  The goal of this paper is to develop a theory of join and slices for strict
$\infty$-categories. To any pair of strict $\infty$-categories, we associate a
third one that we call their join. This operation is compatible with the usual
join of categories up to truncation. We show that the join defines a monoidal
category structure on the category of strict $\infty$-categories and that it
respects connected inductive limits in each variable. In particular, we obtain
the existence of some right adjoints; these adjoints define
$\infty$-categorical slices, in a generalized sense. We state some conjectures
about the functoriality of the join and the slices with respect to higher lax
and oplax transformations and we prove some first results in this direction.
These results are used in another paper to establish a Quillen Theorem A for
strict $\infty$-categories. Finally, in an appendix, we revisit the Gray tensor
product of strict $\infty$-categories. One of the main tools used in this paper
is Steiner's theory of augmented directed complexes.
