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1. • CommentRowNumber1.
   • CommentAuthordomenico_fiorenza
   • CommentTimeApr 16th 2012
   • (edited Apr 16th 2012)
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   Author: domenico_fiorenza
   Format: MarkdownItexWhich is the exact original formulation of May recognition theorem? At [[Ek-Algebras]] Lurie's general forumlation is given and it is said that for the $\infty$-topos being $Top$ this reduces to May's theorem. But it seems to me that even for $Top$ Lurie's formulation tells much more than May's theorem as it is stated there. namely in Lurie's formulation we have an equivalence, which is missing in the formulation of May's theorem given there. Moreover the equivalence in Lurie's verison is an equivalence of $\infty$-categories and I would not be able to say that in a pre-higher categories languages (maybe it is an equivalence of homotopy categories in the original version?). Another thing puzzling me is that if $X$ is homotopy equivalent to an $n$-fold loop space I don't see any natural action of the little $n$-cubes operad on $X$, but only an action up to coherent homotopy. And again, to speak of actions up to coherent homotopy it seems I need a language of higher operads (but I may be wrong here).

   Which is the exact original formulation of May recognition theorem? At Ek-Algebras Lurie’s general forumlation is given and it is said that for the $\infty$-topos being $Top$ this reduces to May’s theorem. But it seems to me that even for $Top$ Lurie’s formulation tells much more than May’s theorem as it is stated there. namely in Lurie’s formulation we have an equivalence, which is missing in the formulation of May’s theorem given there. Moreover the equivalence in Lurie’s verison is an equivalence of $\infty$-categories and I would not be able to say that in a pre-higher categories languages (maybe it is an equivalence of homotopy categories in the original version?). Another thing puzzling me is that if $X$ is homotopy equivalent to an $n$-fold loop space I don’t see any natural action of the little $n$-cubes operad on $X$, but only an action up to coherent homotopy. And again, to speak of actions up to coherent homotopy it seems I need a language of higher operads (but I may be wrong here).

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 17th 2012
   • (edited Apr 17th 2012)
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   Author: Urs
   Format: MarkdownItex> Which is the exact original formulation of May recognition theorem? See in * Peter May, _The geometry of iterated loop spaces_ ([pdf](http://people.virginia.edu/~mah7cd/Foundations/May%20-%20Geometry%20of%20Iterated%20Loop%20Spaces.pdf)) First version there is theorem 1.3, which goes > There exist $\Sigma$-free operads $\mathcal{C}_n$, $1 \leq n \leq \infty$, such that every $n$-fold loop space is a $\mathcal{C}_n$-space and every connected $\mathcal{C}_n$-space has the weak homotopy type of an $n$-fold loop space. We should add more details about all this to the $n$Lab entries. But I won't do it tonight :-)

   Which is the exact original formulation of May recognition theorem?

   See in

   • Peter May, The geometry of iterated loop spaces (pdf)

   First version there is theorem 1.3, which goes

   There exist $\Sigma$-free operads $\mathcal{C}_n$, $1 \leq n \leq \infty$, such that every $n$-fold loop space is a $\mathcal{C}_n$-space and every connected $\mathcal{C}_n$-space has the weak homotopy type of an $n$-fold loop space.

   We should add more details about all this to the $n$Lab entries. But I won’t do it tonight :-)

3. • CommentRowNumber3.
   • CommentAuthorjim_stasheff
   • CommentTimeApr 17th 2012
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   Author: jim_stasheff
   Format: TextAnother thing puzzling me is that if X is homotopy equivalent to an n-fold loop space I don't see any natural action of the little n-cubes operad on X, but only an action up to coherent homotopy. Agreed And again, to speak of actions up to coherent homotopy it seems I need a language of higher operads No, you just need an appropriate notion of a map of C_n -spaces, hopefully still parameterized by cubes, cf. A_\infty maps, using the `good'direction for the homotopy equivalence X <--> \Omega^n Y. it should be in May somewhere or else in Lada of Cohen, Lada and May. I have no idea what jazzy version Lurie came up with; do you need it or only htat it might as well be filled in?

   Another thing puzzling me is that if X is homotopy equivalent to an n-fold loop space I don't see any natural action of the little n-cubes operad on X, but only an action up to coherent homotopy.
   
   Agreed
   
   And again, to speak of actions up to coherent homotopy it seems I need a language of higher operads
   
   No, you just need an appropriate notion of a map of C_n -spaces, hopefully still parameterized by cubes,
   cf. A_\infty maps, using the `good'direction for the homotopy equivalence X <--> \Omega^n Y. it should
   be in May somewhere or else in Lada of Cohen, Lada and May.
   
   I have no idea what jazzy version Lurie came up with; do you need it or only htat it might as well be filled in?
4. • CommentRowNumber4.
   • CommentAuthorMay
   • CommentTimeApr 17th 2012
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   Author: May
   Format: TextRemember the dates: Geometry appeared in 1972. That only stated the result for connected C_n-spaces X. In a 1974 paper E_{\infty}-spaces, group completions, and permutative categories, I generalized to non-connected spaces by showing that there is a natural group completion X >--> E_nX (n\geq 2), where E_n denotes my functor from C_n-spaces to n-fold loop spaces. In that generality, grouplike C_n-spaces and n-fold loop spaces are equivalent. Back in those days, we weren't striving for Quillen equivalences (although Quillen's introduction of model categories dates to 1967). I do now know how to prove a kind of Quillen adjoint equivalence between grouplike C_n-spaces and (n-1)-connected based spaces. (for the ``kind of'' caveat, compare with Theorem 0.10 in ``Model categories of diagram spectra" by Mandell, Schwede, Shipley, and myself. [All of my papers are available on my web site.] Of course, an equivalence of model categories is more precise and convenient (when available!) than an equivalence of (\infty,1)-categories. In any case saying something in that language adds no information relevant to the applications. (Computations first, theoretical addenda later). As to homotopy invariance, the paper of Lada in Cohen-Lada-May needs correction but is basically right. The better alternative is to use Boardman and Vogt: replace C_n by WC_n, in modern language a cofibrant approximation in the category of operads. A space Y homotopy equivalent to a C_n space X has a (compatible) action by WC_n$. It is well understood (at least informally) that this use of cofibrant approximation obviates the need for explicit use of coherent homotopy theory (it is a conceptual version thereof). Higher operads are quite unnecessary. For my original applications, use of just cartesian products of operads obviated the need to consider the question of homotopy invariance.

   Remember the dates: Geometry appeared in 1972. That only stated the result for connected C_n-spaces X. In a 1974 paper E_{\infty}-spaces, group completions, and permutative categories, I generalized to non-connected spaces by showing that there is a natural group completion
   X >--> E_nX (n\geq 2), where E_n denotes my functor from C_n-spaces to n-fold loop spaces. In that generality, grouplike C_n-spaces and n-fold loop spaces are equivalent. Back in those days, we weren't striving for Quillen equivalences (although Quillen's introduction of model categories dates to 1967). I do now know how to prove a kind of Quillen adjoint equivalence between grouplike C_n-spaces and (n-1)-connected based spaces. (for the ``kind of'' caveat, compare with Theorem 0.10 in ``Model categories of diagram spectra" by Mandell, Schwede, Shipley, and myself. [All of my papers are available on my web site.] Of course, an equivalence of model categories is more precise and convenient (when available!) than an equivalence of (\infty,1)-categories. In any case saying something in that language adds no information relevant to the applications. (Computations first, theoretical addenda later). As to homotopy invariance, the paper of Lada in Cohen-Lada-May needs correction but is basically right. The better alternative is to use Boardman and Vogt: replace C_n by WC_n, in modern language a cofibrant approximation in the category of operads. A space Y homotopy equivalent to a C_n space X has a (compatible) action by WC_n$. It is well understood (at least informally) that this use of cofibrant approximation obviates the need for explicit use of coherent homotopy theory (it is a conceptual version thereof). Higher operads are quite unnecessary. For my original applications, use of just cartesian products of operads obviated the need to consider the question of homotopy invariance.
5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeApr 17th 2012
   • (edited Apr 17th 2012)
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   Author: DavidRoberts
   Format: MarkdownItexNever mind.

   Never mind.

6. • CommentRowNumber6.
   • CommentAuthorjim_stasheff
   • CommentTimeApr 17th 2012
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   Author: jim_stasheff
   Format: Text`Never mind' - what?

   `Never mind' - what?
7. • CommentRowNumber7.
   • CommentAuthordomenico_fiorenza
   • CommentTimeApr 17th 2012
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   Author: domenico_fiorenza
   Format: MarkdownItexThanks a lot for the comments and the answers! I'm still a bit puzzled by the homotopy invriance issue. I mean, I'd like to have a statement of the form "A (nice) topolological space has the homotopy type of an $n$-fold loop space if and only if..." where "(nice)" includes all the needed features like "grouplike" or "$(n-1)$-connected" for this statement to be true. But I don't clearly see what should be in place of the "..." in the statement. Namely, I know the following (in the notations from post #4 above), always assuming $X$ to be as nice as needed: i) if $X$ is homeomorphic to an $n$-fold loop space, then there is a $C_n$-action on $X$; ii) if there is a $C_n$-action on $X$, then $X$ is (weakly?) homotopy equivalent to an $n$-fold loop space; iii) if $X$ is (weakly?) homotopy equivalent to an $n$-fold loop space then there is a $WC_n$ action on $X$. how does this continues? I'd like the next step stabilizes, i.e., iv) if there is a $WC_n$ action on $X$ then $X$ is (weakly?) homotopy equivalent to an $n$-fold loop space but I do not know whether this is true or not

   Thanks a lot for the comments and the answers! I’m still a bit puzzled by the homotopy invriance issue. I mean, I’d like to have a statement of the form "A (nice) topolological space has the homotopy type of an $n$-fold loop space if and only if…" where "(nice)" includes all the needed features like "grouplike" or "$(n-1)$-connected" for this statement to be true. But I don’t clearly see what should be in place of the "…" in the statement.

   Namely, I know the following (in the notations from post #4 above), always assuming $X$ to be as nice as needed:

   i) if $X$ is homeomorphic to an $n$-fold loop space, then there is a $C_n$-action on $X$;

   ii) if there is a $C_n$-action on $X$, then $X$ is (weakly?) homotopy equivalent to an $n$-fold loop space;

   iii) if $X$ is (weakly?) homotopy equivalent to an $n$-fold loop space then there is a $WC_n$ action on $X$.

   how does this continues? I’d like the next step stabilizes, i.e.,

   iv) if there is a $WC_n$ action on $X$ then $X$ is (weakly?) homotopy equivalent to an $n$-fold loop space

   but I do not know whether this is true or not

8. • CommentRowNumber8.
   • CommentAuthorMay
   • CommentTimeApr 18th 2012
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   Author: May
   Format: MarkdownItexAnswer to last question iv): Yes it is true, by my original arguments as cited in my post 4. The operad WC_n maps by an equivalence to the operad C_n. That is all that is needed to carry out the argument: I crossed C_n with an E_{\infty}$ operad in many of my applications and exploited the resulting map to C_n. You exploit the map from WC_n in exactly the same way.

   Answer to last question iv): Yes it is true, by my original arguments as cited in my post 4. The operad WC_n maps by an equivalence to the operad C_n. That is all that is needed to carry out the argument: I crossed C_n with an E_{\infty}$ operad in many of my applications and exploited the resulting map to C_n. You exploit the map from WC_n in exactly the same way.

9. • CommentRowNumber9.
   • CommentAuthordomenico_fiorenza
   • CommentTimeApr 18th 2012
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   Author: domenico_fiorenza
   Format: MarkdownItexI see: so the idea is to replace the projection $\pi_n:\mathcal{C}\times \mathcal{C}_n\to \mathcal{C}_n$ used in your "The Geometry of Iterated Loop Spaces" with any cofibrant resolution $\phi_n:\mathcal{Q}_n\to \mathcal{C}_n$ (in particular one can use Boardman-Vogt $\mathcal{W}\mathcal{C}_n\to \mathcal{C}_n$) and then verbatim follow your argument. Thanks a lot!

   I see: so the idea is to replace the projection $\pi_n:\mathcal{C}\times \mathcal{C}_n\to \mathcal{C}_n$ used in your “The Geometry of Iterated Loop Spaces” with any cofibrant resolution $\phi_n:\mathcal{Q}_n\to \mathcal{C}_n$ (in particular one can use Boardman-Vogt $\mathcal{W}\mathcal{C}_n\to \mathcal{C}_n$) and then verbatim follow your argument. Thanks a lot!

10. • CommentRowNumber10.
    • CommentAuthorDavidRoberts
    • CommentTimeApr 18th 2012
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    Author: DavidRoberts
    Format: MarkdownItex@jim - I asked a stupid question, then realised it was stupid rather quickly, so deleted it.

    @jim - I asked a stupid question, then realised it was stupid rather quickly, so deleted it.

11. • CommentRowNumber11.
    • CommentAuthorRenato V V
    • CommentTimeMar 26th 2023
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    Author: Renato V V
    Format: TextHi, I know this is an ancient conversation, but I'd like to point out that the model theoretical issues pointed out by May have been handled in [[recognition of relative loop spaces]], by a generalization of Quillen adjunction inspired by May's original proof.

    Hi, I know this is an ancient conversation, but I'd like to point out that the model theoretical issues pointed out by May have been handled in recognition of relative loop spaces, by a generalization of Quillen adjunction inspired by May's original proof.
12. • CommentRowNumber12.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMar 26th 2023
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    Author: Dmitri Pavlov
    Format: MarkdownItexRe #11: Why not add a comment about this to the article?

    Re #11: Why not add a comment about this to the article?