  In this paper, we introduce the (co)homology group of a multiple conjugation
biquandle. It is the (co)homology group of the prismatic chain complex, which
is related to the homology of foams introduced by J. S. Carter, modulo a
certain subchain complex. We construct invariants for $S^1$-oriented
handlebody-links using $2$-cocycles. When a multiple conjugation biquandle
$X\times\mathbb{Z}_{\operatorname{type}X_Y}$ is obtained from a biquandle $X$
using $n$-parallel operations, we provide a $2$-cocycle (or $3$-cocycle) of the
multiple conjugation biquandle $X\times\mathbb{Z}_{\operatorname{type}X_Y}$
from a $2$-cocycle (or $3$-cocycle) of the biquandle $X$ equipped with an
$X$-set $Y$.
