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Here is a nice simple test case that should serve to provide a good short example entry of the nJournal:
Carlos Simpson had had occasion to cite the $n$Lab entry geometric realization of simplicial topological spaces as citaton [77] in his recent preprint.
It would be in everybody’s interest if instead of just the URL to the $n$Lab, he could cite this as a stable peer-reviewed entry in the $n$Journal.
Would anyone volunteer to formally referee the entry? It is very short and the material should be pretty standard. But it may be a good example and manifestly there is already need for it.
(Since I am not a member of the editorial board to-be-formed, my call for referees here is not fully qualified, so I assume it would be useful in this case if we had non-anonymous referees, so that their name would serve as sufficient indication of expertise. )
Urs: the link does not work.
can I edit geometric realization of simplicial topological spaces before we sumit it to a referee? (nothing crucial, just cosmetics)
The correct link is here
Urs: the link does not work.
Fixed it.
can I edit geometric realization of simplicial topological spaces before we sumit it to a referee? (nothing crucial, just cosmetics)
CERTAINLY! And even once it is submitted. And always. And not only are you allowed to, but you are encouraged to.
All that changes with the submission for the $n$Lab entry is that there will be a remark being inserted that notifies whether some and which of its versions have been formally peer-reviewed.
CERTAINLY! And even once it is submitted. And always. And not only are you allowed to, but you are encouraged to.
I know, it’s just I’d like to clean that entry a bit before we freeze a version by peer-reviewing. The changes I have in mind are minimal: making two definitions of good in the sense of Segal and proper in the sense of May, a statement that goodnees implies properness citing David Roberts-Danny Stevenson unpublished work and possibly providing a link to it. Also we need a reference for the fact that if $X$ is proper then $||X||\to |X|$ is a homotopy equivalence, and be precise whether this is a weak homotopy equivalence or a homotopy equivalence. Finally we should add a reference to the Tammo tom Dieck work where the proof for the “$X$ good case” can be found.
I’ve started editing the entry. Have to break now for a while, will complete editing later (in a short time).
I’ve now done my edits. There’s a weird problem with the display of $||X||$ I’ve been unable to solve (the two $|$’s appear extremely spaced to me).
Use \Vert
: $\Vert X \Vert$
produces $\Vert X \Vert$
Done. It’s better, but within the proposition enviroment (or the like) the result seems still not to be completely satisfactory.
I have added a bit more formatting (the missing Definition- and the missing Proof-Environment), made RobertStevenson a reference and added pointers to it (this way, once it is published, we don’t need to change the entry text, but just update the list of references) and added some cross-hyperlinks within the entry and some out of it. We need closed cofibration, by the way.
${\Vert X \Vert}$
with braces gets the spacing better, just as for single verts ${|X|}$
.
I changed the link to simplicial topological space to point to nice simplicial topological space, which seems more relevant. I observe that there is also substantial duplication between geometric realization of simplicial topological spaces and nice simplicial topological space. I would like to add a comment to the effect that proper simplicial topological spaces are just those that are Reedy cofibrant relative to the Strøm model structure, but I’m unsure which of those pages would be more appropriate for it.
${\Vert X \Vert}$
with braces gets the spacing better, just as for single verts${|X|}$
.
Yes. This is basically an iTeX problem (possibly implicit in MathML, possibly something that could be fixed with complicated coding).
As at nice simplicial topological space, it would be worth pointing out that good $\Rightarrow$ proper has its roots in a paper by Lewis, but is treated like a folk theorem. Danny and I have a generalisation of this result for a bicomplete topological concrete category. I’ve also linked ’closed cofibration’ to Hurewicz cofibration, where it is discussed and defined.
We still don’t have a referee, by the way.
It seems that having the eyes of a potential referee pointed on geometric realization of simplicial topological spaces and related entries is improving them at a remarkable rate :)
I have added some further material:
stated the homeomorphism $|Sing(X_\bullet)| \simeq_{iso} | d Sing(X_\bullet)_\bullet |$
split off the section with discussion of nice simplicial spaces as a separate subsection, so that we can later on easily sync it with nice simplicial topological space;
added the statement that proper = cofibrant in $[\Delta^{op}, Top_{Strom}]_{Reedy}$;
added in a Proof-environment the remark that this implies that realizaiton of proper spaces is their homotopy colimit;
added the statement that $|Sing X_\bullet| \to X_\bullet$ is cofibrant replacement in the Reedy structure (hence “properification”)
added the reference for “good implies proper”.
stated the homeomorphism $|X_\bullet| = | diag (Sing X_\bullet)_\bullet |$
I don’t think this is true. If $X$ is a space considered as a constant simplicial space, then $|X| = X$, $diag(Sing X) = Sing X$, and we know that $X$ is not homeomorphic to $|Sing X|$, only weakly homotopy equivalent.
Sorry, I meant the homeo $|Sing(X_\bullet)| = |diag Sing(X_\bullet)_\bullet|$.
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