In 1991, Kapranov and Voevodsky [published a proof][1] of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity categories that are weak infinity groupoid.

In 1998 Carlos Simpson [showed][2] that their main result could not be true, but did not explain what was precisely wrong in the paper of Kapranov and Voevodsky.

In fact, as explained by Voevodsky [here][3], for a long time after that, Voevodsky apparently thought his proof was correct and that Carlos Simpson made a mistake, until he finally found a mistake in his paper in 2013 !


Despite being false, the paper by Kapranov and Voevodsky contains a lot of very interesting things, moreover, the general strategy of the proof to use Johnson's Higher categorical pasting diagram as generalized Moore path to strictify an infinity groupoid sound like a very reasonable idea and it is a bit of a  surprise, at least to me, that it does not work.

In fact when Carlos Simpson proved that the main theorem of Kapranov and Voevodsky's paper was false he conjectured that their proof could allow one to obtain that the homotopy category of spaces is equivalent to the homotopy category of strict non unital infinity category that are weak (unital) infinity groupoid (this is now known as Simpson's conjecture).

So:

Can someone explain what precisely goes wrong in this paper ?



  [1]: https://eudml.org/doc/91469
  [2]: http://arxiv.org/abs/math/9810059
  [3]: http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf