  We construct a left semi-model category of ``marked strict
$\infty$-categories'' for which the fibrant objects are those whose marked
arrows satisfy natural closure properties and are weakly invertible. The
canonical model structure on strict $\infty$-categories can be recovered as a
left Bousfield localization of this model structure. We show that an
appropriate extension of the Street nerve to the marked setting produces a
Quillen adjunction between our model category and the Verity model structure
for complicial sets, generalizing previous results by the second named author.
Finally, we use this model structure to study, in the setting of strict
$\infty$-categories, the idea that there are several non-equivalent notions of
weak $(\infty,\infty)$-categories - depending on what tower of
$(\infty,n)$-categories is used. We show that there ought to be at least three
different notions of $(\infty,\infty)$-categories.
