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started an entry geometric realization of simplicial topological spaces.
I decided this is a topic big enough to justify splitting it off from geometric realization (of simplicial sets).
But not much there yet. I just wanted to record for the moment that this realization too, does preserve pullbacks.
Put in some more sketchy detail, the relation to homotopy colimits, fat realisation, roughly the definitions of ’good’ and ’proper’ simplicial spaces. I imagine these last two could go at a page called nice simplicial space. Also there is a result (good $\Rightarrow$ proper) that is probably folklore that Danny and I proved a few years back that is still floating around in our mythical unpublished work.
The definition of ’proper’ goes back to May’s
The Geometry of Iterated Loop Spaces, Lecture Notes in Mathematics, 1972, Volume 271(1972), 100-112,
and that of ’good’ to Segal’s
Configuration-Spaces and Iterated Loop-Spaces, Inventiones math. 21,213-221 (1973)
these references would have to go into the putative nice simplicial space, but ’I have not time’.
Just found
L. Gaunce Lewis Jr., When is the natural map $X\to \Omega\Sigma X$ a cofibration?, Trans. Amer. Math. Soc. 273 (1982), 147–155.
which Baez-Stevenson cite as the source of good implies proper.
Thanks, David!
I have started editing your additions to geometric realization of simplicial spaces a bit.
You have this sentence here which breaks off before it is finished:
When these conditions are not met, then the fat realization $||X||$ of the simplicial space, i.e. the coend of the diagram given by restricting to the subcategory of $\Delta$ with only the coface maps.
There is some relation between proper simplicial space and Reedy cofibrant object in $[\Delta^{op}, Top]$. That would also put thestatement about realization being the homotopy colimit in perspective,. But I am not quite sure right now about the different notions of cofibrations in the game.
Goerss-Schemmerhorn say
If C = CGH is the category of compactly generated weak Hausdorff spaces, the notion of a Reedy cofibrant object is a variation on the notion of a proper simplicial space. For proper, one only requires that LnX → Xn be a Hurewicz cofibration or, perhaps, only a closed inclusion.
Got to run home now…
Okay, I put the material that you had listed here into the newly created entry nice simplicial topological space.
But have to take care of something else myself now.
I’m pretty sure that proper = Reedy h-cofibrant, i.e. Reedy cofibrant relative to (perhaps strong/closed) Hurewicz cofibrations. I prefer the latter term, since it is more descriptive.
Sure - all these overused words need a good look at once in a while.
@Urs,
When these conditions are not met, then the fat realization $||X||$ of the simplicial space, i.e. the coend of the diagram given by restricting to the subcategory of $\Delta$ with only the coface maps,
should continue with
computes the homotopy colimit, rather than the geometric realisation.
at home at the moment, so can’t edit. I’ll plug this in when I get to work. Thanks for picking that up, I tied myself up a bit with the convolution of the clause starting ’i.e. the coend…’ so I just forgot to finish the sentence.
And ’good’ is sometimes replaced with ’satisfies the Segal condition’, even though that is no more informative, it at least disambiguates from some generic property, and give you a tiny clue as to where to start.
While I still have not begun reacting to the refree’s comments on the entry geometric realization of simplicial topological spaces (nor has anyone else, as far as I can see), I have now added a remark on the compatibility of the fat geometric realization with limits: it should preserve all finite limits up to homotopy, and more precisely, preserve all finite limits on the nose when regarded as a functor
$\Vert - \Vert : Top^{\Delta^{op}} \to Top/\Vert*\Vert \,.$I have been working on geometric realization of simplicial topological spaces.
I have tried to refine a bunch of things that had been there before. The main bit is that I have now given a detailed discussion that
$|X_\bullet| \simeq hocolim_n X_n$in $Top_{Quillen}$ when $X$ is good and degreewise CW.
Previously the entry had said (I had written this) that this follows immediately given that good implies proper and proper means cofibrant in $[\Delta^{op}, Top_{Strom}]_{Reedy}$ – but then I kept feeling unsure about the Strom model structure and how the derived coend $|X_\bullet|$ computed there really relates to the homotopy colimit in $Top_{Quillen}$,
Maybe I am being silly here and missing the obvious, please let me know. In any case, I have now spelled out at Geometric realization – Relation to homotopy colimit a more lenghty argument.
I have (I think) now reacted to all the referee’s suggestions…
except point 3, where it is suggested to expand the discussion to bisimplicial spaces; that’s a good suggestion, but I appeal to my right to be too lazy to look after that at the moment;
and where it comes to the homotopy colimit I have left my proof (see above comment) and only added a “see also” to Dugger’s notes (which mainly point out the role of Reedy cofibrant replacement and “Bousfield-Kan map” which I think I have covered in some detail).
and I had already added (see above) the whole new section on geometric realization of topological principal $\infty$-bundles.
I am going to hand this in soon now in the context of the entry’s refereeing process as the revised version. If anyone has any further suggestions for modifications, please react now.
I’ve been meaning to do some little fixes here and there too; thanks for spurring me to get around to it. I’m not done yet, but I thought I’d mention that I’m in progress. So far I reversed the order of some of the $X_n \times \Delta^n$ to make them more consistent, tried to improve the formatting in places with {|-|}
and using \colon
and \coloneqq
where appropriate, and I fixed the displayed equation in Prop. 7 which made no sense as it was.
I would like to get the statement about the Strom model structure right; I feel like that is probably the best way to think about it. Hopefully in the next day or two I’ll get to look at that.
Is the notion of “well-sectioned” defined anywhere on the nLab?
well-sectioned appears in my paper with Danny. It is the analogue for well-pointed in the parameterised homotopy setting.
I know what it means; I was asking whether it’s defined anywhere on the nLab. (-: I’m also a little unclear what it means for a simplicial topological group to be “well-sectioned”; is it talking about the degeneracy maps regarded as sections of a parametrized space somehow?
I would like to get the statement about the Strom model structure right;
Which statement precisely is wrong, currently?
I didn’t mean that anything there currently is wrong; I was referring to your comment in #12 about the previous version of the argument that ${|X|}\simeq \hocolim X$, using the Strom model structure, that you removed because it made you unsure. I’d like to clarify that version so that we can agree that it works.
I see, okay.
I am generally lacking a good feeling for the Stron model structure. For instance: is it simplicially enriched, does it have a simplicial Quillen adjunction to $sSet_{Quillen}$, etc.
But I gather I can find exhaustive information on this, generalized to the parameterized case, in May-Sigurdson?
By the way, I have added now the definition of “well-pointed” and “well-sectioned” to simplicial topological group. Or I hope I did – because I see in the David-Danny notes the definition of well-sectioned object in $Top/B$, but not the definition of well-sectioned simplicial group object! I think I know what is meant, but somebody should double-check.
Yes, there is some stuff about the Strom model structure in May-Sigurdsson, although they don’t care about simplicial enrichments. Since the identity functor is left Quillen from the Quillen model structure to the Strom model structure, you can compose that with the adjunction to sSet, and probably that (+ monoidality etc) is enough to give you a simplicial enrichment as well.
probably that (+ monoidality etc) is enough to give you a simplicial enrichment as well.
Okay, thanks, we should then try to make that statement precise, because we need simplicial enrichment for any argument that the coend $\int^{[n]} \Delta^n \cdot (-)[n]$ computes a homotopy colimit, I think.
I’m also a little unclear what it means for a simplicial topological group to be “well-sectioned”;
I have changed all “well-sectioned simplicial topological groups” back to “well-pointed simplicial topological group”. I had thought since “well-sectioned” is the more general term, it is more robust to use this, but here it is probably misleading, unless and until we really boost up the discussion to the parameterized case.
Thanks! I think that helps.
Aren’t there size issues in considering $[Top^{op},sSet]_{proj}$?
Aren’t there size issues
Yes, right, I have changed to $Top_s \hookrightarrow Top$ some small full subcategory.
I have also added (in the section on realization of topological simpliical bundles) a few more details:
added details of the argument that global Kan fibrations of simplicial topological spaces map to fibrations in $[Top_s^{op}, sSet]_{proj}$
added a bunch of pointers to propositions and lemmas in Roberts-Stevenson.
for instance for the proof that for $G$ well-pointed both $\bar W G$ and $W G$ are good (the first statement they make explicit, the second follows from two of their lemmas and a side-remark in some other proof).
We still need to add a reference that $\bar W G$ is a globally Kan simplicial topological space – or else the brief argument (we have an explicit algebraic formula for the horn fillers in Set and hence this gives continuous horn fillers in Top, right?). Currently it says “see RobertsStevenson”, but I don’t actually find the statement there.
On the other hand it’s not so important for the homotopy-fiber argument, because what counts is that $\bar W G$ is fibrant in $[Top_s^{op}, sSet]_{proj}$ and that’s classical.
have added in the Examples-section the statement for geometric realization of Cech-nerves
By the way, on something else: do we have any resources of classes of examples for good simplicial spaces, beyond those coming from well-pointed simplicial topological groups? Any further statement we could make here?
Urs #26:
I think there are some examples in Dugger-Isaksen’s A^1 hypercovers paper. They say a simplicial space has free degeneracies if there are subspaces $N_k \to K_k$ such that $\coprod_{[k]\to [n]} N_n \to X_k$ is an isomorphism where the coproduct is over surjective maps. Such a space is Reedy cofibrant (i.e. proper) if all the $N_k$ are cofibrant in the Strom model structure. For example, consider a hypercover where each level $U_n$ is a cover of the appropriate matching object by contractible opens - I think this should be easily seen to be good, in addition to being proper.
Urs #25:
I think this might be folklore - one can write down the fillers of horns, and they are exactly the same as in the topologically discrete case (but then we’d need a reference to that). A nice argument would be to show that the Artin-Mazur codiagonal took globally Kan bisimplicial spaces (by which I mean each $X_{\bullet k}$ and $X_{k\bullet}$ are globally Kan) to globally Kan simplicial spaces, and then it would follow from the definition of $\bar W G$ and the easy observation that $G$ is globally Kan and so are action groupoids internal to $Top$.
@Urs #26: How about the two-sided bar construction $B(Y,G,X)$ when G is any well-pointed topological monoid, or more generally a well-sectioned internal-topological category? Probably when G is a well-pointed monad too. I think those are the examples May was using in The Geometry of Iterated Loop Spaces and Classifying Spaces and Fibrations.
Thanks, Mike, these are good examples that we should mention.
Thanks, David, while with my question I was looking for “natural” examples, the type of example that you mention – “freely force everything to be cofibrant” – is very much on my mind.
Indeed, by Dugger’s cofibrant replacement theorem in the projective model structure, every simplicial presheaf $F$ has a cofibrant resolution by a split hypercover (which is what you mention). Simply take $Q F : [n] \mapsto \coprod_{U_{i_0} \to \cdots \to U_{i_n} \to F_n} U_{i_0}$, where the $U_\cdots$ are objects of the site.
So if our site is a subsite of that of topological spaces, then every simplicial presheaf has a cofibrant resolution by a good simplicial topological space. These resolutions are big and unwieldy and we don’t want to work with them in concrete computations, but it is good to know that they exist.
In fact, this should allow to make the following claim, let’s see:
Claim In the cohesive $\infty$-topos $(\Pi \dashv \Disc \dashv \Gamma \dashv coDisc) : ETop\infty Grpd \to \infty Grpd$ the functor $\Pi$ preserves the homotopy fibers of all morphisms whose codomain is of the form $\mathbf{B}G$ for $G$ presented by a well-pointed topological group that is degreewise paracompact.
Idea of proof Let $X \to \bar W G$ be any such morphism. Write $Q X \to X \to \bar W G$ for the precomposition with Dugger’s cofibrant replacement functor. The composite is then a morphism between good simplicial spaces such that $\Pi$ is modeled on them by the ordinary geometric realization. By our previous discussion, on morphisms between good simplicial spaces with codomain $\bar W G$ geometric realization preserves homotopy fibers. QED.
@Urs #26
Another one which comes to mind (reminded by, and subsumed by, Mike’s example) is the nerve of a topological category such that $X_0 \to X_1$ is a fibrewise closed cofibration over $X_0 \times X_0$ (via the diagonal and (source,target) maps). I guess that this is what Mike means by ’well-sectioned’. This would correspond to $B(\ast,X,\ast)$ in his example.
Okay, great, I don’t have the leisure right now, would you feel like adding at least a rough reminder on these classes of examples to the Examples-section of the entry? Thanks!
Ah, one more question, probably a case for Tim’s Menagerie : what’s a reference for the interaction of $\bar W : sGrp \to sSet$ with the Dold-Kan $\Xi : Ch_\bullet^+ \to sAb \hookrightarrow sGrp$ and $U : sGrp \to sSet$. It ought to be true that
$\bar W (\Xi V) \simeq U (\Xi (V[1])) \,.$Right?
Idea of proof
Ah, no, that’s too simple: while $Q X$ is a good simplicial space it will not be globally Kan, of course.
Ah, no,
Oh, but that’s okay. (It’s too late at night for me, I shouldn’t be trying to do fiddly proofs at this time, but you’ll forgive me):
We work in the global projective model structure $[CartSp_{top}^{op}, sSet]_{proj}$ and have given a morphism of simplicial presheaves $X \to \bar W G$ where $X$ is fibrant and $G$ is a well-pointed simplicial topological group, hence also $\bar W G$ and $W G$ is fibrant.
Then consider the pasting of pullbacks
$\array{ K &\stackrel{\simeq}{\to}& P &\to& W G \\ \downarrow && \downarrow && \downarrow \\ Q X &\stackrel{\simeq}{\to}& X &\to& \bar W G }$Here $Q X \to X$ is Dugger’s cofibrant replacement, which makes $Q X$ non-fibrant but makes it a good simplicial space. Since the global structure (in which we may compute all our finite homotopy limits, since these are preserved by $\infty$-sheafification) is right-proper, we have that also $K \to P$ is a weak equivalence. That’s the crucial point:
we have that $P$ is a model for the homotopy fiber of $X \to \bar W G$ because it is the pullback of a fibrant replacement diagram, but then $K$ is also a model for the homotopy fiber by right properness. But $K$ is also the pullback of a diagram of good simplicial spaces, so under geometric realization this maps to $|K| = |Q X| \times_{|X|} |W G|$, which is indeed a model for the homotopy fiber of $\Pi(X) \to \Pi(\bar W G)$.
Urs #32
you should get something stronger, since for $G\in sAb$, $\bar W G \in sAb$.
Urs #31 - ok, will do.
you should get something stronger
right, I should be able to drop the forgetful $U$. But do you have a reference? Or did you do the combinatorics?
right, I should be able to drop the forgetful $U$. But do you have a reference? Or did you do the combinatorics?
No to both questions. I’m not saying you can just drop the $U$, that would require a computation (it could be true, though), just that you’ll get something better than a simplicial set.
A formatting question: The titles of “Proposition 5” and “Proposition 6” appear to me indented to the same level as the bullet points in the proposition statements, which doesn’t seem right. Anyone know why that is or how to fix it?
I added a remark about how the fat geometric realization is equivalent to the ordinary geometric realization of a fattened up simplicial space.
I have the feeling that the fact that $\overline{W}$ does what you says on simplicial abelian groups is from right back in Kan’s original paper, but cannot check. I would need to check the dates of the loop group paper and his results on what became the Dold Kan correspondence.
Thanks, Tim. I’ll try to dig through the literature then.
I have added to Strøm model structure some comments leading up to why it is a simplicial model category.
Ah, thanks, so $Top_{Strom}$ is a monoidal model category, good.
Okay, I have added the proof I was thinking of that the natural Bousfield-Kan map from hocolim X to |X| is a weak homotopy equivalence whenever X is proper. (I took the liberty of replacing the previous proof, since this one has a stronger conclusion.) Actually, though, I guess it doesn’t really use the fact that properness is Reedy-cofibrancy for the Strom model structure explicitly. I feel like Lemma 1 ought to follow from facts 1 and 2 in its proof using some abstract Reedy nonsense, but I don’t know how.
Thanks. I see it takes a little bit of work, well, at least it is not quite immediate. I find that the part in Dugger’s notes that is being referenced (I had added the reference following the referee’s suggesion) can give a maybe misleading impression on how obvious this is.
I have added to the section Examples - Classifying spaces the statement of the classical fact hat $\Omega B G \simeq G$ for a topological group $G$ and stated the proof as an immediate corollary of the claim in the section Examples - topological oo-bundles that geometric realization on good simplicial spaces preserves homotopy fibers.
I did a bit more tweaking, in particular I added #anchors to all the numbered propositions, since having numbered propositions without #anchors messes up the automatic numbering of backreferences. Are we still waiting on anything to be done?
Thanks, Mike.
Are we still waiting on anything to be done?
There was one suggestion in the referee’s report on generalization to bisimplicial spaces. This is a good suggestion, but beyond my energy budget at the moment. If nobody particularly feels the energy to get into this, we should try to get a second referee report on the entry in its present form, I guess.
Hmm, that is a good suggestion; it’s something we should definitely have written down somewhere. I just wrote out an argument for the statement the referee wanted to see, but you should have a look at it; I’m not entirely positive that I kept correct track of all the simplicial directions.
I have a question about Proposition 4. It says that for all topological spaces $X$ the counit $|\underline{Sing} X| \to X$ is a homotopy equivalence. I find this rather fishy. In the cited Seymour’s paper you can read that $\underline{Sing} X$ is proper, he even says that it’s “clear”. To me it’s not clear at all and in fact I doubt that this holds for arbitrary $X$. The first degeneracy map of $\underline{Sing} X$ is the diagonal map $X \to X^I$. We know that it’s not always true that the diagonal $X \to X \times X \cong X^{\partial I}$ is a cofibration, so why should we expect this to hold when we replace $\partial I$ by $I$?
My questions:
I agree, that does look like the “same old problem” that you’re so good at pointing out. (-:
Oh right, I forgot that we once discussed something very similar. But then the problem was that when we factor a map of spaces $X \to Y$ through its mapping path object, then the inclusion of $X$ into the mapping path object is not necessarily a cofibration. This was not such a big deal since we have other factorizations that do the job. Here, we’re given a specific simplicial space and it just doesn’t seem to be cofibrant.
I don’t have the leisure to look into this right now. But if there is a mistake, we should fix it. I’m contacting my local expert on these matters, maybe he can help out to rectify the discussion here.
In geometric realization of simplicial topological spaces, it is mentioned with no proof that $sTop \to Top/||\ast||$ preserves finite limits, pointing to Gepner and Henriques’ Homotopy theory of orbispaces, where again it is stated with no proof. I’ve been twice asked for a proof of this fact on MO (possibly by the same person!), and I don’t know an actual place it is proved. Since I keep saying this, I wish I did have a reference. Possibly the article On the homotopy type of classifying spaces (paywall) by tom Dieck has methods one could use to prove this, but I haven’t sat down and tried to figure it out. One could also try to show that fat realisation preserves connected limits directly.
By the way, this reminds me that the question I asked in #51 was resolved last year in this MO question. Chris Schommer-Pries and I figured out a counterexample to the claim that the singular simplicial space is Reedy cofibrant (it is described in the linked question) so Seymour’s argument is wrong. However, Neil Strickland provided a hands on proof of the theorem that doesn’t need this assumption.
I have incorporated this into the nLab page. Feel free to edit formatting or wording.
added these two references on sufficient conditions for geometric realization to preserve homotopy pullbacks
Charles Rezk, When are homotopy colimits compatible with homotopy base change?, 2014 (pdf)
Edoardo Lanari, Compatibility of homotopy colimits and homotopy pullbacks of simplicial presheaves (pdf)
(expanded version of Rezk 14)
Need to dig into this again:
I am looking at a configuration space of points in Euclidean space, being an H-space under disjoint union of configurations.
Does its geometric realization commute with the free loop space construction $\mathcal{L}$
$\mathcal{L} \, B_{\sqcup} Conf \overset{??}{\simeq} B_{\sqcup} \mathcal{L} Conf$??
Is the diagonal $Conf^{\times \bullet} \overset{\Delta}{\to} Conf^{\times \bullet} \times Conf^{\times \bullet}$ representable by a “realization fibration”?
Dear Dmitri,
thanks for looking into it!
Yes, right, the configuration space I have in mind is not connected, its connected components correspond to the number $n$ of points in a configuration.
Let me think about what you write a bit. Thanks again.
Now I finally looked at it, thanks again for the remarks.
So you checked Charles’ Def. 2.10, which is sufficient to decide the realization-fibration property since his Lemma 2.11 applies in the present case (and Theorem 2.13 applies anyways).
Okay, I see. But maybe one gap remains:
While the diagonal on a non-connected monoid is thus not a realization-fibration, the condition of being a realization-fibration could be stronger than what I was after:
It says that geometric realization is preserved by homotopy pullback along every morphism, but what I’d need is only that it’s preserved for one particular morphism, namely for the diagonal itself.
Can we conclude that also this special case does not work? I mean, does it now follow that the delooping of a non-connected monoid does not commute with forming free loop spaces?
(I am prepared to accept that it doesn’t, I am just wondering if we have already established this now.)
Thanks a million! Very helpful. Let me think about this argument…
No problem.
If it bugs you you can (not delete but) clear your message: On the right over your own messages here you should find a link “edit”.
I have added pointer to
and then I wrote out the following claim from that article, which, if true, is an important addition to what we (me, at least) were after with this entry here, in that it replaces the nuisance with properness by a condition that is actually pertinent in practice: degreewise cofibrancy.
The following is what I have added (here) – but I realize that neither Wang 18 nor the precursor Wang 17 have been published, and I spent zero minutes with the article beyond extracting this statement:
Let $X_\bullet$ be a simplicial topological space which 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.
Hm, what’s a good sufficient condition on a topological groupoid $\mathcal{X}$ such that for every finite group $G$ the mapping groupoid $\mathrm{Maps}( \mathbf{B}G, \mathcal{X} )$ is degreewise cofibrant?
added pointer to:
above the definition of fat geometric realization (here) I added pointer to semi-simplicial set. (Much room left to expand/beautify the text at this point…)
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