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1. • CommentRowNumber1.
   • CommentAuthornLab edit announcer
   • CommentTimeMar 21st 2023
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   Author: nLab edit announcer
   Format: MarkdownItexThis is a first draft for a page on the relative recognition principle, as proved in https://msp.org/agt/2020/20-3/agt-v20-n3-p08-s.pdf I intend to finish it in the following couple of weeks. Renato V V <a href="https://ncatlab.org/nlab/revision/recognition+of+relative+loop+spaces/1">v1</a>, <a href="https://ncatlab.org/nlab/show/recognition+of+relative+loop+spaces">current</a>

   This is a first draft for a page on the relative recognition principle, as proved in https://msp.org/agt/2020/20-3/agt-v20-n3-p08-s.pdf

   I intend to finish it in the following couple of weeks.

   Renato V V

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 21st 2023
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   Author: Urs
   Format: MarkdownItexHi Renato, thanks for contributing, am looking forward to seeing the bulk of the entry. Some quick hints on editing: - if you want to explore editing you can use the *[[Sandbox]]* for experiments and also for drafts. It saves all your edits in the page history, nothing is lost. - the idea of the wiki is to enclose all technical keywords in double square brackets, which makes them automatically be hyperlinked to their respective nLab entry. For instance typing `[[pointed topological space]]` produces the link *[[pointed topological space]]* - our software is bad with rendering the combination `:=`, better to code it as `\coloneqq`

   Hi Renato,

   thanks for contributing, am looking forward to seeing the bulk of the entry.

   Some quick hints on editing:

   • if you want to explore editing you can use the Sandbox for experiments and also for drafts. It saves all your edits in the page history, nothing is lost.

   • the idea of the wiki is to enclose all technical keywords in double square brackets, which makes them automatically be hyperlinked to their respective nLab entry. For instance typing [[pointed topological space]] produces the link pointed topological space

   • our software is bad with rendering the combination :=, better to code it as \coloneqq

3. • CommentRowNumber3.
   • CommentAuthorGuest
   • CommentTimeMar 22nd 2023
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   Author: Guest
   Format: TextHello Urs. Thanks for the hints. This is my first contribution, so I'm still getting the hang of the editor. I will add the hyperlinks shortly. Renato V V

   Hello Urs. Thanks for the hints. This is my first contribution, so I'm still getting the hang of the editor. I will add the hyperlinks shortly.
   
   Renato V V
4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 22nd 2023
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   Author: Urs
   Format: MarkdownItexI know, that's why I provided some hints. No rush.

   I know, that’s why I provided some hints. No rush.

5. • CommentRowNumber5.
   • CommentAuthorRenato V V
   • CommentTimeMar 24th 2023
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   Author: Renato V V
   Format: TextThis should be a final first version. I'll check back to see if there is any feedback that requires changes. Thanks for the hints, Urs!

   This should be a final first version. I'll check back to see if there is any feedback that requires changes.
   
   Thanks for the hints, Urs!
6. • CommentRowNumber6.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMar 26th 2023
   • (edited Mar 26th 2023)
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   Author: Dmitri Pavlov
   Format: MarkdownItexA really stupid question: would it be possible to define what a “relative loop space” is? Is this just the loop space of the homotopy fiber? (Is the “relative N-loop pair functor” defined in the article the same as the “relative loop space”?)

   A really stupid question: would it be possible to define what a “relative loop space” is? Is this just the loop space of the homotopy fiber? (Is the “relative N-loop pair functor” defined in the article the same as the “relative loop space”?)

7. • CommentRowNumber7.
   • CommentAuthorRenato V V
   • CommentTimeMar 27th 2023
   • (edited Mar 27th 2023)
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   Author: Renato V V
   Format: MarkdownItexI think the more classic notion of relative loop space is as the space of maps of the form $(I^N,\partial I^N,J^N)\rightarrow (X,A,x_0)$ for $X$ a pointed space, $x_0\in A\subset X$, and $J^N=\partial I^n - \{0\}\times I^{N-1}$. This is what is usually used to define relative homotopy groups of pairs. To get the model theoretical version of the recognition theorem, the first problem with this definition is that there isn't a model structure on the categories of topological pairs (I learned this from the answers to this mathoverflow question: https://mathoverflow.net/questions/128976/can-one-make-the-category-of-pairs-of-topological-spaces-a-model-category). The proper model category to consider is the category of maps of spaces, equipped with the projective model structure. If you start with the Quillen model structure the cofibrant objects will be inclusions of CW-pairs, and if start with the mixed model structure you get the maps homotopy equivalent to those. For an inclusion of topological pairs the definition of relative loop space as a loop space of the homotopy fiber is homeomorphic to the classical one above. As for the last question, the “relative N-loop pair functor” is not exactly the same as the “relative loop space”. The first functor outputs a pair of spaces of the form $(\Omega^N(X),\Omega^N(X,A))$, while the second outputs just $\Omega^N(X,A)$. The theorem is about the structure of a relative loop space plus the action of the loop space of the total space.

   I think the more classic notion of relative loop space is as the space of maps of the form $(I^N,\partial I^N,J^N)\rightarrow (X,A,x_0)$ for $X$ a pointed space, $x_0\in A\subset X$, and $J^N=\partial I^n - \{0\}\times I^{N-1}$. This is what is usually used to define relative homotopy groups of pairs.

   To get the model theoretical version of the recognition theorem, the first problem with this definition is that there isn’t a model structure on the categories of topological pairs (I learned this from the answers to this mathoverflow question: https://mathoverflow.net/questions/128976/can-one-make-the-category-of-pairs-of-topological-spaces-a-model-category). The proper model category to consider is the category of maps of spaces, equipped with the projective model structure. If you start with the Quillen model structure the cofibrant objects will be inclusions of CW-pairs, and if start with the mixed model structure you get the maps homotopy equivalent to those.

   For an inclusion of topological pairs the definition of relative loop space as a loop space of the homotopy fiber is homeomorphic to the classical one above.

   As for the last question, the “relative N-loop pair functor” is not exactly the same as the “relative loop space”. The first functor outputs a pair of spaces of the form $(\Omega^N(X),\Omega^N(X,A))$, while the second outputs just $\Omega^N(X,A)$. The theorem is about the structure of a relative loop space plus the action of the loop space of the total space.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMar 27th 2023
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   Author: Urs
   Format: MarkdownItexBest to add that explanation to the entry! Or better, the term *[[relative loop space]]* should point to an entry with more explanation.

   Best to add that explanation to the entry! Or better, the term relative loop space should point to an entry with more explanation.

9. • CommentRowNumber9.
   • CommentAuthorRenato V V
   • CommentTimeMar 28th 2023
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   Author: Renato V V
   Format: MarkdownItexAdded a remark on the definition of relative loops pairs functor and created a page for relative loop spaces that goes into some details of why we use the definition using homotopy fibers. <a href="https://ncatlab.org/nlab/revision/diff/recognition+of+relative+loop+spaces/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/recognition+of+relative+loop+spaces/14">v14</a>, <a href="https://ncatlab.org/nlab/show/recognition+of+relative+loop+spaces">current</a>

   Added a remark on the definition of relative loops pairs functor and created a page for relative loop spaces that goes into some details of why we use the definition using homotopy fibers.

   diff, v14, current