  Given a Lie groupoid $\mathcal{G}$ over $M$, $A$ the tangent Lie algebroid of
$\mathcal{G}$, and $\rho: A\rightarrow TM$ the anchor map, we provide a formula
that decomposes an arbitrary multiplicative $k$-form $\Theta$ on $\mathcal{G}$
into two parts. The first part is $e$, a $1$-cocycle of
$\mathfrak{J}\mathcal{G}$ valued in $\wedge^k T^*M$, and the second part is
$\theta\in \Gamma(A^*\otimes (\wedge^{k-1} T^*M))$ which is $\rho$-compatible,
meaning that $\iota_{\rho(u)}\theta(u)=0$ for all $u\in A$. We call this pair
of data $(e,\theta)$ the $(0,k)$-characteristic pair of $\Theta$. Next, we
prove that if $\mathcal{G}$ is a Poisson Lie groupoid, then the space
$\Omega^{\bullet}_{\mathrm{mult}}(\mathcal{G})$ of multiplicative forms on
$\mathcal{G}$ has a differential graded Lie algebra (DGLA) structure.
Furthermore, when combined with $\Omega^\bullet(M)$, which is the space of
forms on the base manifold $M$, $\Omega^{\bullet}_{\mathrm{mult}}(\mathcal{G})$
forms a canonical DGLA crossed module. This supplements a previously known fact
that multiplicative multivector fields on $\mathcal{G}$ form a DGLA crossed
module with the Schouten algebra $\Gamma(\wedge^\bullet A)$ stemming from the
tangent Lie algebroid $A$.
