Take the tour ×   Recall (from Cisinski's Astérisque volume 308)
that given a small category $A$, we define an $A$-localizer to
be a class $W$ of morphisms of $\mathrm{Psh}(A)$ satisfying the
following axioms:    • The class $W$ satisfies 2-for-3    • The
class $W$ contains $\mathrm{rlp}(\mathrm{Mono}(A))$, where
$\mathrm{Mono}(A)$ denotes the class of all monomorphisms in
$Psh(A)$.    • The class $W\cap \mathrm{Mono}(A)$ is closed
under transfinite composition and pushouts.  Given a class $C$
of morphisms in $\mathrm{Psh}(A)$, define $W(C)$, the localizer
generated by $C$, to be the intersection of all $A$-localizers
containing $C$. We say that a localizer $W$ is accessible if
there exists a small set $S$ such that $W=W(S)$.  Let $(A,W)$ be
a pair comprising a small category $A$ and an $A$-localizer $W$.
We define the simplicial completion of $(A,W)$ to be the pair
$(A\times \Delta, W_\Delta)$, where we define the $A\times
\Delta$-localizer $W_\Delta$ as follows: The class $W_\Delta$ is
the localizer generat