  We prove Carlos Simpson's "semi-strictification" (or "weak unit") conjecture
in the case of infinity-groupoids. More precisely, we introduce two precise
versions of the conjecture, the "general" and the "regular" conjecture,
involving two different notions of "non-unital categories". The "general"
version involve infinity-categories where absolutely all composition operations
(horizontal, vertical and whiskering) are defined and compatible, the "regular"
version involve infinity-categories where all the composition operations
corresponding to "regular" pasting diagram are defined and compatible. In both
case we construct (weak) model structures on these categories such that fibrant
objects have weak units and weak inverse. We prove the regular version of the
conjecture using the original strategy of Kapranov and Voevodsky, together with
our previous work on polygraphs. The general version cannot be proved by these
methods and is still open. In order to do this we also study some subtle
property of the combinatorics of polygraphs, and we construct a new counting
function for polygraphs, inspired by previous work of Makkai.
