYf \colon X \to Y in [Top s op,sSet] proj[Top_s^{op}, sSet]_{proj} is a fibration if for all UTopU \in Top and all nn \in \mathbb{N} and 0in0 \leq i \leq n diagrams of the form

Λ[n] iU X f Δ[n]U Y \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(Δ[n]U,X)Hom(Δ[n]U,Y)× Hom(Λ[n] iU,Y)Hom(Λ[n] iU,X) 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

Hom [Top s op,sSet](Δ[n]U,X) =Hom sTop(Δ[n]U,X) = [k]ΔHom Top(Δ[n] k×U,X k) = [k]ΔHom Top(U,[Δ[n] k,X k]) =Hom Top(U, [k]Δ[Δ[n] k,X k]) =Hom Top(U,sTop(Δ[n],X)) =Hom Top(U,X n) \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 nY n× sTop(Λ[n] i,Y)sTop(Λ[n] i,X)) 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:XYf \colon X \to Y is a global Kan fibration of simplicial topological spaces, def. 4, we have a section σ:Y n× sTop(Λ[n] i),YsTop(Λ[n] i,X)X n\sigma \colon Y_n \times_{sTop(\Lambda[n]_i), Y} sTop(\Lambda[n]_i,X) \to X_n. Therefore Hom Top(U,σ)Hom_{Top}(U, \sigma) is a section of our function.

Proposition

The homotopy colimit operation

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

preserves homotopy fibers of morphisms τ:XW¯G\tau \colon X \to \bar W G with XX good and globally Kan and GG well-pointed.

Proof

By prop. 11 and prop. 15 we have that WGW¯GW G \to \bar W G is a fibration resolution of the point inclusion *W¯G* \to \bar W G in [Top op,sSet] proj[Top^{op}, sSet]_{proj}. By the general discussion at homotopy limit this means that the homotopy fiber of a morphism τ:XW¯G\tau \colon X \to \bar W G is computed as the ordinary pullback PP in

P WG X τ W¯G \array{ P &\to& W G \\ \downarrow && \downarrow \\ X &\stackrel{\tau}{\to}& \bar W G }

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

hofib(τ)P. hofib(\tau) \simeq P \,.

By prop. 11 and prop. 14 it follows that all objects here are good simplicial topological spaces. Therefore by prop. 10 we have

hocolimP P hocolim P_\bullet \simeq {|P_\bullet|}

in Ho(Top). By prop. 8 we have that

=X × W¯GWG. \cdots = {|X_\bullet|} \times_{|\bar W G|} {|W G|} \,.

But prop. 13 says that this is again the presentation of a homotopy pullback/homotopy fiber by an ordinary pullback

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

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

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

Finally by prop. 10 and using the assumption that XX and W¯G\bar W G are both good, this is

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

In total we have shown

hocolim(hofibτ)hofib(hocolimτ). hocolim (hofib \tau) \simeq hofib (hocolim \tau) \,.

Classifying spaces

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

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

NBG:[n]G × n. N \mathbf{B}G \colon [n] \mapsto G^{\times_n} \,.
Proposition

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

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

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

Corollary

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

There is a weak homotopy equivalence

ΩBGG. \Omega B G \stackrel{\simeq}{\to} G \,.
Proof

By prop. 17 we have that BGBGB G \simeq |\mathbf{B}G| (we suppress the nerve operation notationally, which injects groupoids into ∞-groupoids).

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

G WG * W¯G \array{ G &\to& W G \\ \downarrow && \downarrow \\ * &\to& \bar W G }

exhibiting the homotopy pullback

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

Under geoemtric realization this maps to

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

in Top. By prop 16 this is still a homotopy pullback, and hence exhibits GG as the loop space of BGB G.

Cech nerves

Proposition

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

Then the canonical map

C(π)X {|C(\pi)|} \to X

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

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

References

The first occurence of the definition of geometric realization of simplicial topological spaces seems to be

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

but the construction was implicit in earlier discussion of classifying spaces. The observation that this is a coend was noted in

  • Saunders MacLane, The Milgram bar construction as a tensor product of functors SLNM Vol. 168 (1970)

The definition of good simplicial topological spaces goes back to

  • Graeme Segal, Configuration-Spaces and Iterated Loop-Spaces , Inventiones math. 21,213-221 (1973)

An original reference on geometric realization of simplicial topological spaces is is appendix A of

A standard textbook reference is chapter 11 of

A proof that good simplicial spaces are proper is implicit in the proof of lemma A.5 in (Segal74). Explicitly it appears in

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

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

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

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

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

  • Peter May, E E_\infty-spaces, group completions, and permutative categories. London Math. Soc. Lecture Notes No. 11, 1974, 61-93.

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

The (fat) geometric realization of (nerves of) topological groupoids is discussed in section 2.3 of

Globally Kan simplicial spaces are considered in

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

The right adjoint to geometric realization of simplicial topological spaces is discussed in

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

Geometric realization of general Cech nerves is discussed in

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

The behaviour of fibrations under geometric realization and the preservation of homotopy pullbacks under geometric realization is discussed in

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

Refereeing

This entry is under review. See geometric realization of simplicial topological spaces at nLab (reviewed).

Revised on March 15, 2012 14:26:16 by Urs Schreiber (82.172.178.200)