I have to apologise, because my example is incorrect - firstly, the proof of symmetry is wrong, and secondly, the proof of transitivity is wrong, and neither can be rectified - There is a homotopy from any edge to its end point, and a homotopy from the start of any edge to the edge itself. This kills symmetry - and transitivity fails because there is no homotopy between edges if they cover all three vertices, yet if (say) we have an edge from 0 to 1 and an edge from 1 to 2, we have a homotopy from 01 to 1 and a homotopy from 1 to 12. So the whole example fails - fairly comprehensively! As a matter of interest, why was the question asked in the first place? - also, has anyone asked Prof. Kan? I'll check my counter-examples more carefully next time! - Phil Ehlers ======================================