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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 21st 2010

    I wrote out a proof that geometric realization of simplicial sets valued in compactly generated Hausdorff spaces is left exact, using essentially the observation that simplicial sets are the classifying topos for intervals, combined with various soft topological arguments. I left a hole to be plugged, that geometric realizations are CW complexes. I also added a touch to filtered limit, and removed a query of mine from triangulation.

    I wanted a “pretty proof” for this result on geometric realization, centered on the basic topos observation (due to Joyal). I was hoping Johnstone did this himself in his paper on “a topological topos”, but I couldn’t quite put it together on the basis of what he wrote, so my proof is sort of “homemade”. I wouldn’t be surprised if it could be made prettier still. [Of course, “pretty” is in the eye of the beholder; mainly I want conceptual arguments which avoid fiddling around with the combinatorics of shuffle products (which is what I’m guessing Gabriel and Zisman did), decomposing products of simplices into simplices.]

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJun 21st 2010

    Very nice. Are you using the fact that geometric realizations are CW complexes anywhere other than to conclude that they land in CGHaus?

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 21st 2010

    I did want it around as a general useful fact, but for the direct purpose of proving left exactness, that was the only application.

    If you know of a cleaner way to argue for left exactness of realization as valued in CGHausCGHaus, please let me know! (Johnstone comes tantalizingly close in that paper, but stops just short from what I was able to extract.)

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJun 22nd 2010

    Your argument is the cleanest one I’ve seen (although I haven’t thought much about it). Why not write your argument in terms of an arbitrary convenient category? For instance, many people (particularly those of Peter May’s school) prefer to use weak Hausdorff CG spaces rather than Hausdorff ones, or occasionally to impose no separation condition at all.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2010

    Very nice. Thanks Todd (and Mike)!

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 22nd 2010

    Is there a technical definition of “convenient category (of topological spaces)” that I can use? Edit: I see we have an entry in the Lab, which for all I know is the universally accepted definition, but it mentions just cartesian closure. It feels like something like completeness and cocompleteness should be thrown as well to make it truly “convenient”.

    Thanks for the kind words, guys, but this will require just a bit more polishing. I’ll keep working on it.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJun 22nd 2010

    Yeah, completeness and cocompleness should probably be assumed too. I don’t think I’ve ever seen a technical definition.

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 22nd 2010

    One working notion might be “full subcategory of TopTop (all topological spaces) which is complete, cocomplete, cartesian closed, and includes the category of CW complexes as a full subcategory”. There might be nicer ways of putting that.

    Another question is whether that’s satisfyingly general enough. That working notion excludes other nice categories like subsequential spaces, Johnstone’s topological topos, and others (of which none may be of interest to the average working topologist – I don’t know).

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 23rd 2010
    • (edited Jun 23rd 2010)

    Based on Mike’s suggestion, I just finished rewriting the proof of left exactness of geometric realization so that it applies to any convenient category of topological spaces, at least according to my proposed definition of that notion. I’m actually a bit pleased how it all worked out.

    For those that care: the proof doesn’t actually need that a convenient category is closed under open and closed subspaces, so that axiom should be regarded as optional and up for discussion. On the one hand, we get a more general notion if we drop it. But on the other, all examples of convenient categories that I am aware of actually satisfy that axiom, and anyway it seems to me that it would be “convenient” for topologists to assume that \mathbb{R} is around, which I don’t see is necessarily the case if the axiom is dropped. There are other technical reasons for why it might be a convenient axiom.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2012
    • (edited Jun 19th 2012)

    I would like to boost the entry geometric realization to say something about geometric realization of simplicial objects in an \infty-topos H\mathbf{H} (or else splitt off an entry that does).

    Can we say something about

    lim :H Δ opH \lim_\to : \mathbf{H}^{\Delta^{op}} \to \mathbf{H}

    preserving finite \infty-limits?

    I was thinking one might invoke a variation of this proposition.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJun 19th 2012

    Hmm, in H Δ op\mathbf{H}^{\Delta^{op}} we have a pullback

    Δ 0 d 0 Δ 0 d 1 Δ 1 \array{ \emptyset & \to & \Delta^0 \\ \downarrow &&\downarrow^{d^0} \\ \Delta^0 & \underset{d^1}{\to} & \Delta^1 }

    but since colimΔ 0=colimΔ 1=1\colim \Delta^0 = \colim \Delta^1 = 1, it doesn’t seem to be preserved.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2012
    • (edited Jun 19th 2012)

    Yeah, that’s precisely using the failure of Δ\Delta to satisfy the assumptions of that proposition.

    But something useful must be possible to say here. My motivation is this:

    I am looking at pullbacks of groupoid objects of the form

    (P×V)//G V//G P//G *//G \array{ (P \times V) // G &\to& V //G \\ \downarrow && \downarrow \\ P//G &\to& *//G }

    and I can show that these are preserved, using a presentation by simplicial presheaves, then using Borel construction etc. pp. I am a bit annoyed that I can’t see how to deduce the preservation of these pullbacks more abstractly.

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeJun 19th 2012

    I don’t know. Might be a good question for MO: to what extent does geometric realization preserve finite homotopy limits?

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2012

    I should say that it seems that we solved this. We are busy finalizing a pre-publication version of some writeup, hopefully done by this Weekend. Then I’ll say more about this.

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 26th 2013

    I have done some slight rearranging and reformatting of geometric realization, particularly the section on left exactness. I wanted to bring out the precise topological properties of the unit interval II that make the proof work (viz. that II is a compact Hausdorff space equipped with a closed interval order), in view of an answer I have just posted at MO.

    • CommentRowNumber16.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 28th 2013

    I added some proofs of lemmas (hopefully not too painfully) that had been missing in the section on left exactness.

    • CommentRowNumber17.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 29th 2013

    Very nice construction, Todd! Note that your example in the MO post involving /2\mathbb{Z}/2, up to the part where you take the realisation functor, is the contractible simplicial ring W Ring/2W_{Ring}\mathbb{Z}/2 given by this paper. The apparent contradiction that NK(/2)NK(\mathbb{Z}/2) is contractible (in sRingsRing) and R LNK(/2)R_L NK(\mathbb{Z}/2) isn’t (in CGHausCGHaus), is that the latter is contractible for the interval object LL, but not for II. Do you mind if I mention your example (with attribution) in the final version of the paper?

    • CommentRowNumber18.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 29th 2013

    Sure, feel free to mention the example! And thanks.

    You might remember that the same W Ring/2W_{Ring} \mathbb{Z}/2 came up in another MO answer of mine, here. It’s always a pleasure when the ’soft’ methods of category theory can be put to use solving what might look at first like ’hard’ problems! :-)

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeMar 2nd 2016
    • (edited Mar 2nd 2016)

    just if you watch the logs and are wondering:

    I touched the section where Todd proves the left exactness, doing just some minor editing of formatting, such as fixing “.num_corollary” to “.num_cor” and adding some hyperlinks to keywords.

    • CommentRowNumber20.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 2nd 2016

    Thanks for the touch-up. I’m still not completely satisfied by the account because Lemma 1 still looks like working too hard. I don’t like the line of argument where one iteratively and transfinitely attaches cells; I’m hoping there is yet an easier and softer way to see that RR takes monomorphisms to subspace inclusions.

    (Every other account I’ve seen also looks likes it’s working too hard, but still.)

    • CommentRowNumber21.
    • CommentAuthorKarol Szumiło
    • CommentTimeMar 3rd 2016

    Todd, I don’t know whether you will find it more satisfying but every monomorphism comes with a canonical filtration of length ω\omega using relative skeleta. The relative kk-skeleton of a monomorphism XYX \to Y is Sk X kY=XSk kYSk^k_X Y = X \cup Sk^k Y. We have a filtration X=Sk X 1YSk X 0YX = Sk^{-1}_X Y \to Sk^0_X Y \to \ldots whose union is YY and kkth skeleton is obtained from the (k1)(k-1)st one by attaching all non-degenerate kk-simplices of YY that are not in XX. This can be done with a single pushout of the coproduct of boundary inclusions indexed by these simplices.

    • CommentRowNumber22.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 3rd 2016

    Karol, I do like that way of putting it more than what is currently there; thanks.

    • CommentRowNumber23.
    • CommentAuthorMike Shulman
    • CommentTimeJan 4th 2019

    Added the theorem that geometric realization takes Kan fibrations to Serre fibrations, hence that the induced fibrant replacement functor on sSetsSet preserves fibrations as well as finite limits.

    diff, v37, current

    • CommentRowNumber24.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 4th 2019

    So both geometric realization and singular simplicial set functor preserve all 5 classes of maps in a model category (i.e., weak equivalences, cofibrations, acyclic cofibrations, fibrations, and acyclic fibrations)?

    It may be worth pointing out this fact on appropriate nLab pages.

    • CommentRowNumber25.
    • CommentAuthorMike Shulman
    • CommentTimeJan 4th 2019

    Yes, I think that’s right. Feel free to add it to appropriate pages.

    • CommentRowNumber26.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJan 5th 2019

    Added a remark about the preservation of five classes of maps.

    diff, v38, current

    • CommentRowNumber27.
    • CommentAuthorMike Shulman
    • CommentTimeJan 8th 2019

    Added the additional fact that the geometric realization of a Kan fibration is even a Hurewicz fibration.

    diff, v39, current

    • CommentRowNumber28.
    • CommentAuthorMike Shulman
    • CommentTimeJan 23rd 2019

    Added a mention of geometric realization of simplicial objects in an arbitrary simplicially enriched category.

    diff, v40, current

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