Factorization of Isomorphisms of $(H,θ)$-twisted Lie algebroids

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We study the isomorphism groupoid $\mathcal{T}(M)$ of $θ$-almost twisted Poisson ($θ$-atP) structures on a smooth manifold $M$, focusing on the internal structure of its morphisms. A morphism in $\mathcal{T}(M)$ is a $C^\infty(M)$-linear isomorphism $Φ:\gO^1(M)\to\gO^1(M)$ that simultaneously intertwines the anchor maps and the $(H,θ)$-twisted Koszul brackets associated with two $θ$-atP structures. Every such morphism induces a canonical isomorphism in $θ$-atP cohomology. We define a classifying functor $$ Δ: \mathrm{Mor}(\mathcal{T}(M)) \longrightarrow (Z^1_{\mathrm{dR}}(M) ,+), \qquad Δ(Φ)=θ' - θ, $$ which is additive under composition and partitions the morphisms into two complementary families: the sub-groupoid $\mathcal{T}_{\mathrm{fix}}=\kerΔ$ of isomorphisms preserving $θ$, and the family $\mathcal{T}_{\mathrm{mod}}$ of isomorphisms shifting $θ$. We describe each element in this partition.

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