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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?

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    $\begingroup$ 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. $\endgroup$
    – Yasha
    Mar 28 at 21:13
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    $\begingroup$ These constraints are not independent. $\endgroup$ Mar 29 at 1:59
  • $\begingroup$ 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. $\endgroup$
    – Yasha
    Mar 29 at 18:00

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Yes, see Corollary 4.36 in Christensen–Wu The homotopy theory of diffeological spaces.

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  • $\begingroup$ As far as I remember they have collared simplices, in which case the answer is pretty trivially yes. $\endgroup$
    – Yasha
    Mar 29 at 2:05
  • $\begingroup$ @Yasha: The proofs of 4.36 and its dependencies 4.36, 4.34, 4.30, and 4.28 work equally well for compact simplices. $\endgroup$ Mar 29 at 2:27

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