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The functor $L$ (denoted there by $G$) was introduced by Kan in §7 of
The functor $\bar W$ is essentially due to:
{#MacLane54} Saunders MacLane, Constructions simpliciales acycliques, Colloque Henri Poincaré 1954 (MacLaneConstructionsSimplicialesAcycliques.pdf:file)
{#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)
The article coincides with the article simplicial loop space. Should they be merged?
Not sure I am following. You write in #2 as if pointing out omissions:
The functor $\bar W$ is essentially due to: …
But that’s exactly the references I gave. In fact you seem to have copied them from what I wrote, including my choice of anchors and uploaded pdf-s.
The reference to Kan you added I moved further down, since (a) it’s not the topic of this entry, (b) it comes four years after the original references relevant to this entry; so that it seems weird to have it as the first reference item.
In this vein, the entries should not be merged. Just as the entries on “classifying space” and “loop space” should not be merged! If they appear too close at the moment, that’s because they are waiting for somebody to spell out the definitions and discuss more of the properties.
You write in #2 as if pointing out omissions:
The functor W¯\bar W is essentially due to: … But that’s exactly the references I gave. In fact you seem to have copied them from what I wrote, including my choice of anchors and uploaded pdf-s.
I think the whole adjunction deserves to be mentioned right away.
Also, $\bar W$ is not actually defined in the Eilenberg–MacLane article as far as I can see.
Hence, a slight adjustment in the description (“the functor $\bar W$ is essentially due to…”), which is also how Kan describes it in his article.
Kan’s article is the first one to spell $\bar W$ completely explicitly, as far as I can see.
This was the point of my edit.
In this vein, the entries should not be merged. Just as the entries on “classifying space” and “loop space” should not be merged! If they appear too close at the moment, that’s because they are waiting for somebody to spell out the definitions and discuss more of the properties.
Okay, I can certainly imagine having two separate entries on these two functors, but they are adjoint functors and currently the entries on these two adjoint functors are not even cross-linked.
The adjunction was explicitly spelled out by Kan in
Kan’s paper also appears to be the first reference where $\bar W$ is defined explicitly.
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.
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.
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.
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 ...
***
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.
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:
This was also later discussed in
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
The left adjoint simplicial loop space functor $L$ is also discussed by Kan (there denoted “$G$”) in
The Quillen equivalence was established in
Mentioned that $G$ is the standard notation for the simplicial loop space.
At the risk of self-promotion, I think it might be worth adding a reference to
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.
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?
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.
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: $\pi_0(\phi) \colon \pi_0(\mathcal{G}_1) \twoheadrightarrow{\;} \pi_0(\mathcal{G}_1)$.
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
added pointer to:
for the generalization to simplicial groupoids
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