equivalences in/of $(\infty,1)$-categories
The join of two quasi-categories is the generalization of the join of categories from ordinary categories to quasi-categories.
The join of quasi-categories $C$ and $C'$ is a quasi-category $C \star C'$ which looks roughly like the disjoint union of $C$ with $C'$ with one morphisms from every object of $C$ to every object of $C'$ thrown in.
Two different definitions are used in the literature, which are not isomorphic, but are weakly equivalent with respect to the model structure on quasi-categories.
The join $C \star C'$ of two quasi-categories $C$ and $C'$ is the join of simplicial sets of their underlying simplicial sets.
The alternate join $C \diamondsuit D$ of two quasi-categories should be thought of as something like the pseudopushout
Explicitly (compare mapping cone) it is the ordinary colimit
in sSet.
The join of two simplicial sets that happen to be quasi-categories is itself a quasi-category.
For $C$ and $D$ simplicial sets, the canonical morphism
is a weak equivalence in the model structure on quasi-categories.
Let $* = \Delta[0]$ be the terminal quasi-category. Then for $X$ any quasi-category,
the join $X^{\triangleleft} := (*)\star X$ is the quasi-category obtained from $X$ by freely adjoining a new initial object;
the join $X^{\triangleright} := X \star (*)$ is the quasi-category obtained from $X$ by freely adjoining a new terminal object.
For instance for $X = \Delta[1] = \{ 0 \to 1 \}$ be have
The operation $\star_s$ is discussed around proposition 1.2.8.3, p. 43 of
The operation $\diamondsuit$ is discussed in chapter 3 of
and in section 4.2.1 of
where also the relation between both constructions is established.