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:**

- What do Path $\infty$-Groupoid of a smooth space models?
- 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.

oncein 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 $\endgroup$The homotopy theory of diffeological spacesarxiv.org/abs/1311.6394 New York J. Math. 20 (2014), 1269-1303) $\endgroup$10more comments