# Is the smooth singular simplicial set of a smooth manifold a Kan complex?

It is classical that the singular simplicial set of a topological space is a Kan complex. This is elementary and already due to presumably Kan.

Q: Is the smooth singular simplicial set of a smooth manifold a Kan complex?

More specifically, given a smooth manifold $$Y$$ we have the simplicial set $$Y_{\bullet}$$ whose set of $$k$$-simplices is the set of smooth maps $$\Delta^k \to Y$$, (a smooth map of a manifold with corners). Is $$Y_{\bullet}$$ a Kan complex?

• Thinking about this more carefully, I think the answer is just no. Consider the horn of a 3-simplex mapping to $R^3$. For the map to smoothly extend, there are three constraints for the differential at the vertex point coming from face differentials. We could accommodate 2 constraints but generically could not accommodate all three. Mar 28 at 21:13
• These constraints are not independent. Mar 29 at 1:59
• You are absolutely right, in fact the third constraint is determined by the other 2. This is actually enough to convince me that the answer is yes, heuristically. Mar 29 at 18:00