Subject" Kan Complexes Does anyone know whether it is the case that a simplicial set B is Kan whenever it has the property that, for each A (or for each n-simplex A), homotopy of maps A ---> B is an equivalence relation? Of course here the maps f, g : A ---> B are said to be homotopic iff they are the restrictions of a map A x I ---> B. Michael Barr, in a recent letter to me, said that this is a fact known to topologists; but I don't seem to be able to prove it. I don't see how the property would even allow one to fill in a 1-horn; there doesn't seem to be enough there to get any non-degenerate map on the 2-skeleton. Am I missing something? Max Kelly, 16 Jan. 1111111111111111111111111111