In 1991, Kapranov and Voevodsky published a proof 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 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, 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 ?
Best Answer
Here is my guess. To compare spaces with their notion of strict $\infty$-groupoids (in which everything is strict except inverses) Kapranov and Voevodsky use an intermediate category of Kan diagrammatic sets, which they show to be equivalent to both spaces and strict $\infty$-groupoids (after inverting a suitable collection of weak equivalences). Whatever Kan diagrammatic sets are, they seem to be a non-strict model, and so let's assume that they do form a model for spaces. In this case the mistake must be in the comparison of Kan diagrammatic sets and strict $\infty$-groupoids (Theorem 3.7). This theorem relies on Proposition 3.5 which compares the homotopy groups of a Kan diagrammatic set $X$ and the homotopy groups of the strict $\infty$-groupoid $\Pi(X)$ generated from $X$. This comparison, in turn, is based on Lemma 3.4 which says that any morphism in $\Pi(X)$ can be realized via a single pasting diagram in $X$, which are in some sense the cells of $X$ (since $X$ is a presheaf on pasting diagrams). But this claim doesn't seem to be true, and the reason is that when one generates the $\infty$-groupoid $\Pi(X)$ one doesn't only freely add morphisms, but also identifies pairs of morphisms which are supposed to be the same in a strict $\infty$-category structure. This means, for example, that if two different pasting diagrams coincide after this identification, then the identity morphism between them might not be a pasting diagram in $X$ (or at least, one would have to explicitly argue why this would be the case). The proof of Lemma 3.4 seems to be vague enough to allow for this subtlety to slip. All of this could be wrong of course, but if I had to pick one possibly problematic lemma it would be this Lemma 3.4.