topology (point-set topology)
see also algebraic topology, functional analysis and homotopy theory
nice topological space
Kolmogorov space, Hausdorff space, regular space, normal space
compact space, paracompact space, compactly generated space
connected space, locally connected space, contractible space, locally contractible space
topological vector space, Banach space, Hilbert space
empty space, point space
discrete space, codiscrete space
circle, torus, annulus
loop space, path space
Cantor space, Sierpinski space
long line, line with two origins
For a simplicial object in Top – a simplicial topological space – its geometric realization is a plain topological space obtained by gluing all topological space together, as determined by the face and degeneracy maps.
The construction of 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 -simplices .
Let Top in the following denote either
Let denote the simplex category, and write for the standard cosimplicial topological space of topological simplices.
for the category of simplicial topological spaces.
For a simplicial topological space, its geometric realization is the coend
formed in Top.
This operation naturally extends to a functor
More explicitly, is the topological space given by the quotient
where the equivalence relation “” identifies, for every morphism in , the points and .
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 of the simplex category on the strictly increasing maps (that is, the coface maps only—no codegeneracies).
The corresponding coend in Top is called the fat geometric realization
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
where the equivalence relation “” identifies with only when is a coface map.
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
is an isomorphism, but the morphism
is just a homotopy equivalence.
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.
Let be a simplicial topological space.
Such is called
good if all the degeneracy maps are all closed cofibrations;
proper if the inclusion of the degenerate simplices is a closed cofibration, where .
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).
A good simplicial topological space is 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
for the ordinary geometric realization/singular simplicial complex adjunction (see homotopy hypothesis). Recall that the composite is a cofibrant replacement functor: for any space , the space is a CW complex and comes with a natural map (the counit of the adjunction ) which is a weak homotopy equivalence.
Let be a simplicial topological space. Then the simplicial topological space
obtained by applying degreewise, is good and hence proper. Moreover, we have a natural morphism
which is degreewise a weak homotopy equivalence.
Each space 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 are closed cofibrations.
The second sentence follows directly by the remarks above.
Note that there is nothing special about in the proof; any functorial CW replacement would do just as well (such as that obtained by the small object argument). However, has the advantage that its geometric realization can be computed alternately in terms of diagonals of bisimplicial sets, as we now show.
If is a bisimplicial set, we write for its diagonal, which is the composite
On the other hand, we can also consider a bisimplicial set as a simplicial object in and take its “geometric realization”:
where denotes the -simplex as a simplicial set, i.e. the representable functor .
For a bisimplicial set , there is a natural isomorphism of simplicial sets .
Both functors and are cocontinuous functors between presheaf categories , so it suffices to verify that they agree on representables. The representable is called a bisimplex ; its diagonal is
Now note that geometric realization of a bisimplicial set is left adjoint to the “singular complex” defined by
where denotes the simplicial mapping space between two simplicial sets. But by the Yoneda lemma, , so this shows that has the same universal property as ; 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 , of which the first is representable by the diagonal functor . It follows that if is a bisimplicial object in any cocomplete category, we also have
where the right-hand side is a “realization” functor from bisimplicial objects to simplicial objects in any cocomplete category. On the other hand, if is a bisimplicial space, then we also have the levelwise realization
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.
For any bisimplicial space, there is a homeomorphism between the geometric realizations of the following two simplicial spaces: (1) the diagonal of , and (2) the levelwise realization of .
Applying Lemma 1 as above, we have
If we then take the realization of these simplicial spaces, we find
using the fact that geometric realization of simplicial sets preserves colimits and products, and .
This fact is attributed to Tornehave by Quillen on page 94 of his ‘Higher Algebraic K-theory I’.
Finally, for any simplicial space , we have a bisimplicial set . Applying the previous proposition to this bisimplicial set, regarded as a discrete bisimplicial space, we find a homeomorphism
This also follows from results of (Lewis). Thus, as a resolution of , the levelwise realization of the levelwise singular complex 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 ).
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 is proper. In fact, the first degeneracy map is not in general a cofibration as explained in the linked question.)
Ordinary geometric realization has the following two disadvantages:
If has in each degree the homotopy type of a CW-complex, its realization in general need not.
If a morphism is degreewise a homotopy equivalence, its geometric realization need not be a homotopy equivalence.
See (Segal74, appendix A)
This is different for the fat geometric realization.
If is degreewise of the homotopy type of a CW-complex (i.e. is degreewise m-cofibrant), then so is .
If is degreewise a homotopy equivalence, then also is a homotopy equivalence.
This appears as (Segal74, prop. A.1).
Relation between fat and ordinary geometric realization
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
Geometric realization preserves pullbacks: for a diagram in there are natural homeomorphisms
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 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 for the fat geometric realization of the point. Notice that due to the identification of with its overcategory over the point (the simplicial topological space constant on the point), , we may regard fat geometric gealization as a functor with values in the overcategory over the fat geometric realization of the point.
preserves all finite limits.
See (GepnerHenriques, remark 2.23).
Relation to the homotopy colimit
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 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.
If is an objectwise weak homotopy equivalence between proper simplicial spaces, then the induced map is a weak homotopy equivalence.
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:
Pushouts along Hurewicz cofibrations preserve weak homotopy equivalences, and
Colimits of sequences of Hurewicz cofibrations preserve weak homotopy equivalences.
Let denote the inclusion of the objects , and write . Writing for the th latching object (the subspace of degeneracies in ), we have pushouts
Since is proper, is a cofibration, and of course 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 is the colimit of the sequence of cofibrations
and likewise for . In other words, the geometric realization is filtered by simplicial degree. Thus, by point (2) above, it suffices to show that each map is a weak homotopy equivalence.
Since , this is true for . Moreover, by the above pushout square, is a pushout of along a cofibration. Thus, by point (1) above, since is certainly a weak homotopy equivalence, it will suffice for an induction step to prove that
is a weak homotopy equivalence. However, by definition we have a pushout
This is also a pushout along a Hurewicz cofibration, and cartesian product preserves weak homotopy equivalences, so it will suffice to show that is a weak homotopy equivalence.
Recall that can be written as , where the colimit is over all codegeneracy maps in except the identity . For , write for the corresponding colimit over all codegeneracies which factor through for some . Then and , and for we have a pushout square
We claim that is a cofibration for all , and we prove it by induction on . For it is obvious. If it holds for (and all with ), then by composition and properness of , each map is a cofibration. Hence, by the above pushout square, so is . This proves the claim.
Now, using the above pushout square again and point (1) above, we can prove by induction on , and for fixed , by induction on , that each map is a weak homotopy equivalence. In particular, taking , we find that is a weak homotopy equivalence, as desired.
Recall that one way to compute the homotopy colimit of a diagram , with respect to the standard (Quillen) model structure, is as the tensor product
where sends each object of to its overcategory, denotes the nerve of a small category, and denotes a functorial cofibrant replacement in the Quillen model structure on topological spaces (e.g. CW replacement via singular nerve and realization). When , there is a canonically defined map (where the second denotes the canonical cosimplicial simplicial set) called the Bousfield-Kan map. This map induces, for each simplicial space , a map
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 is Strøm Reedy cofibrant (i.e. proper). Thus we have:
Let be a proper simplicial space. Then the composite
is a weak homotopy equivalence.
Since the definition of doesn’t depend on the choice of an objectwise-cofibrant replacement of , we may as well take to be, instead of the composite with a functorial cofibrant replacement in , rather a cofibrant replacement in the Reedy model structure on with respect to the Quillen model structure on . (Any Reedy cofibrant diagram is in particular objectwise cofibrant.) Then is Quillen-Reedy cofibrant, hence also proper, and so the Bousfield-Kan map is a homotopy equivalence. On the other hand, is a levelwise weak homotopy equivalence, while and are proper, so by Lemma 2 its realization is a weak homotopy equivalence.
By naturality, the above composite is also equal to the composite
hence this composite is also a weak homotopy equivalence.
Examples and Applications
Topological principal -bundles
We discuss aspects of principal ∞-bundles equipped with topological cohesion and their geometric realization to principal bundles in Top.
For , write
for the Top-hom-object, where in the integrand of the end is the internal hom of topological spaces.
We say a morphism of simplicial topological spaces is a global Kan fibration if for all and the canonical morphism
in Top has a section, where sSet is the th -horn regarded as a discrete simplicial topological space.
We say a simplicial topological space is (global) Kan simplicial space if the unique morphism is a global Kan fibration, hence if for all and all the canonical continuous function
into the topological space of th -horns admits a section (in Top, hence a global, continuous section).
Recall from the discussion at universal principal ∞-bundle that for a simplicial topological group the universal simplicial principal bundle is presented by the morphism of simplicial topological spaces traditionally denoted .
Let be a simplicial topological group. Then
is a globally Kan simplicial topological space;
is a globally Kan simplicial topological space;
is a global Kan fibration.
For this is (RobertsStevenson, prop. 19). For this follows with (RobertsStevenson, lemma 10, lemma 11) which says that and the observations in the proof of (RobertsStevenson, prop. 16) that is good if is.
This is (RobertsStevenson, prop. 14).
The bundle is the pullback in
By assumption on and and using prop. 12 we have that , and are all good simplicial spaces.
This means that the degeneracy maps of 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 are closed cofibrations.
Let for the following be any small full subcategory.
Under this embedding a global Kan fibration in maps to a fibration in .
By definition, a morphism in is a fibration if for all and all and diagrams of the form
have a lift. This is equivalent to saying that the function
is surjective. Notice that we have
and analogously for the other factors in the above morphism. Therefore the lifting problem equivalently says that the function
is surjective. But by the assumption that is a global Kan fibration of simplicial topological spaces, def. 4, we have a section . Therefore is a section of our function.
The homotopy colimit operation
preserves homotopy fibers of morphisms with good and globally Kan and well-pointed.
By prop. 11 and prop. 15 we have that is a fibration resolution of the point inclusion in . By the general discussion at homotopy limit this means that the homotopy fiber of a morphism is computed as the ordinary pullback in
(since all objects , and are fibrant and at least one of the two morphisms in the pullback diagram is a fibration) and hence
By prop. 11 and prop. 14 it follows that all objects here are good simplicial topological spaces. Therefore by prop. 10 we have
in Ho(Top). By prop. 8 we have that
But prop. 13 says that this is again the presentation of a homotopy pullback/homotopy fiber by an ordinary pullback
because is again a fibration resolution of the point inclusion. Therefore
Finally by prop. 10 and using the assumption that and are both good, this is
In total we have shown
For a topological group, write for its delooping topological groupoid: the topological groupoid with a single object and , with composition given by the product on .
The nerve of this topological groupoid is naturally a simplicial topological space, with
The geometric realization of is a model for the classifying space of -principal bundles
An early reference for this classical fact is (Segal68).
By prop. 17 we have that (we suppress the nerve operation notationally, which injects groupoids into ∞-groupoids).
By standard facts about the -functor (see simplicial principal bundle) we have a pullback square of simplicial topological spaces
exhibiting the homotopy pullback
Under geoemtric realization this maps to
in Top. By prop 16 this is still a homotopy pullback, and hence exhibits as the loop space of .
For paracompact this goes back to (Segal68). The general case is discussed in (DuggerIsaksen). A generalization to parameterized spaces is in (RobertsStevenson, lemma 22).
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 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 for a simplicial topological group is in
Comments on the relation between properness and cofibrancy in the Reedy model structure on 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, -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 - 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)
This entry is under review. See geometric realization of simplicial topological spaces at nLab (reviewed).