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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 5th 2016
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   Author: David_Corfield
   Format: MarkdownItexJust to point out, without putting any pressure on anyone to do anything, but the various entries for 'bar construction' could do with some coherent organisation. We have: [[bar construction]], [[bar and cobar construction]], [[simplicial bar construction]] and [[two-sided bar construction]]. Perhaps also there's some further wisdom to extract from Todd's [On the Bar Construction](https://golem.ph.utexas.edu/category/2007/05/on_the_bar_construction.html).

   Just to point out, without putting any pressure on anyone to do anything, but the various entries for ’bar construction’ could do with some coherent organisation. We have: bar construction, bar and cobar construction, simplicial bar construction and two-sided bar construction.

   Perhaps also there’s some further wisdom to extract from Todd’s On the Bar Construction.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 5th 2016
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   Author: Todd_Trimble
   Format: MarkdownItexMuch or most of that "wisdom" has been incorporated into [[bar construction]], but maybe there's more along those lines to add to [[two-sided bar construction]] -- I'll check. I would like to understand better bar/cobar as applied to operads and such.

   Much or most of that “wisdom” has been incorporated into bar construction, but maybe there’s more along those lines to add to two-sided bar construction – I’ll check.

   I would like to understand better bar/cobar as applied to operads and such.

3. • CommentRowNumber3.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 5th 2016
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   Author: Todd_Trimble
   Format: MarkdownItexWell, I just checked [[two-sided bar construction]], and what I thought might needed adding was actually there. But the entry needs a little rescuing: I was going on sheer memory on what was in Geometry of Iterated Loop Spaces, and Jesse McKeown seems to think I messed up a bit. Can someone with more expertise (Mike?) take a look, and see if it can be fixed and the query box removed?

   Well, I just checked two-sided bar construction, and what I thought might needed adding was actually there. But the entry needs a little rescuing: I was going on sheer memory on what was in Geometry of Iterated Loop Spaces, and Jesse McKeown seems to think I messed up a bit. Can someone with more expertise (Mike?) take a look, and see if it can be fixed and the query box removed?

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeSep 7th 2016
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   Author: Mike Shulman
   Format: MarkdownItexI don't think I really have any more expertise here than you do, but I do at least have a copy of *The Geometry of Iterated Loop Spaces*. Theorem 13.1 says (as a special case) that if $X$ is a connected $C_n$-algebra, then there is a span of weak homotopy equivalences $$ \Omega^n B(S^n,C_n,X) \leftarrow B(\Omega^n S^n, C_n, X) \leftarrow B(C_n, C_n, X) \to X $$ which looks like your theorem, but has $C_n$'s in place of $\Omega^n S^n$ in the last place (and a corresponding backwards arrow instead of a forwards one). And Theorem 2.7 (which is proven in section 6) says that there is a natural map $\alpha_n : C_n X \to \Omega^n S^n X$ that is a weak homotopy equivalence if $X$ is connected. I don't quite understand what Jesse was worried about.

   I don’t think I really have any more expertise here than you do, but I do at least have a copy of The Geometry of Iterated Loop Spaces. Theorem 13.1 says (as a special case) that if $X$ is a connected $C_n$-algebra, then there is a span of weak homotopy equivalences

   $$\Omega^n B(S^n,C_n,X) \leftarrow B(\Omega^n S^n, C_n, X) \leftarrow B(C_n, C_n, X) \to X$$

   which looks like your theorem, but has $C_n$’s in place of $\Omega^n S^n$ in the last place (and a corresponding backwards arrow instead of a forwards one). And Theorem 2.7 (which is proven in section 6) says that there is a natural map $\alpha_n : C_n X \to \Omega^n S^n X$ that is a weak homotopy equivalence if $X$ is connected. I don’t quite understand what Jesse was worried about.