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).

]]>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? ]]>

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.

]]>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.

]]>I have added pointer to

- Sergey Arkhipov, Sebastian Ørsted, Exp. 6.4 in:
*Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories*(arXiv:1807.03266)

with a link from around this Prop. in the entry

]]>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:

- Tammo tom Dieck, Prop. 1 in:
*On the homotopy type of classifying spaces*, Manuscripta Math 11, 41–49 (1974) (doi:10.1007/BF01189090)

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*”?

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.

]]>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.)

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

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

]]>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.

]]>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?

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

]]>[ duplicate removed ]

]]>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?)

]]>Thanks!

]]>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.

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.

]]>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.

]]>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)

]]>(Thanks to pointers from discordians here!)

]]>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

]]>on this point, I have added also pointer to:

Jacob Lurie,

*Simplicial spaces*, Lecture 7 of:*Algebraic L-theory and Surgery*(pdf)Jacob Lurie, around Lemma 5.5.6.17 in:

*Higher Algebra*

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

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

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.

]]>