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. In particular, any simplicial Lie group is a Kan simplicial manifold.

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.

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 crossed complex whose constituent groupoids are Lie groupoids and the involved homomorphisms and actions are smooth.

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. In particular, any simplicial Lie group is a Kan simplicial manifold.