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.**

definition 1.1of the paperKan replacement of simplicial manifolds by Chenchang Zhuarxiv.org/pdf/0812.4150.pdf $\endgroup$ – Adittya Chaudhuri Dec 24 '20 at 14:18definition 3.1.1ofCech cocycles for differential characteristic classes – An ∞-Lie theoretic constructionbyFiorenza, Schreiber and Stasheffarxiv.org/pdf/1011.4735.pdf, the notion of Smooth $\infty-$ groupoid is defined and it is mentioned that Kan simplicial manifolds is special kind of smooth $\infty-$ groupoids calledrepresentable smooth $\infty-$ groupoids. InPage 17 , in example 3.1.2they mentioned that the Cech $\infty$-groupoid(defined in the same page) and nerves of Lie groupoids are examples of representable smooth $\infty-$ groupoids. $\endgroup$ – Adittya Chaudhuri Dec 24 '20 at 14:54