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1. • CommentRowNumber1.
   • CommentAuthorMatanP
   • CommentTimeApr 2nd 2012
   • (edited Apr 2nd 2012)
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   Author: MatanP
   Format: MarkdownItexIn the works of Bredon, Elmendorf et al. there is the notion of a strong G-equivalence. If $Top^G$ is the category of (nice) G-spaces for G a top. group, then a strong G-equivalence is a G-map which is a homotopy equivalence and admits an inverse which is also a G-map. I was wondering if there is an analogous notion for G a loop space (e.g. G a group-like $A_\infty$-space). The first problem I encountered with is that a strong $G$-equivalence is the same as an ordinary G-equivalence (i.e. a G-map which is a h.e.) if the action is free, and every action can be made free by multiplying with $EG$. Do anyone know of a place where this is done?

   In the works of Bredon, Elmendorf et al. there is the notion of a strong G-equivalence. If $Top^G$ is the category of (nice) G-spaces for G a top. group, then a strong G-equivalence is a G-map which is a homotopy equivalence and admits an inverse which is also a G-map. I was wondering if there is an analogous notion for G a loop space (e.g. G a group-like $A_\infty$-space). The first problem I encountered with is that a strong $G$-equivalence is the same as an ordinary G-equivalence (i.e. a G-map which is a h.e.) if the action is free, and every action can be made free by multiplying with $EG$.

   Do anyone know of a place where this is done?

2. • CommentRowNumber2.
   • CommentAuthorjim_stasheff
   • CommentTimeApr 2nd 2012
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   Author: jim_stasheff
   Format: TextCould you clarify your question? One of the points of the $A_\infty$ point of view is that a G-map that is a homotopy equivalence has n inverse which is at least an $A_\infty$ map. As you point out, freeness of the action is not a homotopy invariant concept.

   Could you clarify your question? One of the points of the $A_\infty$ point of view is that a G-map that is a homotopy equivalence has n inverse which is at least an $A_\infty$ map. As you point out, freeness of the action is not a homotopy invariant concept.
3. • CommentRowNumber3.
   • CommentAuthorMatanP
   • CommentTimeApr 2nd 2012
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   Author: MatanP
   Format: MarkdownItex(I deleted my last remark since to avoid confusion) I guess what I'm asking is if there is a notion of an action of a (group-like) $A_\infty$-space G on a space for which not every G-map that is a h.e. will admit a G-map inverse but rather some inverse and under some conditions on the action, such a map will have G-map inverse. It may be a long shot, but it seems weird to me that a "flexixble" framework cannot differentiate between weak G-equivalence and strong G-equivalence whereas the "rigid" framework does.

   (I deleted my last remark since to avoid confusion) I guess what I’m asking is if there is a notion of an action of a (group-like) $A_\infty$-space G on a space for which not every G-map that is a h.e. will admit a G-map inverse but rather some inverse and under some conditions on the action, such a map will have G-map inverse. It may be a long shot, but it seems weird to me that a “flexixble” framework cannot differentiate between weak G-equivalence and strong G-equivalence whereas the “rigid” framework does.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 2nd 2012
   • (edited Apr 2nd 2012)
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   Author: Urs
   Format: MarkdownItex> it seems weird to me that a "flexixble" framework cannot differentiate between weak G-equivalence and strong G-equivalence whereas the "rigid" framework does. This is however precisely the hallmark of that "flexible" framework: that distinctions cannot be made which are "[[evil]]" in that they are not invariant under the correct notion of weak equivalence. The thing is that an (enriched) 1-functor from the one-object groupoid $\mathbf{B}G$ (which you write just "$G$" above) to $Top$ is a very specific model or presentation for an $\infty$-functor $\mathbf{B}G \to \infty Grpd$. That specific model singles out plenty of structure which is not intrinsic to the notion it models. Hence it makes plenty of distinctions which abstractly are not but also should not be made. You can see illustrating examples of this in much more simple situations alerady. Say in the notion of equvalence of categories. Given any two categories, it always makes sense to ask if they are equivalent. But only if you pick a very specific presentation of them does it make sense to ask if these models are even isomorphic. You could then complain that it is weird that category theory, flexible as it is, cannot even distinguish equivalent but non-isomorphic categories. But it would be besides the point, because the properties of categories that actually matter don't make this distinction. The case that you are looking at is an immediate $(\infty,1)$-categorical generalization of this situation.

   it seems weird to me that a “flexixble” framework cannot differentiate between weak G-equivalence and strong G-equivalence whereas the “rigid” framework does.

   This is however precisely the hallmark of that “flexible” framework: that distinctions cannot be made which are “evil” in that they are not invariant under the correct notion of weak equivalence.

   The thing is that an (enriched) 1-functor from the one-object groupoid $\mathbf{B}G$ (which you write just “$G$” above) to $Top$ is a very specific model or presentation for an $\infty$-functor $\mathbf{B}G \to \infty Grpd$. That specific model singles out plenty of structure which is not intrinsic to the notion it models. Hence it makes plenty of distinctions which abstractly are not but also should not be made.

   You can see illustrating examples of this in much more simple situations alerady. Say in the notion of equvalence of categories. Given any two categories, it always makes sense to ask if they are equivalent. But only if you pick a very specific presentation of them does it make sense to ask if these models are even isomorphic.

   You could then complain that it is weird that category theory, flexible as it is, cannot even distinguish equivalent but non-isomorphic categories. But it would be besides the point, because the properties of categories that actually matter don’t make this distinction.

   The case that you are looking at is an immediate $(\infty,1)$-categorical generalization of this situation.

5. • CommentRowNumber5.
   • CommentAuthorMatanP
   • CommentTimeApr 2nd 2012
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   Author: MatanP
   Format: MarkdownItexthank you for the clarification Urs.

   thank you for the clarification Urs.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeApr 2nd 2012
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   Author: Mike Shulman
   Format: MarkdownItexIt's important to note that the classical subject of *equivariant homotopy theory* for a group $G$, which is what I assume you're referring to with Bredon Elmendorf etc., is *not* a presentation of the $(\infty,1)$-category of spaces with a $G$-action. Rather, it is a presentation of the $(\infty,1)$-category of diagrams on the *orbit category* of $G$. The distinction is, I believe, precisely that between strong/weak equivalence that you mention. To obtain a version of "equivariant homotopy theory" for an $A_\infty$-space, you'd need to invent some analogue of the "orbit category".

   It’s important to note that the classical subject of equivariant homotopy theory for a group $G$, which is what I assume you’re referring to with Bredon Elmendorf etc., is not a presentation of the $(\infty,1)$-category of spaces with a $G$-action. Rather, it is a presentation of the $(\infty,1)$-category of diagrams on the orbit category of $G$. The distinction is, I believe, precisely that between strong/weak equivalence that you mention. To obtain a version of “equivariant homotopy theory” for an $A_\infty$-space, you’d need to invent some analogue of the “orbit category”.

7. • CommentRowNumber7.
   • CommentAuthorMatanP
   • CommentTimeApr 3rd 2012
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   Author: MatanP
   Format: MarkdownItex> To obtain a version of "equivariant homotopy theory" for an $A_\infty$-space, you'd need to invent some analogue of the "orbit category". I'm not sure what you mean here. if we take as an analogue of the orbit category, the category of homotopy quotients (induced by any $A_\infty$-map to the original $A_\infty$-space) I think we'll get that the homotopy theory of G-spaces with weak G-equivalences is equivalent to that of diagrams over this analogues orbit category with w.e. being the ones that induce w.e. on all homotopy fixed points. In this case, the theory becomes trivial.

   To obtain a version of “equivariant homotopy theory” for an $A_\infty$-space, you’d need to invent some analogue of the “orbit category”.

   I’m not sure what you mean here. if we take as an analogue of the orbit category, the category of homotopy quotients (induced by any $A_\infty$-map to the original $A_\infty$-space) I think we’ll get that the homotopy theory of G-spaces with weak G-equivalences is equivalent to that of diagrams over this analogues orbit category with w.e. being the ones that induce w.e. on all homotopy fixed points. In this case, the theory becomes trivial.

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeApr 3rd 2012
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   Author: Mike Shulman
   Format: MarkdownItex> In this case, the theory becomes trivial. Well, then, obviously you made a poor choice of analogue for the orbit category. (-:O Not that I'm claiming there necessarily *exists* a good such choice....

   In this case, the theory becomes trivial.

   Well, then, obviously you made a poor choice of analogue for the orbit category. (-:O

   Not that I’m claiming there necessarily exists a good such choice….