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 $Hom(\Delta^{m},X) \rightarrow Hom(\wedge^{m}_{j}, X)$ 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.

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.

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 $Hom(\Delta^{m},X) \rightarrow Hom(\wedge^{m}_{j}, X)$ 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 any example in each of the above 3 references and also anywhere else. I also could not construct one.

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 examples of Kan simplicial manifolds or suggest some literatures in this direction.

Thanks in advance.

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 $Hom(\Delta^{m},X) \rightarrow Hom(\wedge^{m}_{j}, X)$ 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.