# Is there a Geometric/Smooth version of Homotopy Hypothesis using the path $\infty$-Groupoid of a Smooth Space?

A version of Homotopy Hypothesis says that the Fundamental $$n$$-grupoids model Homotopy $$n$$-types... and if we continue upto $$\infty$$, then the Fundamental $$\infty$$- groupoids or Kan Complexes model Topological spaces upto Weak Homotopy.

Now in Higher Gauge Theory there is a notion of Path Groupoid of a smooth space https://ncatlab.org/nlab/show/path+groupoid which is a refinement of fundamental groupoid of a topological space and which remembers more information(Thin homotopy class) than the fundamental Groupoid (homotopy class). Analogous to the notion of Higher Fundamental Groupoids of a topological space one can define higher Path Groupoids of a smooth space https://ncatlab.org/nlab/show/path+n-groupoid. Also one can go upto $$\infty$$ and can analogously define Path $$\infty$$-Groupoid of a smooth space https://ncatlab.org/nlab/show/path+infinity-groupoid.

My Questions are the following:

1. What do Path $$\infty$$-Groupoid of a smooth space models?
2. Is there an analogous smooth or geometric version of Homotopy Hypothesis which deals with Path $$\infty$$-Groupoid of a smooth space?

Now according to Remark 4.3 in https://ncatlab.org/nlab/show/diffeological+space#OnTopologicalHomotopyTypeAndDiffeologicalShape one can present Path $$\infty$$-Groupoid of a smooth space as a diffeological singular simplicial set. Also they mentioned that the diffeological singular simplicial set $$Sing_{diff}(X)$$ of a smooth space $$X$$ is equivalent to the "Shape" of $$X$$. (Though I could not understand clearly what they actually meant by "shape" of $$X$$).

Now according to https://ncatlab.org/nlab/show/smooth+infinity-stack#in_terms_of_groupoids_internal_to_smooth_spaces an $$\infty$$-groupoid internal to diffeological spaces models Smooth $$\infty$$-stack. Now I guess Path $$\infty$$-Groupoid of a smooth space is also an $$\infty$$-groupoid internal to diffeological spaces. So I am expecting Path $$\infty$$-Groupoid to model Smooth $$\infty$$-stack.

So in that case I am expecting the answer to my questions is:

Path $$\infty$$-Groupoid of a smooth space $$X$$ models Smooth $$\infty$$-stack over the site $$O(X)$$ and a geometric/smooth version of Homotopy Hypothesis will say that the Path $$\infty$$-Groupoid of $$X$$ is same as Smooth $$\infty$$-stack over $$O(X)$$ which is same as a diffeological singular simplicial set $$Sing_{diff}(X)$$.

But still my answer is incomplete in the sense that I still do not know completely what extra information does the Path $$\infty$$-Groupoid of $$X$$ captures about the smooth space $$X$$ other than the homotopical information of the underlying topological space of $$X$$.

Also I am feeling little positive about my expectations because of the following reason:

Roughly the usual Homotopy Hypothesis makes sense because $$Top_{Quillen}$$ is Quillen Equivalent to $$sSets_{Quillen}$$ via the total singular complex functor $$Sing$$ and the geometric realization functor $$|-|$$. Now according to https://ncatlab.org/nlab/show/model+structure+on+diffeological+spaces I am guessing that we may have different model structures on the category of diffeological spaces(Diff). Then for my claim of "Smooth or Geometric Homotopy Hypotheis" to be more meaningful I should expect a Quillen equivalence of Model categories between Diff(with some Model structure) and sSets(with some Model Structure). Now according to https://ncatlab.org/nlab/show/model+structure+on+diffeological+spaces#details there exists a Quillen equivalence between Diff(with some appropriate model structure) and $$Top_{Quillen}$$.(Though it is mentioned in the same page that there may exist some gap in the proof of this Quillen Equivalence). Now If I assume for the time being that there is no gap in the proof then in that case we get that Diff(with appropriate Model Structure) is Quillen equivalent to $$sSets_{Quillen}$$(because $$sSets_{Quillen}$$ is Quillen equivalent to $$Top_{Quillen}$$). But I am not sure whether this Quillen equivalence is given via $$Sing_{diff}$$ and $$|-|$$ or not as mentioned in https://ncatlab.org/nlab/show/diffeological+space#OnTopologicalHomotopyTypeAndDiffeologicalShape.

If this is not the case then I can't say anything. But if it is the case then this is what I expected. Even if it is true still this Quillen Equivalence is not a sufficient condition for my claim of Smooth or Geometric Homotopy Hypotheis" to hold true but definitely it can give a positive feeling about my claim.

Am I misunderstanding anything? Or are my expectations not making much sense?

I am not an expert, so apology in advance if I sound stupid.

• I'm not convinced that the smooth path $\infty$-groupoid is different from the smooth fundamental $\infty$-groupoid, since the whole point of eg the path 1-groupoid is that associativity is forced on path concatenation by a quotient of the space of all paths. Note also that the page ncatlab.org/nlab/show/path+infinity-groupoid has only been updated once in the past 11 years, to fix a bug with page rendering, and to remove material that looks, to me at least, a bit dubious. That's not to say one can't ask about a diffeological/Lie $\infty$-groupoid $\Pi_\infty$ and the Homotopy Hypoth Aug 18, 2020 at 1:27
• People do study homotopy types of diffeological spaces, for instance Dan Christensen (see eg theoreticalatlas.wordpress.com/2011/06/10/… and Christensen–Wu The homotopy theory of diffeological spaces arxiv.org/abs/1311.6394 New York J. Math. 20 (2014), 1269-1303) Aug 18, 2020 at 1:32
• The page you link to about the path $\infty$-groupoid has no definition, it just mentions an analogy, and then gives no hints as to how it might even work. The definition at the other page linked there is the same as the ordinary fundamental $\infty$-groupoid, just with the induced diffeological structure. The underlying Kan complex is the same, unlike the path 1-groupoid case. Aug 18, 2020 at 10:09
• I don't know how to answer your question, but I strongly suggest the 1-groupoid case is too degenerate to get a proper intuition. I will point out that even the underlying algebraic structure of the path 1-groupoid and the fundamental groupoid are completely different, even ignoring the diffeological structures that both naturally carry. Aug 19, 2020 at 0:08
• These should be new and separate questions. Aug 19, 2020 at 7:27