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.
Let Top in the following denote either
for the category of simplicial topological spaces.
formed in Top.
This operation naturally extends to a functor
where the equivalence relation ”” identifies, for every morphism in , the points and .
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).
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.
The ordinary geometric realization can be described as the tensor product of functors , and the fat geometric realization can likewise be described as , where 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 . In other words, the fat geometric realization of is the ordinary geometric realization of a “fattened up” version of , which is obtained by forgetting the degeneracy maps of and then “freely throwing in new ones.”
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:
A good simplicial topological space is proper:
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.
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 .
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 ).
We have the following degenerate case of geometric realization of simplicial topological spaces.
If is simplicially constant on a topological space , then its geometric realization is homeomorphic to that space:
(This is not true for the fat geometric realization, only the ordinary one. The fat geometric realization will be homotopy equivalent to .)
For every there is a weak homotopy equivalence
This appears as (Seymour, prop. 3.1).
Ordinary geometric realization has the following two disadvantages:
See (Segal74, appendix A)
This is different for the fat geometric realization.
This appears as (Segal74, prop. A.1).
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.
We discuss how geometric realization interacts with limits of simplicial topological spaces.
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.
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).
Recall that a simplicial space is proper if it is Reedy cofibrant relative to the Strøm model structure on , 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.
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 (e.g. CW replacement). 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.
For , write
We say a morphism of simplicial topological spaces is a global Kan fibration if for all and the canonical morphism
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 globally Kan also the internal hom is globally Kan, for any simplicial topological space . This is more than we need and want to impose here. For our purposes it is sufficient to observe that if 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 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 corresponding topological simplicial principal bundle over is itself a good simplicial topological space.
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.
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
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
An early reference for this classical fact is (Segal68).
There is a weak homotopy equivalence
exhibiting the homotopy pullback
Under geoemtric realization this maps to
Then the canonical map
The first occurence of the definition of geometric realization of simplicial topological spaces seems to be
The definition of good simplicial topological spaces goes back to
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
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
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
Globally Kan simplicial spaces are considered in
The right adjoint to geometric realization of simplicial topological spaces is discussed in
Geometric realization of general Cech nerves is discussed in
The behaviour of fibrations under geometric realization and the preservation of homotopy pullbacks under geometric realization is discussed in
This entry is under review. See geometric realization of simplicial topological spaces at nLab (reviewed).