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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
I know, that’s why I provided some hints. No rush.
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”?)
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.
Best to add that explanation to the entry! Or better, the term relative loop space should point to an entry with more explanation.
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