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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 4th 2021

am finally giving $\overline{W}G$ its own entry, for ease of hyperlinking to it

• CommentRowNumber2.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 4th 2021

The functor $L$ (denoted there by $G$) was introduced by Kan in §7 of

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

The functor $\bar W$ is essentially due to:

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 4th 2021
• (edited Jun 4th 2021)

The article coincides with the article simplicial loop space. Should they be merged?

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeJun 5th 2021

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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJun 5th 2021

added pointer to:

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJun 5th 2021
• (edited Jun 5th 2021)

added the component-definition (following Goerss-Jardine);

made explicit the example for $G$ an ordinary group (constant simplicial group);

added brief statement of the abstract description via total simplicial sets;

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJun 5th 2021

added brief statement of various properties, with pointers to the literature:

• Kan fibrancy,

• classification of simplicial principal bundles

• relation to $G$-actions by slicing

• CommentRowNumber8.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 5th 2021

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.

• CommentRowNumber9.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 5th 2021

## Related concepts

• CommentRowNumber10.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 5th 2021

The adjunction was explicitly spelled out by Kan in

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

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

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJun 5th 2021

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.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJun 5th 2021

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.

• CommentRowNumber13.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 5th 2021

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.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeJun 5th 2021
• (edited Jun 5th 2021)

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 ...

***

• CommentRowNumber15.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 5th 2021

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.

• CommentRowNumber16.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 5th 2021

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

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)

• CommentRowNumber17.
• CommentAuthorDmitri Pavlov
• CommentTimeJun 5th 2021

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

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeJun 6th 2021

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

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