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at simplicial group I added/expanded the section delooping and simplicial principal bundles
I discuss this in revisionistic terms meant to exhibit the simple general underlying structure, and then try to spell out how it corresponds to vaarious explicit constructions in the literature, trying to point out page and verse in May’s “Simplicial objects in algebraic topology” and discuss how that yields what I am discussing.
By the way, did anyone ever find the time to make a sanity check of my query-box claim at decalage that forming decalage in sSet is nothing but forming the standard based path space object?
Don’t worry, Urs – you’re completely sane. At least in this instance ;-)
I think maybe your argument can be simplified, but I don’t have time to look into this now. Maybe later.
Thanks for the peer review, Todd!
This motivated me to make the statement a formal proposition with formal proof at decalage. Also expanded and edited the entry further, adding in particular an Idea-section that uses this statement to give a good idea of the nature of decalage.
I know – thanks to chat with Danny Stevenson – precisely one point in the literature where the (evident) observation that for G a simplicial group the thing called $W G$ is nothing but decalage of the thing called $\overline{W} G$ is stated more or less explicitly. That’s on page 85 of Duskin’s “Simplicial methods”. Does anyone know of any other reference that does? I would like to list them withj page and verse on this statement in the nLab etry.
Have you tried Tim Porter’s enormous electronic tome on simplicial methods? I remember him talking a lot about decalage in his talks at the Barcelona 2-group conference.
(It’s probably more efficient to ask Tim than to look through that book… but sorry, I’m too lazy to do this.)
Have you tried Tim Porter’s enormous electronic tome on simplicial methods?
In crossed menagerie (timporter) the fact that “twisted Cartesian products” are nothig but pullbacks of a universal twisted product is made explicit, but this fact – maybe with less emphasis or more between the lines – has been in the literature for long. The literature that I am aware of I listed at simplicial group – delooping and bundles – references .
But the fact that I am after, that everythig is much simpler even and all the “twisted” structures nothing but explicit components for the abstractly and easily defined homotopy fiber I don’t see in the menagerie (though maybe it’s there, I haven’t read the full thing).
The only somewhat intermediate step to this statement in the literature that I am aware of is Duskin’s remark on p. 85 of “Simplicial methods…” where he observes that the “universal twisted product” is just the decalage of $\mathbf{B}G$, which unfortunately is written $\overline{W}G$ in the literature.
page 239 mentions the fiber sequence $G \to W G \to \overline{W}G$…
Urs, I owe you an apology. It seems there’s something a little fishy about the claim after all. (And it teaches me that you have to be careful in the simplicial world, because things happen there that you wouldn’t expect from experience with $Top$. I forget this at times.) I discovered the fishiness while following up on my claim that your argument can be simplified.
Both sides of the asserted isomorphism
$Dec(Y)(-) \coloneqq Y(1 + -) \cong Y^{\Delta_1} \times_Y Y_0$has a left adjoint. (Here $Y_0$ is the discrete simplicial set on the set of 0-simplices.) The left adjoint of decalage is the left Kan extension along yoneda $\Delta \to Set^{\Delta^{op}}$ of
$\Delta \stackrel{1 + -}{\to} \Delta \stackrel{yoneda}{\to} Set^{\Delta^{op}},$and the left adjoint of the other side is given by pushout of
$\array{ Y & \stackrel{\{0\} \times 1}{\to} & \Delta_1 \times Y \\ \pi \downarrow & & \\ \pi_0(Y) & & }$(because $\Delta_1 \times -$ is left adjoint to $(-)^{\Delta_1}$, and $\pi(-)$ is left adjoint to $(-)_0)$. Since left adjoints preserve colimits and every simplicial set $Y$ is a colimit of representables, we’d just have to show that the left adjoints are isomorphic when restricted to representables $Y = \Delta_{n} = \Delta(-, [n+1])$ (naturally in $n$). Since $\pi_0(\Delta_n) = 1$, we’d just have to show the following is a pushout (and naturally so in $n$).
$\array{ \Delta_n & \stackrel{0 \times id}{\to} & I \times \Delta_n \\ \downarrow & & \downarrow \\ 1 & \to & \Delta_{1+n} }$The only problem is that this is false (!). For this to be a pushout, we have to check that it is a pushout objectwise, i.e., that we have a pushout of sets
$\array{ \Delta([m], [1+n]) & \stackrel{0 \times id}{\to} & \Delta([m], [2]) \times \Delta([m], [1+n]) \\ \downarrow & & \downarrow \\ 1 & \to & \Delta([m], [2+n]) }$Take $n=1$, $m=2$. If you take cardinalities of sets, you would get a false pushout:
$\array{ 3 & \hookrightarrow & 9 \\ \downarrow & & \downarrow \\ 1 & \to & 6 }$because the lower right corner of the pushout ought be $7$, not $6$.
I can explain in more geometric language what is happening. In dimension $n=1$, we are taking a simplicial square $I \times I$ and then contracting the edge $0 \times I$ to a point. So this contracts the 1-simplex $(0, 0) \to (0, 1)$ to a point. But the quotient does not identify the 1-simplex $(0, 1) \to (1, 1)$ with the diagonal simplex $(0, 0) \to (1, 1)$. Thus there are four nondegenerate 1-simplices in the pushout, whereas we expect just three in the cone $\Delta_2$. Similarly, there are two nondegenerate 2-simplices in the pushout, but only one in $\Delta_2$.
I was pretty shocked when it began to dawn on me that something was wrong, because it upsets something that I have assumed as obviously true for years. But I’m pretty sure that the above is error-free.
Thanks, Todd, of course you are right. Silly me. I’ll fix entry as soon as I have a minute…
Ha, you think you feel silly? I wonder if I’ll be able to track down all the places I’ve been spreading the lie myself! (And my erroneous belief was so deeply ingrained that it took me quite some time to be convinced it really was a mistake.)
Don’t feel silly. I think in fact it’s a very easy trap to fall into. (Not long ago there was an MO thread on falling into mathematical traps…)
While I’m in a confessional mood, I’ll mention that last night I fixed a mistake I had perpetrated quite a while ago at connected space. I had long suspected there was a mistake there, and was just about to break down and ask MO for a counterexample, when I finally managed to find one.
Thanks, Todd, but I feel I should have known better.
But also I feel, most mistakes are there so that we learn somehting. Here is still something to be learned I think:
after all, the main point I found noteworthy was
decalage is a fibration replacemnt of the morphism $X_0 \to X$ for fibrant $X$;
there is a canonical such fibration replacement, namely $X^I \times_X X_0$.
Then I concluded erroneously that both replacments are in fact isomorphic. But as we have seen now, in fact decalage gives a smaller replacement.
For the moment I just rolled back the entry. Eventually I would like to restorre a corrected version of my last expanded version with a discussion of ho decalage factorizes the standard fibration replacement.
added to simplicial group the statement that for a principal action of a simplicial group $G$ on a Kan complex P, the quotient map
$P \to X := P/G$is always a Kan fibration.
I have added statement and proof of the observation that for any simplicial group $G$ there is a fiber sequence of Kan complexes
$G_0 \to G \to B \Omega G/G_0$In the References-section at simplicial group I have added two more items for John Moore with the original proof that simplicial groups are Kan, using the information that is being thrown around currently on the CatTheory mailing list
So 1954 is the year of the theorem…
The decalage is the simplicial version of the path space. So $WG = Dec \overline{W}G$ is morally like $PBG$, the based path space of the classifying space. Except that $Dec \overline{W}G$ is a simplicial group (well, not explicitly - it is iso to the underlying simplicial set of a simplicial group), with $G$ as a subgroup and $WG/G = \overline{W}G$.
At simplicial group I can’t find what I thought was the basic example, i.e., the singular simplicial complex of a topological group. so maybe I’m wrong and that is actually not an example? or have I missed that example from the page on simplicial groups?
It should be an example because singularization is a right adjoint and therefore preserves the finite product diagrams that express something as a topological group.
Does anyone know to what extent the simplicial theory of things like $W G$ and $\overline{W} G$ can be extended from simplicial groups to grouplike simplicial monoids?
In particular, I’m wondering about a simplicial version of May’s theorem 7.6 from Classifying spaces and fibrations that if $G$ is a grouplike topological monoid, then $E G\times_{B G} X = B(*,G,X) \to B G$ is a quasifibration.
The proof of Quillen’s theorem B uses a lemma (cf Lemma 5.7 in [Goerss and Jardine, Ch. IV]), that essentially says that $B (*, \mathcal{C}, X) \to N \mathcal{C}$ is a quasifibration for any category $\mathcal{C}$ and any homotopically constant diagram $X : \mathcal{C} \to sSet$. It seems to me that this generalises to simplicially enriched categories $\mathcal{C}$ as well.
And, of course, any action by a grouplike monoid is homotopically constant. Thanks!
Where Moore’s theorem is stated (here), I added more pointers to references, such as to
and also pointer to where in Weibel’s book this is stated and proven.
added pointer to
for another account of Moore’s theorem
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