added (here) a minimum indication of the generalization of $\overline{W}$ to simplicial groupoids

]]>added pointer to:

- William Dwyer, Daniel Kan, §3.2 of:
*Homotopy theory and simplicial groupoids*, Indagationes Mathematicae (Proceedings)**87**4 (1984) 379-385 [doi:10.1016/1385-7258(84)90038-6]

for the generalization to simplicial groupoids

]]>I am getting some strange behaviour with simplicial classifying space. I clicked on Changes from previous revision and goot an edit page with a load of code above the edit box. Does any one else get this?

I also tried clicking on the Previous Revision tab at the bottom and got 500 Internal Server Error

]]>Typo in index

Anonymous

]]>added also the following statement (here):

Let $\mathcal{G}_1 \xrightarrow{\phi} \mathcal{G}_2$ be a homomorphism of simplicial groups which is a Kan fibration. Then the induced morphism of simplicial classifying spaces $\overline{W}\mathcal{G}_1 \xrightarrow{ \overline{W}(\phi)} \overline{W}\mathcal{G}_2$ is a Kan fibration if and only if $\phi$ is a surjection on connected components: $\pi_0(\phi) \colon \pi_0(\mathcal{G}_1) \twoheadrightarrow{\;} \pi_0(\mathcal{G}_1)$.

]]>

made the statement that $W G \to \overline{W}G$ is Kan fibrant (here) more explicitly a corollary of the fact that $(W G \times X)/G \to \overline{W}G$ is Kan fibrant for fibrant simplicial group actions $X$.

]]>after the statement of $W G \to \overline{W}G$ being a Kan fibration, I added the statement (here) that, more generally, $(W G \times X)/G \to \overline{W}G$ is a Kan fibration for $X$ a Kan complex with simplicial G-action

]]>added (here) statement and proof of $\pi_{n+1}(\overline{W}G) \simeq \pi_n(G)$

]]>to the statement (here) that every $\overline{W}G$ is Kan fibrant I added an abstract argument to see this from lemmas elsewhere on the $n$Lab (which amounts to stating the proof as given in Goerss & Jardine)

]]>added a remark on décalage (here)

$W G \;=\; Dec^0\big( \overline{W}G \big) \,.$but see the comment in the thread there

]]>have now spelled out (here) the face and degeneracy maps of $\overline{W}G$ itself

]]>fixed a typo (here) in the subscripts in the formula for the degeneracy maps (the same typo is on Goerss&Jardine’s p. 269, so I guess I had copied it from there…)

]]>The $i=0$ case for the face maps of the standard $WG$ being included is debatable, sure, but I included it for clarity. But the $i=n$ case … whoops! I think I was trying to parallel the case of $W_{gr}G$, where that division of the cases makes more sense.

Thanks for the hint of where it might fit in.

I was hoping, way back then, that people who worked on rigidifying $\infty$-/weak models for Lawvere theories might be able to do something with the results of this short paper, but that never eventuated. I don’t know anything myself, but it seemed it might be possible.

]]>As long as you are not actually promoting yourself, but your results: that’s what the $n$Lab is for!

There is already an entry essentially geared towards your result, it’s *groupal model for universal principal infinity-bundles*.

But if you feel the simplicial aspect should be amplified further, there would be room for an entry *universal simplicial principal bundle*.

Incidentally, the reason why the present entry, which a priori is about ${\overline W} G$, talks so much about $W G$ is that the special property of the traditional $W G$ is that it’s the natural intermediate step for obtaining/understanding $\overline{W}G$. The point being that for $W G$ the left $G$-action is most simple.

This is in contrast to your $W_{grp} G$, which gets its simplicial group structure at the expense of the left $G$-action having a more complicated component expression. (Nothing wrong with that, just pointing it out in response to you saying in #20 that you are not sure how things fit together.)

By the way, in your Def. 1: Isn’t the case disctinction at $i = 0$ unnecessary, while the necessary case distinction at $i = n$ is missing?

]]>At the risk of self-promotion, I think it might be worth adding a reference to

*The universal simplicial bundle is a simplicial group*, New York Journal of Mathematics, Volume 19 (2013) 51-60, journal version, arXiv:1204.4886.

somewhere, as there’s a reasonable amount of discussion of $WG$. But I’m not sure where to fit it in the article, since the page is meant to be about $\overline{W}G$. I can work something up later, if it’s deemed reasonable to insert.

]]>For what it’s worth, I have added a rendering of the generic 2-simplex in $W G$: here

]]>added the statement that $W G$ is contractible, with pointer to GJ V4.6.

]]>Mentioned that $G$ is the standard notation for the simplicial loop space.

]]>Added more historical details:

The idea of constructing $\overline{W}$ using the bar construction is due to Eilenberg and MacLane, who apply it to simplicial rings with the usual tensor product operation:

- {#EilenbergMacLane53I} Samuel Eilenberg, Saunders Mac Lane,
*On the Groups $H(\Pi,n)$, I*, Annals of Mathematics Second Series, Vol. 58, No. 1 (Jul., 1953), pp. 55-106 (jstor:1969820) (See, in particular, §17.)

This was also later discussed in

- {#MacLane54} Saunders MacLane,
*Constructions simpliciales acycliques*, Colloque Henri Poincaré 1954 (MacLaneConstructionsSimplicialesAcycliques.pdf:file) (See, in particular, §3.)

The first reference where $\bar W$ is defined explicitly for simplicial groups and the adjunction between simplicial groups and reduced simplicial sets is explicitly spelled out is

- {#Kan58} Daniel Kan, Sections 10-11 in:
*On homotopy theory and c.s.s. groups*, Ann. of Math. 68 (1958), 38-53 (jstor:1970042)

The left adjoint simplicial loop space functor $L$ is also discussed by Kan (there denoted “$G$”) in

- Daniel M. Kan, §7 of:
*A combinatorial definition of homotopy groups*, Annals of Mathematics 67:2 (1958), 282–312. doi.

The Quillen equivalence was established in

- {#Quillen69} Dan Quillen, Section 2 of:
*Rational homotopy theory*, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (jstor:1970725)

]]>

The explicit definition does appear on p. 3 of MacLane 54 – only that MacLane insists on using the product in a simplicial ring instead of the product in a simplicial group.

Yes, and MacLane uses tensor products of abelian groups, not cartesian products.

Also, he gives a references to an earlier paper by Eilenberg and MacLane, which I am going to add now.

]]>That’s a good point. For that reason I often add pointers to anchors in the entry where the edit took place. When I feel I need to include actual snippets of edits in the nForum logs, then I usually put them in between horizontal lines after a line announcing an edit, like this:

```
I have touched the following bit of the entry:
***
... Entry text goes here,
which might say things that acquire an unintended meaning
if they'd appear un-escaped in a discussion forum ...
***
```

]]>
And I gather now that what you send through the announcement mechanism are not edit-logs but straight copies of the material that you edited! That caused the confusion in #2: I read this as a message to me/us (which is how we all usually use the nForum, no?) while you meant it to be the uncommented snippet of the entry that you had re-arranged.

Yes. One advantage of this is that it is immediately clear what exactly has been changed, so one doesn’t have to actually look at the article.

]]>but they are adjoint functors and currently the entries on these two adjoint functors are not even cross-linked.

Not that it matters much, but allow me to say that the stub of the entry that I had yesterday (rev 1) contained essentially *nothing else* but the mentioning of this adjunction, with hyperlink. :-)

Anyway, it’s not important, we both agree on what needs to be done to improve the entries.

And I gather now that what you send through the announcement mechanism are not edit-logs but straight copies of the material that you edited! That caused the confusion in #2: I read this as a message to me/us (which is how we all usually use the nForum, no?) while you meant it to be the uncommented snippet of the entry that you had re-arranged.

]]>Oh, I see, you want to sort out the fine-print of the attribution. Then let’s add the explicit pointers to the original definition:

Kan’s paper also appears to be the first reference where $\bar W$ is defined explicitly.

The explicit definition does appear on p. 3 of MacLane 54 – only that MacLane insists on using the product in a simplicial ring instead of the product in a simplicial group.

So after the component-definition in the entry, I have added pointer to p. 3 in MacLan54 and to Def. 10.3 in Kan 58.

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