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I have adjusted and expanded wording and formatting in this entry.
Notice that the definition of the source and target maps that was (and still is) given here differs from that in Dwyer & Kan (1984):
where Dwyer & Kan’s §3.1(ii) “discards vertices from the right”, the definition that was (and still is) given in the entry seems to want to switch to the convention where vertices are discarded “from the left”.
With due care this can probably be made an equivalent definition, but as currently stated
$s = (d_1)^{n+1}$
$t = d_0(d_2)^n$
this must be wrong in itself: the “$d_2$” probably wants to be a “$d_1$”.
If anyone wants to fix this, feel invited. Otherwise I’ll change this to Dwyer & Kan’s definition.
I understood that there were typos in the definition of Dwyer and Kan. For instance they write: If $K\in S$ is reduced, then the above definition clearly reduces to the old one. but in fact the old one twists the $d_0$ and that is omitted by the formula in their paper. There are other porblems as well. If I remember correctly, if $x\in K_2$ then $q(x)$ seems to need to go from $q(d_0d_2 x)\to q(d_1d_2x)$, but the formula for $d_0q(x)$ goes from $q(d_0d_1x) \to q(d_1d_1x)$. This means that it ends up in a different part of the simplicial groupoid.
Why do you say: this must be wrong in itself , in your first post in this thread?
I may be getting confused but I do not understand your reasoning here.
Re #2: I think this is an instance of a situation where following the original definition of Dwyer and Kan is probably not the best idea.
The modern convention is that 1-simplices in a simplicial set are interpreted as morphisms d_1(σ)→d_0(σ). This is what is used for quasicategories, Segal spaces, etc.
Dwyer and Kan use the opposite convention (d_0(σ)→d_1(σ)), which means that we must constantly use the opposite category functor when talking about their construction.
I think at least for the sake of keeping the field consistent and the definition interoperable, it may be best to restore the previous convention and point out the discrepancy with the original definition of Dwyer and Kan instead.
this must be wrong in itself: the “d 2” probably wants to be a “d 1”.
I do not understand this comment, to be honest. d_2 will repeatedly remove the 2nd vertex, so eventually all vertices except 0 and 1 will be gone. This is what we want: we want to get the edge 0→1.
On the other hand, d_1 will remove all vertices except 0 and n+1, so we will get the edge 0→n+1. This is not what we want.
Thanks for saying!
I have edited to show both definitions, with a brief disambiguation comment.
But the discrepancy of the definitions is one thing, the more interesting point is their compatibility: Who checked the alternative definition? What’s a source?
Re #5: Goerss–Jardine, Chapter V, Section 7, page 302.
Oh, I see. :-) All right, I’ll add the reference. Thanks.
Hm, now that I looked again at Goerss & Jardine, it still requires work to check that their definition equals the alternative one that our page claims, if it does.
Getting lazy now, I just dropped a screenshot of what Goerss & Jardine say (here).
Incidentally, the defiition that I’d like to see verified/referenced was added by Tim Porter in revision 1.
And another reference is Section 3.1 in https://arxiv.org/abs/2201.03046.
Thanks, Dmitri! I appreciate it. Have added the reference (here).
Now I don’t want to be trouble, but I admit that this looks like yet another definition to me, not manifestly equal to the other three.
I am not doubting that it is equivalent to Dwyer-Kan, and in any case we have a published reference for it.
The alternative definition which our entry claims seems to have been taken verbatim from p. 99 here, where it is (inaccurately) attributed to Dwyer & Kan, without comment on how to see the equivalence.
This still does not handle the point that (due to a typo which is repeated in the nLab entry!) the Dwyer Kan definition does not work as claimed, due to the $\delta_0$ not working correctly.
BTW see Philip Ehlers, Simplicial groupoids as models for homotopy type, Master’s thesis (1991) (pdf). His work was based on that of Joyal and Tierney (some of it unpublished ) and notes of Duskin and van Osdol. Note part of Joyal and Tierney’s work was published in Journal of Pure and Applied Algebra 149 (2000) 69–100. Historically they seem to have worked on the problem of doing the construction at the same time as Dwyer and Kan (and I think they talked on this at the time in seminars, but did not write it up then). In Phil’s MSc he does the detailed verification that the alternative definition works. His MSc is already mentioned in the entry simplicial groupoid.
I am not sure how much of this should go into the references.
Okay, thanks. I have added pointer to Ehlers (1991), pp. 10 and merged the discussion accordingly.
BTW I attributed that alternative to Dwyer and Kan because i did not want to say that the published version had a serious typo in it! This had been known at the time I think, but I do not know if it was corrected in later work. A corrected version of Dwyer Kan could be written (it probably would just need to correct the $d_0$ ). At present and as published I doubt that it gives a left adjoint to the classifying space. The twisting function for twisted cartesian products might not work.
There is probably a sense in which the corrected version would correspond to the alternative decalage functor so in some sense to the conjugate/dual convention for simplicial set calculations.
A corrected version of Dwyer Kan could be written (it probably would just need to correct the $d_0$).
That seems to be what Ehlers did: His definition is that of Dwyer & Kan except for changing source$\leftrightarrow$target and changing $d_0$.
You wrote into the entry:
Their definition doesn’t quite work, as the $\delta_0$ does not generalise that of Kan’s original loop group definition. This was fixed in work by Joyal and Tierney, as discussed in: [Ehlers (1991)]
Actually Ehlers’ thesis neither says what the problem with the original definition is, nor that Joyal and Tierney were the ones to fix it.
What it does claim is that the Dwyer-Kan loop groupoid is “usually called Joyal-Tierney loop groupoid” (is it really?), which may be a secret reference to input from these authors.
Ehlers’ thesis also does not give a citation to Joyal and/or Tierney. But I see that several years later they have:
Is that maybe what Ehlers had seen a preprint version of? Or what other article by Joyal & Tierney could be meant?
Tim wrote
His work was based on that of Joyal and Tierney (some of it unpublished )
Historically they seem to have worked on the problem of doing the construction at the same time as Dwyer and Kan (and I think they talked on this at the time in seminars, but did not write it up then)
I presume this means we can’t point to an actual source other than Ehlers’ writeup of this unpublished work.
Edit:
What it does claim is that the Dwyer-Kan loop groupoid is “usually called Joyal-Tierney loop groupoid” (is it really?), which may be a secret reference to input from these authors.
Well, perhaps in 1991 it was called that verbally among experts, those who actually went to these seminars, and the practice has fallen out of use, since J+T didn’t publish until a decade later. Maybe. It’s not totally unreasonable :-)
What I am saying that, contrary to the edit in #15, Ehlers claims the alleged correction for himself and does not credit it to Joyal & Tierney. He explicitly writes (p. 10):
we have taken the liberty of correcting what we believe to be typing errors in their [Dwyer & Kan’s] text
The edit from #15 instead sounds like crediting the correction to Joyal & Tierney and claiming that Ehlers said so, which is just not the case as far as I can see.
Sounds like me to be the authorial “we”, as in “this document corrects what we believe” etc. But Tim would know better.
I do not know of the earliest use of the alternative version. I recall that someone (possbly Jack Duskin or Don van Osdol) talked about the Joyal Tierney approach in conference talks in the 1980s. When Ehlers started his MSc I suggested he contact them and they sent him rough notes of their discussions Their notes involved a treatment of the ordinal sum and decalage as well as some connection with Joyal and Tierney’s ideas, but I left it to Phil Ehlers to pour over those notes and extract the relevant bits. I think those notes still exist but am not sure where they are. The Joyal Tierney paper talks of the links with Dec and ordinal sum.
As this is over 30 years ago, I do not guarantee the facts are all correct. As the thesis was ’just’ for an MSc there are some omissions. (iNote that in Ehlers PhD he looked at other aspects of this. )
I suspect that the alternative version was known to Joyal and Tierney. Their paper is from 2000 but I do not know at what stage there was a preprint version. Phil based his MSc on Duskin-van Osdol, but they did not publish their work either.
It is worth noting that the detailed verification that the Dwyer Kan version (as published) satisfies the simplicial identities etc. does not seem to have been published, although it may be that I have forgotten where it was. (My own suspicion is that their definition (as published) fails. Try it out on $K=\Delta[2]$ in all its detail.)
Thanks. This does sound like it is indeed better that our page does not contradict Ehlers’ claim of originality.
Incidentally, also the later publication J & T 2000 does not state the corrected version of the component description of the “loop groupoid”, nor does it comment on any issues in this regard:
In fact, it does not state any component description at all, instead it gives a more abstract definition of some $\mathbb{G}(-)$ (p. 94) and later on p. 96 claims — in passing and without further discussion — that this “is the groupoid of Dwyer and Kan”.
I cannot be sure (time passes!!!!) but my suspicion is that Phil had met the (alternative) version in the documents he had seen in which they were attributed to Joyal and Tierney, so to him, they were ’usually referred to as Joyal-Tierney loop groupoids’. Phil knew that the Dwyer Kan version (as published) did not work exactly as claimed. He had checked it early on in his work on it.
Joyal and Tierney (as far as I remember) developed their approach independently of Dwyer and Kan so did not ’correct’ their error. They did not write it up, possibly, as the Dwyer-Kan appoach had been published. It may be worth trying to clarify if the evident minor adjustment to the (published) Dwyer-Kan approach does satisfy the requirements. I suspect it does. I will try to check this later on if I have time. (In fact, I have a faint memory of seeing that version in some preprint with a mention of the typo. I do not know where nor when so it would be hard to track down.)
One ’final’ point may be that if the two constructions differ by using the conjugate form of Decalage etc. then the resulting classifying spaces may just be ’conjugate isomorphic’, although I do not know if that is the case.
I have done some scratchwork on that formula. My best guess would be that the $\delta_0$ term should be the composite of $q(d_0x)^{-1}$ and $q(d_1(x))$. That seems to fit. I say it like that as I am not sure which convention for composition in a groupoid they are using. (If someone wants to try it, I took $K= \Delta [2]$ and $x$ to be the non-degenerate 2-simplex. If I have followed all their conventions correctly $q(x)\in G(K)(1,0)_1$ whilst $q(d_1x)^{-1} \in G(K)(0,2)_0$, so their formula is not only missing a term, but cannot be fixed simply by adding in something like $q(d_0x)$. I have not checked that with the suggested correction everything works. The calculations are quite easy so if someone has some enthusiastic masters students this could be a ’fun’ exercise to give them!)
Note that the formula Dwyer and Kan give ends up going between the wrong objects of $G(K)$. $q(x)$ goes from 1 to 0, but $d_0q(x)$ goes from 0 to 2.
I note that the result for reduced simplicial sets i.e. giving a simplicial group is given as $d_0(q(x)) = q(d_1x) . q(d_0 x)^{-1}$ in Curtis’ survey article (Simplicial Homotopy Theory, Advances in Math., 6, (1971), 107 – 209), but is given from the other end (conjugate version) in Kan’s original article (Annals of Mathematics, Second Series, Vol. 67, No. 2 (Mar., 1958), pp. 282-312).
You write:
My best guess would be that the $\delta_0$ term should be the composite of $q(d_0x)^{-1}$ and $q(d_1(x))$.
That seems to be just what Ehlers’ thesis (and hence now also our entry) says — except maybe for the order of the composition. Are you suggesting that there is still a mistake to be fixed?
No. Our entry gives what seems to be the standard version used now. The original version of Dwyer and Kan is flawed in the way we discussed. I am sure that was a typo / slip. We would only have a slip if we had kept that original version. We should almost certainly keep the current version unless some mild rewording needs doing.
Okay. Incidentally, since you say
…the standard version used now.
What’s another reference? So far, all the references listed in the entry use a different way to speak about the construction.
Added:
The original reference for the functor $G$ is Section 9 in
Later, Dwyer and Kan constructed the right adjoint $\bar W$ to $G$ in
Just from scanning over that section 9, it seems to advertise the construction of a simplicial group $G(...)$ instead of a simplicial groupoid – no?
Re #31: The point is that Kan performs the construction for connected simplicial sets (and not just reduced simplicial sets).
Since every simplicial set is a disjoint union of connected simplicial sets, this does not result in a loss of generality.
Of course, the simplicial groupoid associated to a connected simplicial set has a single isomorphism class of objects.
Kan constructs the simplicial automorphism group of a single object, which encodes the same data as the simplicial groupoid. As far as I can see, he does not explicitly construct the simplicial set of morphisms between two points.
Sure, but then it’s probably still the Kan loop group over the chosen base vertex, not the “loop groupoid” (which should really be called a “path groupoid”).
Yes, so it could be credited as an intermediate step between the Kan loop group (which is defined for reduced simplicial sets) and the full groupoid. Furthermore, the construction for connected simplicial sets already contains all the additional technical difficulties, and the extension to simplicial groupoids is quite straightforward.
I changed ’claimed’ to ’given’ as the former has a negative sense as well as the sense hopefully intended here. For instance, ‘someone claimed to have a simple proof of the Poincaré conjecture’.
I think it is true to say that the version in the paper of Dwyer and Kan has a mistake in the description of the $d_0$. There is no doubt. The ’apparently’ is misplaced. The mistake is almost certainly a typo but it is there.
Dmitri’s comments about the groupoid case being a simple extension of the original case of Kan’s paper reinforce that. If you use the version of Dwyer and Kan on a reduced simplicial set, you clarly do not get Kan’s original version or something equivalent to it.
In the description of face and degeneracy maps, $\sigma_n$ should probably read just $\sigma$.
I agree with Tim’s comments in #35.
I would call it a “typo”, not a “mistake”, and also there no “apparently”: as written, it is just wrong, since the domain of $d_0 t x$ should be the first vertex of $x$, not the second vertex of $x$. I also support changing “claimed”, which is somewhat loaded, to “given”.
The minimal correction to Dwyer and Kan’s original formula is to write $d_0 t x = (t d_1 x)^{-1} t d_0 x$ instead of their formula for $d_0 t x$.
Up to exchanging 0 and 1, this is precisely Ehlers’s formula. (It may be good to mention the convention with 0 and 1 explicitly.)
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