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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.]
Very nice. Are you using the fact that geometric realizations are CW complexes anywhere other than to conclude that they land in CGHaus?
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 $CGHaus$, please let me know! (Johnstone comes tantalizingly close in that paper, but stops just short from what I was able to extract.)
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.
Very nice. Thanks Todd (and Mike)!
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.
Yeah, completeness and cocompleness should probably be assumed too. I don’t think I’ve ever seen a technical definition.
One working notion might be “full subcategory of $Top$ (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).
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.
I would like to boost the entry geometric realization to say something about geometric realization of simplicial objects in an $\infty$-topos $\mathbf{H}$ (or else splitt off an entry that does).
Can we say something about
$\lim_\to : \mathbf{H}^{\Delta^{op}} \to \mathbf{H}$preserving finite $\infty$-limits?
I was thinking one might invoke a variation of this proposition.
Hmm, in $\mathbf{H}^{\Delta^{op}}$ we have a pullback
$\array{ \emptyset & \to & \Delta^0 \\ \downarrow &&\downarrow^{d^0} \\ \Delta^0 & \underset{d^1}{\to} & \Delta^1 }$but since $\colim \Delta^0 = \colim \Delta^1 = 1$, it doesn’t seem to be preserved.
Yeah, that’s precisely using the failure of $\Delta$ 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
$\array{ (P \times V) // G &\to& V //G \\ \downarrow && \downarrow \\ P//G &\to& *//G }$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.
I don’t know. Might be a good question for MO: to what extent does geometric realization preserve finite homotopy limits?
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.
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 $I$ that make the proof work (viz. that $I$ is a compact Hausdorff space equipped with a closed interval order), in view of an answer I have just posted at MO.
I added some proofs of lemmas (hopefully not too painfully) that had been missing in the section on left exactness.
Very nice construction, Todd! Note that your example in the MO post involving $\mathbb{Z}/2$, up to the part where you take the realisation functor, is the contractible simplicial ring $W_{Ring}\mathbb{Z}/2$ given by this paper. The apparent contradiction that $NK(\mathbb{Z}/2)$ is contractible (in $sRing$) and $R_L NK(\mathbb{Z}/2)$ isn’t (in $CGHaus$), is that the latter is contractible for the interval object $L$, but not for $I$. Do you mind if I mention your example (with attribution) in the final version of the paper?
Sure, feel free to mention the example! And thanks.
You might remember that the same $W_{Ring} \mathbb{Z}/2$ 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! :-)
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.
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 $R$ takes monomorphisms to subspace inclusions.
(Every other account I’ve seen also looks likes it’s working too hard, but still.)
Todd, I don’t know whether you will find it more satisfying but every monomorphism comes with a canonical filtration of length $\omega$ using relative skeleta. The relative $k$-skeleton of a monomorphism $X \to Y$ is $Sk^k_X Y = X \cup Sk^k Y$. We have a filtration $X = Sk^{-1}_X Y \to Sk^0_X Y \to \ldots$ whose union is $Y$ and $k$th skeleton is obtained from the $(k-1)$st one by attaching all non-degenerate $k$-simplices of $Y$ that are not in $X$. This can be done with a single pushout of the coproduct of boundary inclusions indexed by these simplices.
Karol, I do like that way of putting it more than what is currently there; thanks.
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.
Yes, I think that’s right. Feel free to add it to appropriate pages.
I was wondering if the following may be added to the page on geometric realisation. The problem is that it is rather preliminary.
A conceptual explanation of left-exactness of geometric realization is arguably provided by the paper of Drinfeld on geometric realization. This draft (Geometric realisation as a Skorokhod semi-continous path space endofunctor)[http://mishap.sdf.org/Skorokhod_Geometric_Realisation.pdf] attempts to further clarify this by showing that, in a certain precise sense, the geometric realisation is the functor
$Hom ( [0,1], Y )$In a certain category sF which contains both simplicial sets and topological and uniform spaces as full subcategories, and has forgetful functors $sF\to sSets$ and $sF\to Top$. Moreover, this functor seems to have adjoint, defined by the usual construction.
Here are some details.
The category sF may be thought as the category of simplicial sets with extra structure of topological nature, a notion of smallness. Formally it is just the category of simplicial objects in the category of filters. The functor $Hom ([0,1],-)$ above is the inner hom of ssets equipped with extra structure. The geometric realisation of sSets factors as
$sSets \to sF \to Top$Here is a proper link to draft paper (by Misha Gavrilovich and Konstantin Pimenov, not by Drinfeld, as it turns out)
@Guest: the syntax for links is [like this](https://example.com)
: like this
Thanks. And this is the proper link to the paper by Drinfeld On the notion of geometric realisation.
It sounds good to me to add this, no problem if it is preliminary.
Linking to geometric realisation of cubical sets in a couple of places.
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