According to the **definition 1.1** of the paper *Kan Replacement of simplicial manifolds* by *Chenchang Zhu* <https://arxiv.org/pdf/0812.4150.pdf>,


A *Kan simplicial manifold* is a simplicial manifold $X$ such that for all $m \in \mathbb{N} \cup \lbrace 0 \rbrace $ and $0 \leq j \leq m$, the restriction map $Hom(\Delta^{m},X) \rightarrow Hom(\Lambda^{m}\_{j}, X)$ is a surjective submersion.


I also encountered this notion in the **definition 2.24** of the paper *Higher Groupoid Bundles, Higher Spaces, and Self-Dual Tensor Field Equations* by *Branislav Jurco, Christian Samann, and Martin Wolf* <https://arxiv.org/pdf/1604.01639v2.pdf>. and in the **definition 1.2** of *Integrating L∞-Algebras* by *Andre ́ Henriques* (in the name of *simplicial manifold satisfying Kan condition*)<https://arxiv.org/pdf/math/0603563.pdf>.


**But I could not find much good examples in each of the above 3 references and also anywhere else. I also could not construct one.**


(Though I could find some examples like **Cech $\infty$-groupoids** and **internal nerve of Lie groupoids** in references *Cech cocycles for differential characteristic classes – An ∞-Lie theoretic construction by Domenico Fiorenza, Urs Schreiber and Jim Stasheff* and *Kan Replacement of simplicial manifolds* by *Chenchang Zhu* **respectively**. **[Please check my 1st two comments for details]**).


But it seems to me that, this notion is a very direct and natural generalisation of the notion Lie groupoid to Lie $\infty$-groupoid. (Though Lie $\infty$-groupoid is defined sometimes differently in some literatures). Though according to the discussion in <https://ncatlab.org/nlab/show/Kan-fibrant+simplicial+manifold>, it is not clear to me whether this notion is very useful or not from the perspective of homotopy theory, but the notion itself looks very elegant to me.


It would be very helpful for me if someone can suggest some interesting examples of *Kan simplicial manifolds* or suggest some literatures in this direction.


**Thanks in advance.**



What are some "good" examples of Kan simplicial manifolds?
Kan simplicial manifolds are in the same relation to differentiable ∞-stacks
(i.e., locally fibrant simplicial presheaves on the site of cartesian spaces and smooth maps)
as smooth manifolds are to sheaves of sets on the same site.
That is to say, Kan simplicial manifolds can be seen as the ∞-categorification of manifolds.


Some important examples include:


1. Any ordinary manifold, interpreted as a constant simplicial object.
2. The nerve of a Lie groupoid. In particular, the delooping of any Lie group, which represents principal bundles with this Lie group as a structure group.
3. The Dold–Kan functor Γ applied to any nonnegatively graded chain complex of abelian Lie groups.
4. In particular, applying Γ to the chain complexes U(1)[n],
we get the Kan simplicial manifold representing bundle (n-1)-gerbes.
5. The nonabelian analogue of Γ applied to any crossed module whose two constituent groups are Lie groups
and the involved homomorphisms and actions are smooth.
6. The nonabelian analogue of Γ applied to any (hyper)crossed complex whose constituent groupoids are Lie groupoids and the involved homomorphisms and actions are smooth.
7. As a special case of the previous example, any simplicial Lie group is a Kan simplicial manifold.


