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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeAug 24th 2010
• (edited Aug 24th 2010)

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.

• CommentRowNumber2.
• CommentAuthorDavidRoberts
• CommentTimeAug 25th 2010

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that is probably folklore that Danny and I proved a few years back that is still floating around in our mythical unpublished work.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’.

• CommentRowNumber3.
• CommentAuthorDavidRoberts
• CommentTimeAug 25th 2010

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.

• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeAug 25th 2010
May originally said 'strictly proper' instead of 'proper'. Got to run home now...
• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeAug 25th 2010

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.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeAug 25th 2010

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.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeAug 25th 2010

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.

• CommentRowNumber8.
• CommentAuthorMike Shulman
• CommentTimeAug 25th 2010

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.

• CommentRowNumber9.
• CommentAuthorDavidRoberts
• CommentTimeAug 25th 2010
• (edited Aug 25th 2010)

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.

• CommentRowNumber10.
• CommentAuthorDavidRoberts
• CommentTimeAug 25th 2010

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.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeFeb 18th 2011

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

$\Vert - \Vert : Top^{\Delta^{op}} \to Top/\Vert*\Vert \,.$
• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeFeb 19th 2011
• (edited Feb 19th 2011)

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.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeFeb 21st 2011
• (edited Feb 21st 2011)

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.

• CommentRowNumber14.
• CommentAuthorMike Shulman
• CommentTimeFeb 22nd 2011

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?

• CommentRowNumber15.
• CommentAuthorDavidRoberts
• CommentTimeFeb 22nd 2011

well-sectioned appears in my paper with Danny. It is the analogue for well-pointed in the parameterised homotopy setting.

• CommentRowNumber16.
• CommentAuthorMike Shulman
• CommentTimeFeb 22nd 2011

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?

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeFeb 22nd 2011

I would like to get the statement about the Strom model structure right;

Which statement precisely is wrong, currently?

• CommentRowNumber18.
• CommentAuthorMike Shulman
• CommentTimeFeb 22nd 2011

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.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeFeb 22nd 2011
• (edited Feb 22nd 2011)

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.

• CommentRowNumber20.
• CommentAuthorMike Shulman
• CommentTimeFeb 22nd 2011

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.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeFeb 22nd 2011
• (edited Feb 22nd 2011)

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.

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeFeb 22nd 2011

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.

• CommentRowNumber23.
• CommentAuthorMike Shulman
• CommentTimeFeb 22nd 2011

Thanks! I think that helps.

Aren’t there size issues in considering $[Top^{op},sSet]_{proj}$?

• CommentRowNumber24.
• CommentAuthorUrs
• CommentTimeFeb 22nd 2011
• (edited Feb 22nd 2011)

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).

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeFeb 22nd 2011
• (edited Feb 22nd 2011)

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.

• CommentRowNumber26.
• CommentAuthorUrs
• CommentTimeFeb 22nd 2011

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?

• CommentRowNumber27.
• CommentAuthorDavidRoberts
• CommentTimeFeb 22nd 2011
• (edited Feb 22nd 2011)

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 $Top$.

• CommentRowNumber28.
• CommentAuthorMike Shulman
• CommentTimeFeb 22nd 2011
• (edited Feb 22nd 2011)

@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.

• CommentRowNumber29.
• CommentAuthorUrs
• CommentTimeFeb 22nd 2011
• (edited Feb 22nd 2011)

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.

• CommentRowNumber30.
• CommentAuthorDavidRoberts
• CommentTimeFeb 22nd 2011

@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.

• CommentRowNumber31.
• CommentAuthorUrs
• CommentTimeFeb 22nd 2011

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!

• CommentRowNumber32.
• CommentAuthorUrs
• CommentTimeFeb 22nd 2011
• (edited Feb 22nd 2011)

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?

• CommentRowNumber33.
• CommentAuthorUrs
• CommentTimeFeb 23rd 2011

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.

• CommentRowNumber34.
• CommentAuthorUrs
• CommentTimeFeb 23rd 2011
• (edited Feb 23rd 2011)

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 $\Pi(X) \to \Pi(\bar W G)$.

• CommentRowNumber35.
• CommentAuthorDavidRoberts
• CommentTimeFeb 23rd 2011

Urs #32

you should get something stronger, since for $G\in sAb$, $\bar W G \in sAb$.

• CommentRowNumber36.
• CommentAuthorDavidRoberts
• CommentTimeFeb 23rd 2011

Urs #31 - ok, will do.

• CommentRowNumber37.
• CommentAuthorUrs
• CommentTimeFeb 23rd 2011

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?

• CommentRowNumber38.
• CommentAuthorDavidRoberts
• CommentTimeFeb 23rd 2011

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.

• CommentRowNumber39.
• CommentAuthorMike Shulman
• CommentTimeFeb 23rd 2011

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?

• CommentRowNumber40.
• CommentAuthorMike Shulman
• CommentTimeFeb 23rd 2011

I added a remark about how the fat geometric realization is equivalent to the ordinary geometric realization of a fattened up simplicial space.

• CommentRowNumber41.
• CommentAuthorTim_Porter
• CommentTimeFeb 23rd 2011

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.

• CommentRowNumber42.
• CommentAuthorUrs
• CommentTimeFeb 23rd 2011

Thanks, Tim. I’ll try to dig through the literature then.

• CommentRowNumber43.
• CommentAuthorMike Shulman
• CommentTimeFeb 24th 2011

I have added to Strøm model structure some comments leading up to why it is a simplicial model category.

• CommentRowNumber44.
• CommentAuthorUrs
• CommentTimeFeb 24th 2011

Ah, thanks, so $Top_{Strom}$ is a monoidal model category, good.

• CommentRowNumber45.
• CommentAuthorMike Shulman
• CommentTimeFeb 25th 2011

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.

• CommentRowNumber46.
• CommentAuthorUrs
• CommentTimeFeb 25th 2011

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.

• CommentRowNumber47.
• CommentAuthorUrs
• CommentTimeFeb 28th 2011

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.

• CommentRowNumber48.
• CommentAuthorMike Shulman
• CommentTimeMar 7th 2011

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?

• CommentRowNumber49.
• CommentAuthorUrs
• CommentTimeMar 7th 2011
• (edited Mar 7th 2011)

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.

• CommentRowNumber50.
• CommentAuthorMike Shulman
• CommentTimeMar 7th 2011

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.

• CommentRowNumber51.
• CommentAuthorKarol Szumiło
• CommentTimeJan 24th 2013
• (edited Jan 24th 2013)

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:

1. Is it true that $\underline{Sing} X$ is always good or proper or otherwise sufficiently cofibrant so that $|\underline{Sing} X| \to X$ is a (weak) homotopy equivalence for all spaces $X$? If yes, what’s the proof?
2. If not, under what reasonable conditions is $\underline{Sing} X$ “sufficiently cofibrant”?
• CommentRowNumber52.
• CommentAuthorMike Shulman
• CommentTimeJan 25th 2013

I agree, that does look like the “same old problem” that you’re so good at pointing out. (-:

• CommentRowNumber53.
• CommentAuthorKarol Szumiło
• CommentTimeJan 25th 2013

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.

• CommentRowNumber54.
• CommentAuthorUrs
• CommentTimeJan 25th 2013

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.

• CommentRowNumber55.
• CommentAuthorDavidRoberts
• CommentTimeAug 16th 2015

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.

• CommentRowNumber56.
• CommentAuthorKarol Szumiło
• CommentTimeAug 19th 2015
• (edited Aug 20th 2015)

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.

• CommentRowNumber57.
• CommentAuthorUrs
• CommentTimeOct 14th 2019
• (edited Oct 14th 2019)

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)

• CommentRowNumber58.
• CommentAuthorUrs
• CommentTimeOct 14th 2019
• (edited Oct 14th 2019)

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”?

• CommentRowNumber59.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 15th 2019
• (edited Oct 15th 2019)
Re #58: If I correctly understood all the notation involved,
the question can be answered in the negative using Rezk's criterion.

Recall that a map of simplicial spaces X→Y
is a realization fibration if and only if for any chain of maps of simplicial spaces
Δ^0→Δ^n→Y
the induced chain of maps of pullbacks
P_0→P_n→X
the map P_0→P_n is a weak equivalence after applying the homotopy colimit functor.

In the case under consideration, B_⊔ Conf is the homotopy colimit of a simplicial space
whose space of n-simplices is the n-fold product of Conf.

It would seem then that Rezk's criterion is already violated for the case n=1.
Indeed, a 0-simplex * in Conf ×•×Conf ×•
is unique and P_0 has as its space of k-simplices
the k-fold product of Ω_*(Conf).

On the other hand, a 1-simplex
is a pair of configurations A_0, A_1.
The simplicial space P_1 is either empty if A_0 is not homotopic to A_1,
and otherwise has as its space of k-simplices the loop space Ω_{s_0^k(A_0)}(Conf_k)
of the space of k-simplices of Conf^⨯ with respect to the base point
given by the k-fold degeneration of A_0≃A_1.
This, P_1, being the homotopy colimit of this space, does not appear
to be homotopy equivalent to P_0
if * and A_0 lie in different connected components of Conf.

Or perhaps it is? Is Conf connected? Maybe I misunderstood the particular flavor and it is?
In this case, the same computation appears to establish that the diagonal map is a realization fibration.
• CommentRowNumber60.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 15th 2019
If the ⊔-structure is group-like (e.g., using orientations for inverses, or something like that), then the simplicial space that defines B_⊔ Conf is Kan ∞-fibrant:
the homotopy horn filling condition is satisfied for almost trivial reasons,
since the space of k-simplices is the k-fold product of Conf,
and exterior horns can be filled using subtraction.
If so, then the free loop space does commute with realization.
• CommentRowNumber61.
• CommentAuthorUrs
• CommentTimeOct 15th 2019
• (edited Oct 15th 2019)

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.

• CommentRowNumber62.
• CommentAuthorUrs
• CommentTimeOct 16th 2019
• (edited Oct 16th 2019)

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.)

• CommentRowNumber63.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 17th 2019
No, it does not.

In fact, it appears to me that the statement ℒ B_⊔ Conf ≃ B_⊔ ℒ Conf
is true if and only if Conf is group-complete.
That is, not only if Conf is group-complete, then the above equivalence is true,
as I explained in #60, but the opposite is also true: the above equivalence appears to imply group-completeness of Conf.

Let me argue as follows.
I will work in the category of simplicial spaces throughout.
The functor Ex^∞ can be defined just like for simplicial sets.
For a simplicial space X, the canonical morphism X → Ex^∞ X
becomes an equivalence of spaces after taking hocolim.

Now, ℒ B_⊔ Conf ≃ ℒ B_⊔ Ex^∞ Conf,
and the simplicial space Ex^∞ Conf is a Kan simplicial space.
For such spaces, the functors ℒ and B_⊔ commute.
We have ℒ B_⊔ Ex^∞ Conf ≃ B_⊔ ℒ Ex^∞ Conf.

Thus, to show the desired equivalence, we have to show that
the map Conf → Ex^∞ Conf becomes an equivalence of spaces
after applying B_⊔ ℒ.

We use the simplicial Whitehead theorem
and look at the relative-homotopy-lifting property
for ∂Δ^n → Δ^n and the map B_⊔ ℒ Conf → B_⊔ ℒ Ex^∞ Conf.
Moving B_⊔ (i.e., the homotopy colimit functor, or simply the diagonal)
to the left by adjunction amounts to promoting
∂Δ^n → Δ^n to a morphism of simplicial discrete spaces.
Moving ℒ to the left by adjunction amounts to looking at the appropriate r-h-l property
for ∂Δ^n⨯S^1 → Δ^n⨯S^1 with respect to the map Conf → Ex^∞ Conf.

Both objects ∂Δ^n⨯S^1 → Δ^n⨯S^1 are compact simplicial spaces,
so maps out of them factor through some finite stage Ex^k of Ex^∞.
Moving Ex^n to the left by adjunction amounts to working with
Sd^k (∂Δ^n ⨯ S^1) → ∂Δ^n ⨯ S^1
↓ ↓
Sd^k (Δ^n ⨯ S^1) → Δ^n ⨯ S^1
where the bottom left and top right entry both map to Conf
and we have to find a map from the bottom right entry to Conf
that makes the triangle with two right vertices and Conf commute up to a homotopy,
whereas the triangle with two bottom vertices and Conf must commute
only up to a simplicial homotopy (of simplicial spaces).
Furthermore, when constructing the map, we are allowed to increase k
as much as we please by further subdividing things.

Now let's look at the easiest nontrivial case k=1, n=0 (so that ∂Δ^n=∅ and half of the diagram disappears).
The diagram now reduces to
Sd^k (Δ^n ⨯ S^1) → Δ^n ⨯ S^1,
i.e.,
Sd (S^1) → S^1,
which must map to Conf and the triangle must commute, but only up to a simplicial homotopy.

Furthermore, let's observe that mapping a simplicial set (or a simplicial discrete space) to Conf
amounts to assigning points in Conf to each 1-simplex in such a way that
for each 2-simplex the 1st face is the product of the 2nd and 0th face.
(Recall that the space of 0-simplices is contractible, so no data is assigned to vertices.)

So a map S^1 → Conf is a single edge labeled with a point in Conf
and a map Sd(S^1) → Conf is two opposite-oriented edges, each with a point in Conf.
The composition Sd(S^1) → S^1 → Conf will migrate an existing label to one of the 1-simplices
and put the identity element on the other.

Now, we have to show that any circle with arbitrary labels
a, b
is simplicially homotopic to a circle with labels
p, 1
where 1 is the identity element.

It appears to me (after drawing some pictures of simplicial homotopies of circles,
i.e., maps Δ^1 ⨯ S^1 → Conf)
that such a homotopy exists if and only if
b+c+p = a+c, for some arbitrary c.
This appears to suggest that Conf must be group-complete.

I can provide more details if desired.
• CommentRowNumber64.
• CommentAuthorUrs
• CommentTimeOct 17th 2019

Thanks a million! Very helpful. Let me think about this argument…

• CommentRowNumber65.
• CommentAuthorGuest
• CommentTimeJun 15th 2020
I'm pretty sure that a lot of things are wrong with the section "Bisimplicial sets and good resolutions"
The functor S_\bullet|-| is a *fibrant* replacement functor, not a cofibrant one (in fact every simplicial set is cofibrant in the usual model structure). The counit map mentioned should be the unit map $X\toS_\bullet|X|$
• CommentRowNumber66.
• CommentAuthorGuest
• CommentTimeJun 15th 2020
Sorry, actually I misread things, but cannot delete my comment - sorry!
• CommentRowNumber67.
• CommentAuthorUrs
• CommentTimeJun 15th 2020

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”.

• CommentRowNumber68.
• CommentAuthorUrs
• CommentTimeJul 9th 2020
• (edited Jul 9th 2020)

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.

• CommentRowNumber69.
• CommentAuthorUrs
• CommentTimeJul 10th 2020

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?

• CommentRowNumber70.
• CommentAuthorUrs
• CommentTimeJun 25th 2021

added pointer to:

• CommentRowNumber71.
• CommentAuthorUrs
• CommentTimeJun 25th 2021
• (edited Jun 25th 2021)

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…)

• CommentRowNumber72.
• CommentAuthorUrs
• CommentTimeAug 23rd 2021
• (edited Aug 23rd 2021)

Made explicit in a new Proposition (here) that geometric realization (into convenient spaces) preserves all finite limits.

• CommentRowNumber73.
• CommentAuthorUrs
• CommentTimeAug 24th 2021
• (edited Aug 24th 2021)

I have added DOI-link, pdf link, and section pointer to our reference:

This is actually a really good early account (except for the choice of dimension shift in defining $\Delta^\bullet_{top}$! which, until one spots it, makes all following formulas seem evidently wrong…).

The article by Segal which the entry currently gives first hardly provides a definition of anything relevant to the entry here – and doesn’t claim to (it keeps to refer, vaguely, to known constructions). So I think I will go and move Segal’s paper down and tone down the commentary that goes with it.

• CommentRowNumber74.
• CommentAuthorUrs
• CommentTimeAug 24th 2021

Okay, I have polished up the list (here) of early references a little (Segal, MacLane, May, Segal, Segal), adjusting commentary to be more informative (I hope) and ordering more strictly by appearance.

What’s the deal with these two items that follow:

• {#Wang17} Yi-Sheng Wang, Fat realization and Segal’s classifying space (arXiv:1710.03796)

• {#Wang18} Yi-Sheng Wang, Geometric realization and its variants (arXiv:1804.00345)

Both seem to remain unpublished?

• CommentRowNumber75.
• CommentAuthorUrs
• CommentTimeSep 9th 2021
• (edited Sep 9th 2021)

Question:

The first lines of Anderson 1978 speak of a “folk theorem” that geometric realization of simplicial spaces sends degreewise “fibrations” to “fibrations”.

What’s the precise statement that this might be alluding to?

The next lines say that “a useful form” of this statement is in May 1972, but again I am not sure what this means to be pointing to (May’s Thm. 12.7?)

In any case, what’s some good sufficient conditions that geometric realization of simplicial spaces produces a Serre fibration?

• CommentRowNumber76.
• CommentAuthorUrs
• CommentTimeSep 10th 2021
• (edited Sep 10th 2021)

Have added (here) the following statement from Anderson 1978, hope this is right:

If $f_\bullet \,\colon\, X_\bullet \xrightarrow{\;} Y_\bullet$ is a morphism of simplicial spaces such that

1. on simplicial sets of connected components it is a Kan fibration;

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

then the geometric realization of any homotopy pullback-square of $f_\bullet$ is a homotopy pullback-square in topological spaces.

• CommentRowNumber77.
• CommentAuthorUrs
• CommentTimeSep 10th 2021

on this point, I have added also pointer to:

• CommentRowNumber78.
• CommentAuthorUrs
• CommentTimeSep 10th 2021
• (edited Sep 10th 2021)

enhanced (here) the statement of the proposition a little more, now referencing also the sufficient conditions due to Charles Rezk and due to Lurie’s Higher Algebra

• CommentRowNumber79.
• CommentAuthorUrs
• CommentTimeSep 10th 2021

(Thanks to pointers from discordians here!)

• CommentRowNumber80.
• CommentAuthorDavidRoberts
• CommentTimeSep 10th 2021

What’s the name of that discord channel and how does one see it? (I do have a discord account, but I can see anything)

• CommentRowNumber81.
• CommentAuthorUrs
• CommentTimeSep 10th 2021

Don’t really know. (Does the link not work?) I followed instructions I found on the MO Homotopy chat, which say that chat room has been abandoned.

• CommentRowNumber82.
• CommentAuthorDavidRoberts
• CommentTimeSep 11th 2021

The link works, but I get nothing, in that the no channels on that server are available to me. I didn’t know about the MO homotopy chat instructions, I can check to see if I can find those.

• CommentRowNumber83.
• CommentAuthorUrs
• CommentTimeSep 12th 2021

This here looks like the official start page: Algebraic Topology Discord. Clicking on the link https://nodorek.net at the bottom should take you to a sign-up and log-in dialogue.

• CommentRowNumber84.
• CommentAuthorDavidRoberts
• CommentTimeSep 12th 2021

Thanks!

• CommentRowNumber85.
• CommentAuthorUrs
• CommentTimeSep 13th 2021

added mentioning of the “Kan-like fibration” condition (here), though without details yet (Is Mazel-Gee’s condition the same as in Lurie’s lecture note, as referenced behing the above link?)

• CommentRowNumber86.
• CommentAuthorUrs
• CommentTimeSep 13th 2021
• (edited Sep 13th 2021)

[ duplicate removed ]

• CommentRowNumber87.
• CommentAuthorUrs
• CommentTimeSep 13th 2021

and added mentioning (here) of Bousfield-Friedlander’s $\pi_\ast$-Kan condition

• CommentRowNumber88.
• CommentAuthorUrs
• CommentTimeSep 14th 2021

keep making minor formatting improvements to the older material in this entry (such as replacing $\bar W G$ by $\overline{W}G$, but also fixing a couple of spurious such bars altogether)

also added more precise links to lemmas in other entries that are being referenced.

Question. Is the Borel construction $X \times_G W G$ of a good simplicial space acted on by a good simplicial topolgical group $G$ itself a good simplicial space?

• CommentRowNumber89.
• CommentAuthorUrs
• CommentTimeSep 14th 2021

Actually, the question I should first be wondering about is much simpler: For $G$ a well-pointed topological group acting on a topological space $X$, is the nerve of the corresponding action groupoid a good simplicial space? That should be real easy: The product of a closed cofibration with a space needs to still be a closed cofibration.

• CommentRowNumber90.
• CommentAuthorDavidRoberts
• CommentTimeSep 14th 2021
• (edited Sep 14th 2021)

One only needs to check that $X\to X\times G$ is a closed cofibration, as you say (I know I proved for myself a general result about nerves of “well-pointed” groupoids, not sure if this is in the literature), but wouldn’t this follow from using the description as an NDR-pair?

(added: I recorded the observation on nerves here, and have a pdf written when I was a PhD student that goes through the result on this page in detail, just something I worked out for my own benefit that never went anywhere

• CommentRowNumber91.
• CommentAuthorUrs
• CommentTimeSep 14th 2021

But what’s the argument. I have now written out one here at Hurewicz cofibration. (Announced in the relevant thread here.)

• CommentRowNumber92.
• CommentAuthorUrs
• CommentTimeSep 14th 2021
• (edited Sep 14th 2021)

I have expanded out a little more (here) the “homotopy Kan fibration”-condition due to Lurie 2011 and Mazel-Gee 2014.

(This is maybe the most useful in this list of sufficient conditions, for purposes of classifying-space theory. Too bad that both references seem to remain unpublished.)

• CommentRowNumber93.
• CommentAuthorDavidRoberts
• CommentTimeSep 15th 2021
• (edited Sep 15th 2021)

Well, it was too late for me to write it down at the time. It was essentially an even more low-key version of that one, just defining the retraction to be constant.

• CommentRowNumber94.
• CommentAuthorUrs
• CommentTimeSep 15th 2021
• (edited Sep 15th 2021)

The entry used to attribute, without any reference, to tom Dieck a proof of the comparison lemma between fat and plain realization for proper simplicial spaces. Have now added the relevant pointer:

Regarding references, I have a question:

A series of propositions in this entry (from long time back) derives that fat realization, and hence realization of good simplicial spaces, is a model for the homotopy colimit. While this proof is fairly straightforward using, as it does, assorted propositions found in the literature, is there any place in the published (citable) literature which makes the conclusion explicit? i.e. which explicitly says something like: “Prop. $n$: Plain topological realization of good simplicial spaces models their homotopy colimit, up to equivalence”?

• CommentRowNumber95.
• CommentAuthorUrs
• CommentTimeSep 18th 2021

I have added pointer to

with a link from around this Prop. in the entry

• CommentRowNumber96.
• CommentAuthorUrs
• CommentTimeSep 18th 2021
• (edited Sep 18th 2021)

The references for the proposition (here) that “good implies proper” was entirely broken:

First it claimed that the proof is in Gaunce Lewis’, Cor. 2.4 (b), but that is the statement that degreewise locally equiconnected simplicial spaces realize to locally equiconnected spaces.

Then it claimed that a more general proof is Roberts & Stevenson’s Prop. 16, which is again something else entirely.

Not sure what happened here. But I have now fixed the second pointer to saying “Appendix A”, which is obvious enough. But not sure yet to which statement in Gaunce Lewis’s article the reader really needs to be pointed to. Maybe Lemma 3.2?, but this would still require work.

• CommentRowNumber97.
• CommentAuthorUrs
• CommentTimeSep 19th 2021
• (edited Sep 19th 2021)

where the text (here) gives a good resolution and then says that there others, I have added brief pointer to the two constructions $\tau(-)$ and $simp(-)$ that Segal 74 uses in Appendix A.

• CommentRowNumber98.
• CommentAuthorjim stasheff
• CommentTimeJan 8th 2022
What is a link for when the realization of the singular simplicial set is homotopy equivalent? |Symp X| to X is the easy part.
Ditto for the cubical case.

If E \to B us a good fibre bundle, what about
|Symp E| to E
over
|Symp B| to B?
• CommentRowNumber99.
• CommentAuthorUrs
• CommentTime7 days ago

Hi Jim,

the point is that plain topological realization preserves finite limits, in particular pullbacks. For simplicial cg-spaces this is due to May, referenced here, but it seems (if by “Symp” you mean “Sing”) that your question concerns just realization of simplicial sets, for which the statement is more classical still (here).

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